# Thread: Gravity according to GR? Force? Geometry? Other?

1. It came up in the trash, but it appears there are a few members interested in the following question.

According to GR is gravity a force? And, is that even the right question?

My understanding is that gravity is a consequence of spacetime geometry and hence would not be specifically a force.

And what about the relationship of energy and gravity? Which may be a better question.

Obviously the energy that accelerates a dropped object came from the object being lifted in the first place. As such it would seem that gravity has the potential to store energy. But think of two objects in space approaching each other from initial great distance, such as two rogue planets from different stellar systems. As they approach each other they should accelerate to even greater speeds from gravity. Where does the energy for this acceleration come from?

2.

3. Originally Posted by GiantEvil
It came up in the trash, but it appears there are a few members interested in the following question.

According to GR is gravity a force?
No, gravity is not a force. Gravity is a field, in both GR and Newtonian physics.

4. Originally Posted by GiantEvil
According to GR is gravity a force?
That depends on what you mean. The term Gravity is defined as a natural phenomena by which all physical bodies attract each other. In GR There is something called a Gravitational Force just like in electrodynamics there's something called an electromagnetic force. If you find the right textbook then you'll find the expression for the gravitational force. I know that Mould's text has it. Moller's text might have it too. However its not what a relativist would call a 4-force though. It's what's called an inertial force like the Coriolis force.

Originally Posted by GiantEvil
And, is that even the right question?
That all depends on what it is that you're seeking to learn here.

Originally Posted by GiantEvil
My understanding is that gravity is a consequence of spacetime geometry and hence would not be specifically a force.
The gravitational force is determined by something called the Christoffel symbols (Christoffel symbols - Wikipedia, the free encyclopedia) whereas the tidal force is determined by the Riemann curvature tensor (Riemann curvature tensor - Wikipedia, the free encyclopedia). Nowadays things are a bit different with most GRists. When Max Von Laue sent Einstein a book on GR he was working on to be proof read. Einstein wrote back to him with the following protest
.
.. what characterizes the existence of a gravitational field from the empirical standpoint is the non-vanishing of the components of the affine connection], not the vanishing of the [components of the Riemann tensor]. If one does not think in such intuitive (anschaulich) ways, one cannot grasp why something like curvature should have anything at all to do with gravitation. In any case, no rational person would have hit upon anything otherwise. The key to the understanding of the equality of gravitational mass and inertial mass would have been missing.
That quote is in the paper written by a GR historian and GR expert John Stachel, in

General Relativity and Gravitation Proceedings of the 11th International Conference on General Relativity and Gravitation, (Stockholm,Cambridge University Press, Jul 6-12, 1986), How Einstein Discovered General Relativity: A Historical Tale With Some Contemporary Morals, John J. Stachel

Originally Posted by GiantEvil
And what about the relationship of energy and gravity?

Originally Posted by GiantEvil
Obviously the energy that accelerates a dropped object came from the object being lifted in the first place. As such it would seem that gravity has the potential to store energy. But think of two objects in space approaching each other from initial great distance, such as two rogue planets from different stellar systems. As they approach each other they should accelerate to even greater speeds from gravity. Where does the energy for this acceleration come from?
From potential energy due to their separation and their gravitational attraction. Have you ever learned about how mechanical energy works and how there's a trade off between the kinetic energy and potential energy as a body falls? It's difficult to help someone when you don't know anything about their educational background.

5. Ahh... Never mind. English is just too inadequate a language for understanding GR. I'll just wait till I've maybe learned some differential geometry. Though that will probably take a few more years.

Originally Posted by Galileo Galilei
The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
Opere Il Saggiatore p. 171.

6. Originally Posted by GiantEvil
Ahh... Never mind. English is just too inadequate a language for understanding GR. I'll just wait till I've maybe learned some differential geometry. Though that will probably take a few more years.
This is an excellent approach, rarely seen in this forum.

7. Originally Posted by GiantEvil
According to GR is gravity a force? And, is that even the right question?

My understanding is that gravity is a consequence of spacetime geometry and hence would not be specifically a force.
What relativity is about is applying the principle of general covariance which demands that all observers see the same laws of physics. What therefore relativity does is evaluate those things that are common to all observers and those that simply are 'fictitious' notions belonging to a certain class of observers. In essence identifying that which is "real" or common to all observers and that which is frame dependent. Relativity seeks to define those invariants of ontological difference and in that way determine valid laws that are applicable throughout the entire universe.

When it comes to gravity, and indeed all the forces, is that what appears as a force is frame dependent. All the forces essentially arise in the same way specifically to account for discrepancies that arise between mismatched reference frames. When it comes to the other forces aside from gravity, the mismatched reference frames relate to phase differences of quantum wavefunctions as opposed to coordinate references mapped in space and time but all are associated with the viewpoint taken from different frames of references. When you switch frames these supposed forces can disappear. They are all gauge dependent which is another way of saying observer dependent.

Real to Einstein meant observer independent and the only way to determine what was observer independent was to compare all possible points of view and hope to find the invariants that are common to all observers. What was real to Einstein therefore was that which is invariant.

It doesn't matter what you choose as spacetime coordinates in describing reality as it is structure that is important as opposed to objective properties that may arise as most of these objective properties can be found to be frame dependent. You can describe a curved world line in a flat spacetime or a stright world line in a curved spacetime. You can talk of a warped manifold with a simple grid-like metric or about a distorted metric and no manifold at all. You can arbitraily label spacetime points in any manner and in infinitely varied ways and none of it makes any difference to the underlying invariant structure that remains the same.

The key lies in determining the underlying structure and not being distracted by the associated illusions that arise from seeing things in the context of preferred points of view and this is where relativity shines as a theoretical theory.

As Einstein wrote "The requirement of general covariance....takes away from space and time the last remnant of physical objectivity...... For example We are able to 'produce' a gravitational field merely by changing the system of coordinates"

In stark contrast to Newton's description of gravity as a force between massive objects and the force of acceleration of a given mass being two independent behaviours which arose from Newton's retention of a notion of absolute space, GR shows that what looks like an accelerated frame from one point of view looks from another like an inertial frame with gravity. There is no ontological difference between the two. You therefore don't need space for something to be real.

Why does a ping pong ball and billiard ball dropped from an equivalent height in a gravitational field hit the ground at the same time (assuming of course they are falling in a vacuum)? If a heavier mass fell at a faster rate than lighter things, you would then be able to tell whether you were in an accelerated reference frame as opposed to an inertial frame with gravity. It is only because both objects of different mass fall at the same rate that the equivalence principle holds. If you could tell the difference between acceleration and gravity, then space would have to mean something or be real. But it's not.

8. Originally Posted by Implicate Order
Real to Einstein meant observer independent and the only way to determine what was observer independent was to compare all possible points of view and hope to find the invariants that are common to all observers. What was real to Einstein therefore was that which is invariant.
Sorry for cutting up your post, IO. I'm just in the process of copying it over to a draw program, so I can print it out on a nice big poster and stick it on my wall !

9. Originally Posted by GiantEvil
It came up in the trash, but it appears there are a few members interested in the following question.

According to GR is gravity a force? And, is that even the right question?

My understanding is that gravity is a consequence of spacetime geometry and hence would not be specifically a force.

And what about the relationship of energy and gravity? Which may be a better question.

Obviously the energy that accelerates a dropped object came from the object being lifted in the first place. As such it would seem that gravity has the potential to store energy. But think of two objects in space approaching each other from initial great distance, such as two rogue planets from different stellar systems. As they approach each other they should accelerate to even greater speeds from gravity. Where does the energy for this acceleration come from?
I've never quite understood that. In those terms, what causes the, say two planets, to move towards each other along that geometry ? Another gravity ?

10. Originally Posted by Implicate Order
If you could tell the difference between acceleration and gravity, then space would have to mean something or be real. But it's not.
Space is not real ? Please explain.

11. Originally Posted by marcbo
Originally Posted by Implicate Order
If you could tell the difference between acceleration and gravity, then space would have to mean something or be real. But it's not.
Space is not real ? Please explain.
For a relativist who has rejected notions of any independent absolute space, the position of objects can only be defined on a relative basis, not by some absolute coordinate system. When we define the distance between objects then we define a geometry and we get an impression of space emerging from these relationships without space having to exist independenly in its own right. By removing all the objects, we therefore remove the notion of any space existing. This actually goes back to the thinking of Leibniz and is referred to as eliminative rationalism.

Let's imagine 3 objects and instead of assigning each object a coordinate reference in association with an external grid, we simply assign coordinates from the frame of reference of each object itself. We still preserve exactly the same information but we can define this geometry by relationships between the things as opposed to an external grid comprising empty space. What we must however do in relativity is disband the idea of spatial seperation as being fixed but rather hold onto the structural relationships between these objects.

Edit: Now imagine we are trying to identify relationships between more things in this universe (or a system). For every new object introduced, each objects position would need to hold the corresponding referential information to each other object. Soon the growing list of information held by each object would become virually intractable. But note that this information held by each object would be hugely redundant (duplicated). To make this tractable we can introduce a notion of simply finding the distance between each point in an arbitrary space and the point infintesimally close to it and for objects seperated in that arbitrary space by distance these small differences can be summed. This contrived entity which defines distances and is defined at every single point forms a metric field. It is simply an artificial construction used to conveniently define measurements without having to store that information for each object introduced to the system. So when relativists talk of metric fields this is simply a convenient device used to referentially relate all the objects described in that system.

12. Originally Posted by Implicate Order
Originally Posted by marcbo
Originally Posted by Implicate Order
If you could tell the difference between acceleration and gravity, then space would have to mean something or be real. But it's not.
Space is not real ? Please explain.
For a relativist who has rejected notions of any independent absolute space, the position of objects can only be defined on a relative basis, not by some absolute coordinate system. When we define the distance between objects then we define a geometry and we get an impression of space emerging from these relationships without space having to exist independenly in its own right. By removing all the objects, we therefore remove the notion of any space existing. This actually goes back to the thinking of Leibniz and is referred to as eliminative rationalism.

Let's imagine 3 objects and instead of assigning each object a coordinate reference in association with an external grid, we simply assign coordinates from the frame of reference of each object itself. We still preserve exactly the same information but we can define this geometry by relationships between the things as opposed to an external grid comprising empty space. What we must however do in relativity is disband the idea of spatial seperation as being fixed but rather hold onto the structural relationships between these objects.

Edit: Now imagine we are trying to identify relationships between more things in this universe (or a system). For every new object introduced, each objects position would need to hold the corresponding referential information to each other object. Soon the growing list of information held by each object would become virually intractable. But note that this information held by each object would be hugely redundant (duplicated). To make this tractable we can introduce a notion of simply finding the distance between each point in an arbitrary space and the point infintesimally close to it and for objects seperated in that arbitrary space by distance these small differences can be summed. This contrived entity which defines distances and is defined at every single point forms a metric field. It is simply an artificial construction used to conveniently define measurements without having to store that information for each object introduced to the system. So when relativists talk of metric fields this is simply a convenient device used to referentially relate all the objects described in that system.
Hi IO. A post like this of yours requires the exersise of a lot more brain cells than I can call upon this lazy Sunday morning, and as usual, it requires a great deal of consideration. I am going to try and formulate a reply, as what you said fascinates me.

In the interim, I find it a good practice that when folk discuss a particular concept or word, it is worthwhile following up the etymology of the word in the ancient languages, such as Latin and Greek. I am good with the Ancient and Modern Greek, so I looked at that.

And it happens that the translation of the word 'space' (as in outer space) that we are considering the existance of here, is, both in ancient and modern Greek ..

διάστημα .. diastema

diastema - Wiktionary
From Late Latin, from Ancient Greek διάστημα (diástēma , “interval, space between”). Compare diastasis.

.. which goes to what you say above (and probably where the word 'distance' comes from).

Fascinating. Hope to contribute more soon.

13. Hi marcbo. Enjoy your lazy day. I am having one myself :-))

As a bit of background refer to this and as part of the points raised above focus on the section on Einstein's Equations dealing with the notion of a metric. I have only a basic understanding of what it is trying to achieve but if you want to get into the details then others here have the mathematical robustness to help you out.

This also might help in working from ground up.

PS Interesting word that...."diastema". Thanks for the info. I must admit I am fascinated by history and in particular the 'history of science'. It is the machinations of the mind and the conclusions drawn that really intrigue me.

14. Originally Posted by marcbo
I've never quite understood that. In those terms, what causes the, say two planets, to move towards each other along that geometry ? Another gravity ?
Time. Empirically, everything ages into the future - that is what drives the dynamics of the universe as we observe it. Just as longitudinal lines on the surface of the earth converge and meet at the poles, the world lines of massive test particles converge as they age into the future - we see this as test particles being gravitationally attracted, and GR models it as geodesics converging over time, due to a curved background geometry. Gravity is relative acceleration between test particles, a phenomenon called geodesic deviation :

A globe is far from a perfect analogy for gravity, but it does convey the general idea of what geodesic deviation is all about.

15. Okay, I might have some understanding now. (Thank you Markus for your generally superior explications!)

Let's say that the spacetime manifold is a sheet of paper. Further I will consider "force" and "energy" to be synonymous. Now forces will be arrows of various length drawn upon the paper (a vector). Test particles on the paper follow these arrows. Now if the paper is bent, a test particle will furthermore also have its path effected by the bend in the paper. Also that the paper is bent only by the presence of arrows and or test particles. (We might consider the paper as a rubber sheet.) Now a test particle may be moving because of a bend in the paper and there is no arrow present (gravity). Such that neither force nor energy is involved. Such that gravity is not an arrow and doesn't involve force or energy, although a test particle could follow a sufficiently long arrow against the bend of the paper.

Although now I've said that gravity has no intrinsic energy, which seems as if it could be horribly wrong. (Once DrRocket told me that gravitational energy is factored into the stress energy tensor, i.e. gravity gravitates.) My whole analogy is too simple. It doesn't really cut it, does it?

16. Originally Posted by GiantEvil
Okay, I might have some understanding now. (Thank you Markus for your generally superior explications!)

Let's say that the spacetime manifold is a sheet of paper. Further I will consider "force" and "energy" to be synonymous. Now forces will be arrows of various length drawn upon the paper (a vector). Test particles on the paper follow these arrows. Now if the paper is bent, a test particle will furthermore also have its path effected by the bend in the paper. Also that the paper is bent only by the presence of arrows and or test particles. (We might consider the paper as a rubber sheet.) Now a test particle may be moving because of a bend in the paper and there is no arrow present (gravity). Such that neither force nor energy is involved. Such that gravity is not an arrow and doesn't involve force or energy, although a test particle could follow a sufficiently long arrow against the bend of the paper.

Although now I've said that gravity has no intrinsic energy, which seems as if it could be horribly wrong. (Once DrRocket told me that gravitational energy is factored into the stress energy tensor, i.e. gravity gravitates.) My whole analogy is too simple. It doesn't really cut it, does it?
What bends the paper?

17. Originally Posted by Robittybob1
What bends the paper?
Originally Posted by "Me, post #14
Also that the paper is bent only by the presence of arrows and or test particles. (We might consider the paper as a rubber sheet.)
*The editor seems a little broke, it wants a greater than a single character change to save.

18. Originally Posted by GiantEvil
Originally Posted by Robittybob1
What bends the paper?
Originally Posted by "Me, post #14
Also that the paper is bent only by the presence of arrows and or test particles. (We might consider the paper as a rubber sheet.)
*The editor seems a little broke, it wants a greater than a single character change to save.
I did read that but I wasn't sure if the arrows were on the plane of the paper or separate from it. So is the bending always in one direction so they never cancel each other out?

19. Originally Posted by GiantEvil
Let's say that the spacetime manifold is a sheet of paper. Further I will consider "force" and "energy" to be synonymous. Now forces will be arrows of various length drawn upon the paper (a vector). Test particles on the paper follow these arrows. Now if the paper is bent, a test particle will furthermore also have its path effected by the bend in the paper. Also that the paper is bent only by the presence of arrows and or test particles. (We might consider the paper as a rubber sheet.) Now a test particle may be moving because of a bend in the paper and there is no arrow present (gravity). Such that neither force nor energy is involved. Such that gravity is not an arrow and doesn't involve force or energy, although a test particle could follow a sufficiently long arrow against the bend of the paper.
I have a personal dislike for graphic explanations of gravity because they fail to address questions such as yours, "Where does the energy for this acceleration come from?"

In GR, gravity is a matter of curved spacetime geometry and stress energy tensors but these meaningless terms in human languages because they go beyond human experience. Curved space makes sense but how can you curve time? Gravity can be represented as a geometry but that is not what gravity is.

Lengths grow shorter in a gravitational field and clocks run slower so I prefer to think of gravity as, "Shorter space and slower time."

All material objects are in a state of rapid motion even when they are "sitting still". We have Brownian motion, zitterbewung ZPE, and all sorts of spins and shakes. When two planets move together, they are moving through a gradient from a point where their internal motions are rapid to a point where their internal motions are slower. They are moving from a high energy level to a lower energy level.

Gravity is a lower energy level because clocks run slower in a gravitational field so all the internal motions of a massive body are slowed. Gravity can be thought of as negative energy.

When an object falls, the potential energy stored in its internal motions becomes the kinetic energy that we observe as acceleration so energy is conserved. In answer to your question, the energy to accelerate two planets as they move together comes from within the planets themselves.

20. Originally Posted by Robbitybob1
I did read that but I wasn't sure if the arrows were on the plane of the paper or separate from it. So is the bending always in one direction so they never cancel each other out?
Think of the arrows as drawn on the paper. And they only bend the paper a very tiny bit. The test particles bend the paper more so. And the paper only bends in one direction. Remember also this is just an analogy, and has it's limits of application.

21. Originally Posted by bangstrom
I have a personal dislike for graphic explanations of gravity because they fail to address questions such as yours, "Where does the energy for this acceleration come from?"
Yes, I understand. Please go ahead and explain it using the differential geometry. I'll try to follow if I can.
Originally Posted by bangstrom
Lengths grow shorter in a gravitational field and clocks run slower
From the perspective of some other frames of reference, yes.
Originally Posted by bangstrom
All material objects are in a state of rapid motion even when they are "sitting still". We have Brownian motion, zitterbewung ZPE, and all sorts of spins and shakes. When two planets move together, they are moving through a gradient from a point where their internal motions are rapid to a point where their internal motions are slower. They are moving from a high energy level to a lower energy level.

Gravity is a lower energy level because clocks run slower in a gravitational field so all the internal motions of a massive body are slowed. Gravity can be thought of as negative energy.

When an object falls, the potential energy stored in its internal motions becomes the kinetic energy that we observe as acceleration so energy is conserved. In answer to your question, the energy to accelerate two planets as they move together comes from within the planets themselves.
Oh dear! We have thermodynamics being ad hock mixed with quantum mechanics. And properties of quantum objects being assigned to macroscopic bodies. Although it might be possible that different temperatures would be measured from different frames of reference. I'm afraid however that I have no choice but to reject your fantasies and replace them with a common reality. Sorry dude.

22. Originally Posted by GiantEvil
Yes, I understand. Please go ahead and explain it using the differential geometry. I'll try to follow if I can.
There is no difference at all in the geometry but it is necessary to understand that changes in space also involve changes in time so the geometry is in spacetime and not just space. When you bend the paper in space you are also bending the model in time and you can't bend one side without bending the other.

On the "rubber sheet" model, a greater gravitational field is represented by smaller squares and smaller squares represent shorter space but they also also represent slower time on the same model. So greater gravity is shorter space and slower time in the model as well as beyond the model.

Originally Posted by GiantEvil
Oh dear! We have thermodynamics being ad hock mixed with quantum mechanics. And properties of quantum objects being assigned to macroscopic bodies. Although it might be possible that different temperatures would be measured from different frames of reference. I'm afraid however that I have no choice but to reject your fantasies and replace them with a common reality. Sorry dude.
My explanation has nothing to do with thermodynamics because the observed speed of atomic particles (their temperature) can only change with an exchange of energy from outside the system.

When an object moves about in a gravitational field the internal speeds change but the change is reference frame dependent so no change in temperature is observed locally. Faster or slower speeds are relative to a faster or slower moving clock.

If you put a rock on an elevator and take it to the top of a building, it gains in energy but it does not grow hotter or gain in mass. Where does the energy go and how is it stored? This is your question in reverse and I am saying that time at the top of a building progresses at a faster rate than time at the bottom so the rock is moving through a time gradient where its energy is greater at the top than at the bottom.

23. Originally Posted by bangstrom
There is no difference at all in the geometry
"Differential geometry" is the body of mathematics in which the theory of GR is based.

Oh, bangstrom. You really should do some research into the topic of sarcasm, come back and read my post #20 again, and then reply.

24. Originally Posted by GiantEvil
Originally Posted by bangstrom
There is no difference at all in the geometry
"Differential geometry" is the body of mathematics in which the theory of GR is based.

Oh, bangstrom. You really should do some research into the topic of sarcasm, come back and read my post #20 again, and then reply.
If you were being sarcastic, I missed it.
I know GR involves differential geometry. This is not new.

My explanation used words other than the more "common curved spacetime" etc. to describe gravity so the differences were semantic rather mathematical. My point is that changes in gravity involve changes in both space and time and it is mainly changes in time that provide the mechanism by which kinetic energy is stored and released when a massive body rises or falls in a gravitational field. This is much the same as storing energy in a flywheel.

25. Originally Posted by bangstrom
If you were being sarcastic, I missed it.
I'm guessing that English isn't your primary language?

Well, anyhow. You act as if you do know the uber-geek technical answer to the question, let's hear it. With the equations and the affine spaces and the Ricci flow tensors, all in laTEX. You can do that, right?

26. While the temptation is probably quite strong for some people, it is not generally a good idea to attempt to understand GR in terms of forces, accelerations and energies - these purely Newtonian concepts generalise only locally, but they are meaningless and not well defined over extended regions in the presence of sources of gravity.

In essence, GR is a theory of local measurements and of how those measurements are related in space-time. What we find is that measurements of space, time, energy and acceleration are all purely local notions, and that observers located at different events therefore may disagree on such measurements. The classic textbook example here is something falling into a black hole - an external stationary observer will, using his own clock, argue that it takes an infinite amount of his local "far-away" time for anything to reach the event horizon; however, someone who falls together with the object disagrees, since he measures a finite amount of his own local "free fall" time to reach and cross the horizon. The concepts of "time" that these observers use are not the same ! Likewise, a far away observer can draw concentric circles around a black hole and finds that they are all evenly spaced according to his own local rulers, but someone physically falling from one circle to another will disagree and find that the "free fall" ruler he carries will measure a very much larger and non-constant distance between them. Again - the concepts of "space" which these observers use aren't the same. There is no "right" or "wrong" in this - all observers are right, but only in their own local frames of reference.

So - observers disagree on local measurements, but what they all agree on is how those measurements at different events are related to one another. These relationships are tensorial quantities, which means they do not depend on what method you use to label events in space-time. It's a little like your daily commute to work - you can describe your route in terms of (x,y,z,t) coordinates measured in kilometres with the origin being your house, or you could use polar long/lat coordinates as read off a globe in degrees, but the actual relationship between your house and your workplace never changes - it is entirely independent of the method you use to describe it. You have to commute the same physical distance, no matter what labels you give to your house and your workplace. This is the true essence of GR - to find a description of observers and their measurements that captures only their physical relationship and ignores the ( entirely arbitrary ) choice of event-labelling that any specific observer might employ. That's the concept of diffeomorphism invariance ( aka general covariance ) - the form our laws of physics take must not intrinsically depend on arbitrary coordinate choices, i.e. all classical observers experience the same laws of physics, irrespective of how they move, and where they are in relation to sources of gravity. All measurements become local and observer-dependent, but the relationships between measurements are global and invariant.

What does this have to do with gravity ? Well, it turns out that measurements between events are intrinsically related to the presence and structure of sources of energy-momentum. Issues of synchronisation and comparability aside, clocks in free space and clocks near massive bodies don't measure the same separations in time; rulers in free space and rulers close to massive bodies do not measure the same separations in space. Balls of test particles in free space maintain both their volume and shape into all eternity; balls of test particles in vacuum near a massive body deform in such a way that their volume remains constant, and balls of test particles in the interior of energy-momentum sources both deform in shape and also change their overall volume. Measurements of the numerical value of the volume and shape are observer-dependent, but the way it changes is not - all observers agree on this. Geometrically, the rate at which a ball of test particles in free fall changes its volume is measured by the Ricci tensor; in vacuum the volume is always constant, so

which are the Einstein equations in vacuum. Their intuitive meaning is simply that a ball of test particles in free fall may get deformed in shape, but only in such a way that its total volume remains always constant. Inside regions with sources of energy-momentum, both the volume and shape of our ball can change, and the way it does that is directly related to the energy-momentum source :

which are the inhomogenous Einstein equations in trace-reversed form. So, the difference between vacuum and matter/radiation lies in the geometry of space-time itself, and can be visualised by what happens to the shape and volume of balls of test particles in free fall. Of course, this simplistic view obscures much of the complicated mathematics underlying all of this, but it does allow us to grasp the intuitive meaning behind the equations, instead of just having to stare at a series of meaningless symbols. Similar visualisations can be developed for other objects in GR as well - for example, the Riemann tensor measures what happens when we transport a vector along a small closed loop, the Weyl tensor tells us how the shape ( as opposed to the volume ) of our ball of test particles changes, the metric tensor defines the separation of two events in space-time, the energy-momentum tensor is what remains conserved when we translate any arbitrary system to a different point in space-time, etc etc. So at least in a limited sense there are ways to make GR somewhat intuitive, one just has to look hard enough

Obviously though, there's a lot of stuff that simply cannot be visualised in any meaningful way - so then it is down to the old adage : "Shut up and calculate !"

27. That was a superb post Markus. A huge thumbs up.

28. Originally Posted by Implicate Order
That was a superb post Markus. A huge thumbs up.
Why, thank you I am a very visual person, so it is important for me to be able to see behind the equations and visualise what they actually mean, instead of just tossing about algebraic expressions. The problem with this - and the reason you don't normally find this explanation in textbooks - is that it isn't mathematically rigorous; in fact a real mathematician would ridicule and dismiss what I have written. However, in my opinion it enables a beginner to develop an intuitive picture of what physically happens in the field equations, so it serves a purpose even if it is not mathematically exact and rigorous. After all, not everyone here is proficient in tensor calculus, so the brute force "shut up and calculate" approach just won't do much for most amateurs - while everyone can intuitively visualise a ball of test particles freely falling, it is unfortunately impossible to visualise a tensor in the same way as one can do with vectors. So hopefully my post is helpful to some.

29. I don't think that we should dismiss the intuitive nature of visualizations too lightly as the core concepts such as general covariance and symmetry lie at the heart of the mathematical formalism and it is easy to lose sight of what the maths is trying to achieve without these bedrocks firmly in front of mind. Sometimes I wonder whether we can actually lose sight of reality when we adopt abstract mathematical structures of convenience to describe our universe as they tend to take on an independent life of their own that can lead us down the garden path (thinking of strings).

Your post was powerful in it's expose of what GR seeks to achieve without reverting to traditional one liners that are past their use-by-date and fail to strike at the heart of the matter.

30. Originally Posted by GiantEvil
According to GR is gravity a force?
4-force? No. Inertial force? Yes.

31. Originally Posted by Markus Hanke
Originally Posted by marcbo
I've never quite understood that. In those terms, what causes the, say two planets, to move towards each other along that geometry ? Another gravity ?
Time. Empirically, everything ages into the future - that is what drives the dynamics of the universe as we observe it. Just as longitudinal lines on the surface of the earth converge and meet at the poles, the world lines of massive test particles converge as they age into the future - we see this as test particles being gravitationally attracted, and GR models it as geodesics converging over time, due to a curved background geometry. Gravity is relative acceleration between test particles, a phenomenon called geodesic deviation :

A globe is far from a perfect analogy for gravity, but it does convey the general idea of what geodesic deviation is all about.
Hi Markus.

Time ? Time causes the (say) two planets .. or the two test particles (is that the sceintific way to say it ?) to move toward each other ?

Fair enough. It also causes the all too noticable greying of the hair around my ears .. and those eaves haven't beem painted for some years now .. must find some time to get to them.

In fact, we could say time causes everything, because without time, nothing would happen. So 'time causes gravity' is rather a redundancy than anything else.

Yes, the globe IS far from a perfect analogy for gravity, because, what is it other than an arbitrary choice of curved lines ? It really answers nothing. How did the two particles know or decide to travel along a couple of meridians of longitude so as to meet at a point, rather than along two parralels of latutude so as to never meet ? And do you see (in another post you said you are a very visual person) that the pole itself is purely an arbitrary choice - a construct of the mind ? The pole enjoys no exclusive position on the globe - it exists only because we imagine it there. In fact, I would say that the globe is a much worse analogy than the trampoline analogy, because it merely extends the aporia to an additional dimension.

32. Hi Implicate Order

Originally Posted by Implicate Order
For a relativist who has rejected notions of any independent absolute space, the position of objects can only be defined on a relative basis, not by some absolute coordinate system. When we define the distance between objects then we define a geometry and we get an impression of space emerging from these relationships without space having to exist independenly in its own right. By removing all the objects, we therefore remove the notion of any space existing. This actually goes back to the thinking of Leibniz and is referred to as eliminative rationalism.
I do get the general drift of what you are saying, but let me play the devils advocate here. There are two objects on my desk, a tea cup and my phone. They are seperated by a space of 2 feet. I remove them. The 2 foot space is, however, still there. To make your statement true, we would have to remove ALL objects in the universe, including ourselves. Then what ?

Let's imagine 3 objects and instead of assigning each object a coordinate reference in association with an external grid, we simply assign coordinates from the frame of reference of each object itself. We still preserve exactly the same information but we can define this geometry by relationships between the things as opposed to an external grid comprising empty space. What we must however do in relativity is disband the idea of spatial seperation as being fixed but rather hold onto the structural relationships between these objects.
This is somewhat murky. A snapshot of any event WILL show spacial seperation being fixed.

Edit: Now imagine we are trying to identify relationships between more things in this universe (or a system). For every new object introduced, each objects position would need to hold the corresponding referential information to each other object. Soon the growing list of information held by each object would become virually intractable. But note that this information held by each object would be hugely redundant (duplicated).
Yes, I see. And I would also take the step of saying it becomes self referential.

To make this tractable we can introduce a notion of simply finding the distance between each point in an arbitrary space and the point infintesimally close to it and for objects seperated in that arbitrary space by distance these small differences can be summed.
This I don't understand. Why take an infintesimal number of points, only to then sum them ?

This contrived entity which defines distances and is defined at every single point forms a metric field. It is simply an artificial construction used to conveniently define measurements without having to store that information for each object introduced to the system.
Yes, the metric field is an articficial construction.

So when relativists talk of metric fields this is simply a convenient device used to referentially relate all the objects described in that system.
OK, an artificially constructed convenience to position objects. And without it, they would still be there, and I could artificially construct another convenience to do a similar job.

I am not been trite here - I have thought about this over the last couple of days and still cannot see how space is not real.

33. Originally Posted by marcbo
Time ? Time causes the (say) two planets .. or the two test particles (is that the sceintific way to say it ?) to move toward each other ?
No. What causes the two test particles to approach is the fact that the geometry of space-time is such that their world lines cease to remain parallel, in the same way as longitudinal lines on a sphere cease to remain parallel as you move away from the equator. Time is simply what corresponds to the direction perpendicular to the equator, and the relationship between the particles is no longer constant over time.
Reading back to the relevant posts I realise that the fault here is largely mine, since I shouldn't have replied the way I did without properly qualifying my answer. Hopefully this makes it somewhat clearer now.

Yes, the globe IS far from a perfect analogy for gravity
Yes, as I had said. It is good demonstration for the concept of geodesic deviation though, which was the point in bringing it up.

How did the two particles know or decide to travel along a couple of meridians of longitude so as to meet at a point
Once again, the idea behind mentioning the globe was to illustrate geodesic deviation, not to give a complete model for gravity - clearly there are a number of degrees of freedom missing, and also the sphere does not reflect the time-orientability of the manifold, which is an important property. But if you wish to extend the analogy nonetheless, then the direction perpendicular to the equator ( i.e. either north or south ) corresponds to the future; the particles' ageing into the future then corresponds to the motion along longitudinal lines.

rather than along two parralels of latutude so as to never meet ?
The lines of latitude are surfaces of constant time; a particle travelling perpendicularly to the time direction would mean that it moves instantaneously and hence at infinite speed. That is obviously unphysical, so it isn't an option, not even in this imperfect analogy. To be honest, I thought that was quite clear, so I am a bit surprised at this comment. To keep things simple, this visualisation aid just plots separation ( angular distance ) against time ( north-south direction, i.e. latitude surfaces ).

And do you see (in another post you said you are a very visual person) that the pole itself is purely an arbitrary choice - a construct of the mind ?
Of course I do, that's elementary. You are free to pick your coordinates and their origin, so you can choose the poles to be wherever you want them to be, so long as your surface is spherically symmetric and stationary; but you have to understand that the important thing is the existence of poles, not where they are, because it is their existence that implies geodesic deviation. It is not possible to cover the surface of a sphere with a coordinate chart that does not exhibit geodesic deviation; that is what differentiates the surface of a sphere ( which is intrinsically curved ) from - say - a piece of paper rolled up into a cylinder ( which is intrinsically flat, since there is no geodesic deviation ).

The pole enjoys no exclusive position on the globe
I didn't say that it did, nor does it need to in order to demonstrate geodesic deviation. This nicely demonstrates another important feature of GR - diffeomorphism invariance. You can choose whichever coordinate basis you like without affecting the underlying geometry of the manifold. That is because the geodesic structure - and hence curvature - arises not from the metric, but from the connection.

In fact, I would say that the globe is a much worse analogy than the trampoline analogy
The trampoline is not even a proper analogy, it is just an embedding diagram for a particular coordinate system ( usually Schwarzschild coordinates ). The difference between the trampoline and the globe is that the trampoline plots coordinate radial distance versus proper radial distance ( which only works for static, stationary, spherically symmetric geometries ), whereas the globe plots ( oriented ) time versus separation of geodesics. Both of them fail as models of gravity, because they depict only a small subset of the actual degrees of freedom present in the phenomenon of gravity.

34. Originally Posted by marcbo
This I don't understand. Why take an infintesimal number of points, only to then sum them ?
Because otherwise you would have to explicitly define the distance between every possible set of points, so you would end up with a system of infinitely many equations, which is physically and mathematically useless, and also not diffeomorphism invariant. The advantage of using a quadratic differential form as your metric is that you need to only specify how distances change from event to event, rather than the actual distances themselves; knowledge of the relationship between events then allow you to define the physical distances via simple integration over the required domain, plus boundary conditions. So, instead of specifying all possible distances ( a set with infinitely many elements ), all you need to do now is to specify the 10 independent components of the metric tensor, and you are done.

I am not been trite here - I have thought about this over the last couple of days and still cannot see how space is not real.
I think you might be confusing space with measurements performed in it and their relationships; the metric merely defines measurements of distances, angles, areas and volumes, and how they are related. How you do these measurements is arbitrary, but the relationship between them at different points is not. So, space-time is very real, but how you label individual events in it is completely arbitrary.

And without it, they would still be there, and I could artificially construct another convenience to do a similar job.
Yes, exactly. Remember the example about your commute to work - you can label your house and your workplace on a map any way you want, but that doesn't change the distance you have to travel, i.e. their relationship on the map is always the same.

35. Hi Marcbo.

Here is a question for you. Imagine a single white ink spot drawn on a black page of infinite extent. I then ask you to define it's location. How do you do this using its position in space as your guide. I then add another white ink spot arbitrarily placed on the black canvas. How can you define that location. The rationale is that the only tools you have is the relative position of the other inkspot.

The important thing to consider is that position can be defined only with respect to the other things in the system. If it has motion, that motion can only be identified by comparison to the changes in its position against other things in the system. There is no meaning to space that is independent of the relationships that exist between things in a system.

Lee Smolin had a great analogy where he likened space to a sentence. There is no sense to a sentence without words in it. Each sentence has a grammatical structure that is defined by the relationships between the words. Just as a sentence has no structure and no existence apart from the relationships between the words, space has no existence apart from the relationships between the things in the universe.

If we change the words of a sentence we then change its grammatical structure. So too does the geometry of space change when the things in the universe change their relationship to each other.

36. Markus, thanks for your posts #33 and 32 above, and your earnest and admirable efforts. I am, however, none the wiser as to what gravitry really is or how it works. And I suspect neither is science, as it appears it only removes the question to a higher order of complexity, with it's own, equivalent, higher order issues.

Anyway, thanks again.

37. Hi IO;

Originally Posted by Implicate Order
Hi Marcbo.

Here is a question for you. Imagine a single white ink spot drawn on a black page of infinite extent. I then ask you to define it's location. How do you do this using its position in space as your guide. I then add another white ink spot arbitrarily placed on the black canvas. How can you define that location. The rationale is that the only tools you have is the relative position of the other inkspot.

The important thing to consider is that position can be defined only with respect to the other things in the system.
This requires some mental gymnastics, but lets go with it. There is ALWAYS some things in the system, if in the least, YOU, contemplating this. If there was NO THING in the system, then this arguement is MOOT.
If it has motion, that motion can only be identified by comparison to the changes in its position against other things in the system.
Well, this goes to the relative motion thread, so I won't move too much here .. :-)
There is no meaning to space that is independent of the relationships that exist between things in a system.
To me, IO, this is a completely meaningless statement, as to be considering relationships, necessitates there being some thing or person doing the considering.

Lee Smolin had a great analogy where he likened space to a sentence. There is no sense to a sentence without words in it. Each sentence has a grammatical structure that is defined by the relationships between the words. Just as a sentence has no structure and no existence apart from the relationships between the words, space has no existence apart from the relationships between the things in the universe.

If we change the words of a sentence we then change its grammatical structure. So too does the geometry of space change when the things in the universe change their relationship to each other.
Yes, and we can have nonsense sentences, as well !

Edit spelling

38. Originally Posted by marcbo
Yes, and we can have nonsense sentences, as well !
You are a hard man to please marcbo

39. Originally Posted by marcbo
Markus, thanks for your posts #33 and 32 above, and your earnest and admirable efforts. I am, however, none the wiser as to what gravitry really is or how it works.
Don't give up marcbo. There is an epiphany to be had in GR with the equivalence principle that 'smacks you over the head with an apple orchard' when you compare what you think is stationery against that which is free falling. Once you 'get it', you can literally see how in the presence of a massive body, spacetime curvature is manifested everywhere. But to get to this epiphany you need to release this notion of an absolute space and climb on board the relativity train.

40. Originally Posted by Markus Hanke
No. What causes the two test particles to approach is the fact that the geometry of space-time is such that their world lines cease to remain parallel, in the same way as longitudinal lines on a sphere cease to remain parallel as you move away from the equator. Time is simply what corresponds to the direction perpendicular to the equator, and the relationship between the particles is no longer constant over time.
I prefer to visualize things from a more Machian point of view that tries to avoid any of the imaginary sort of things that Mach referred to as "metaphysicals." Time can't talk so time can't "tell" and things like world lines are strictly imaginary so they can't play an active role directing the path of particles. I prefer to leave them out of the discussion unless referring to actual lines or arrows on a piece of paper.

My understanding of the two particles goes like this and I hope we are saying much the same thing.

The two test particles in the example are responding to their immediate space-time environment where time on the near side of the particles progresses at a slightly slower rate than time on the far side. The particles respond by moving through a time gradient from a point of high energy where time is fast to a point of lower energy where time is slow.

The energy lost as the two particles move closer becomes evident to us as an acceleration of the particles towards each other so energy is conserved.

41. Originally Posted by marcbo
No problem, you are welcome

I am, however, none the wiser as to what gravitry really is or how it works.
I think after all is said and done, and we end up with a complete model of gravity, we will find that it is merely an emergent phenomenon of something much more fundamental. That's my two cents' worth anyway. At present, all we have is a classical description ( GR ), the domain of which is limited, that can tell us how gravity acts, but not why it works the way it does. There is much speculation out there about what happens when we try to extend that domain into the quantum world, but few real answers - but even the answers we have so far seem to hint towards a far-ranging and fundamental paradigm shift, on par with the paradigm shift from Newtonian mechanics to relativistic physics a century ago. I certainly hope I will be able to witness ( and understand ) it in my lifetime !

42. Originally Posted by bangstrom
The energy lost as the two particles move closer becomes evident to us as an acceleration of the particles towards each other so energy is conserved.
The problem with this that you cannot easily define the concept of "conservation of energy" across curved regions of space-time. In flat space-time the situation is straightforward :

by virtue of the definition of the energy-momentum tensor of the system of particles, so in flat space-time the divergence vanishes and energy-momentum is automatically conserved. However, see what happens once we introduce curvature into the picture - we now have to replace the ordinary derivative with a covariant derivative, and get :

We get extra curvature-related terms in the divergence integral, so energy-momentum is not automatically conserved across extended regions of curved space-time. Fixing this is possible, but it requires introducing new objects ( such as the Landau-Lifschitz pseudotensor ) which are not covariant, and hence observer dependent - it makes the whole scenario very difficult to visualise and understand correctly.

This is just one example of the complications which arise if we try to understand GR in terms of Newtonian concepts such as energy and acceleration, hence my personal preference in trying to avoid this.

43. Originally Posted by Markus Hanke

The problem with this that you cannot easily define the concept of "conservation of energy" across curved regions of space-time. In flat space-time the situation is straightforward
I can see what you are saying bangstrom (as I can't help but share those Machian viewpoints) but while visually I can sense what's going on with two test particles, the situation becomes exceedingly complicated when examining a collection of test particles which therefore necessitates the use of these abstract mathematical objects in their resolution as Markus has emphasised above. I have quite a bit of grief dealing with the notions of what 'local' and 'global' really mean and in my awkward way I sense the difference as one comparing a viewpoint from a single frame of reference (an observer dependent viewpoint) to that comparing a viewpoint of all possible frames of reference (all viewpoints). I have often pondered on Markus's comments regarding the way he uses the definition of local versus global and wonder whether this may be perhaps an alternate way to view this.

Wheeler had some interesting thoughts regarding the 'local' conservation of energy and was clearly on the case in his latter years to attempt to understand why mass curves spacetime as part of his more exotic ruminations. He continually emphasised the notion that 'the boundary of a boundary is zero' almost as an obsession. What he was referring to was that in any 'local region', the local curvature of spacetime 'exactly' cancels out the energy and momentum of the mass that is present there which in his view was 'why mass curves spacetime'. So of course I wonder whether this means that our interpretation should simply be restricted to one frame at a time as opposed to all frames at once.

It is only when the boundary of a boundary is closed that everything cancels out. He furthermore extended this principle as one that is valid not only for general relativity but all the quantum field theories, or in other words all the gauge theories. He formed that view that this principle was a universal notion and created a reason for why for example fields respond to masses or charges.

What this is reminiscent of is the holographic principle (the outside reveals the inside) and his suggestion that spacetime is an emergent condition based on a simpler basic pregeometry which itself has no dimensionality and ultimately according to Wheeler is information theoretic in origin.

He was particularly interested in elasticity and the notion that 'the outside reveals the inside'. For example in calculating the forces applied to some deformed body, you only need to calculate what is going on at the surface as everything inside cancels out.

44. Originally Posted by Implicate Order
I have quite a bit of grief dealing with the notions of what 'local' and 'global' really mean
Local = Any region of space-time that can be considered approximately flat, so that Minkowskian physics apply
Global = Any extended region of space-time in which curvature effects cannot be neglected

It really is that simple

He continually emphasised the notion that 'the boundary of a boundary is zero' almost as an obsession.
And for good reason - it is basic fundamental principle due to which GR works the way it does. The maths in my signature captures this, which is why I chose it

45. Thanks Markus. This forum is becoming an addiction for me given the insights I am getting from posts such as yours.

But like every addiction I better chill out for a while to collect my thoughts. Be back soon.

46. Originally Posted by Implicate Order
Originally Posted by marcbo
Yes, and we can have nonsense sentences, as well !
You are a hard man to please marcbo
Lol .. no, not at all, IO. I was commenting on your analogy concerning sentences. You must agree, that though some sentences may be quite proper, Cicerian even, they can nonetheless be nonsense. Thus the Smolin analogy is not a good one.

Also, I saw no responses to my responses to your earlier post (other than the above) but no matter. It was getting circular, anyway, as these things do tend to get.

47. Originally Posted by Implicate Order
Originally Posted by marcbo
Markus, thanks for your posts #33 and 32 above, and your earnest and admirable efforts. I am, however, none the wiser as to what gravitry really is or how it works.
Don't give up marcbo. There is an epiphany to be had in GR with the equivalence principle that 'smacks you over the head with an apple orchard' when you compare what you think is stationery against that which is free falling. Once you 'get it', you can literally see how in the presence of a massive body, spacetime curvature is manifested everywhere. But to get to this epiphany you need to release this notion of an absolute space and climb on board the relativity train.
I think I've probably had that epiphany some time ago, IO, and then another one subsequently, and that being, that all knowledge is provisional.

A while back, in the other thread, you (I'm almost sure it was you) said to the effect that 'the shackles of Newtonism are hard to break'. And I fully agree. Any shackles of the mind are hard to break.

It would then be a logical, even a necessary conclusion, that the shackles of Einsteinism are as hard to break, perhaps harder, given the much greater complexity, dedication, and investment involved. Note, I made no comment about whether they should be broken or not, I just commented on the probable existance, and relative (heh) strength of said shackles.

48. Hi marcbo. Unlike most others here, I like you am on a steep learning curve and consequently am not in a confident position to respond to all your queries and therefore rely on the experts here to fill in any gaps. Please don't take this as avoiding questions. I really enjoy these discussions. As I mentioned above however I need to have a spell for a bit to get back to the pile of books sitting next to me and focus on the dreaded day to day necessities.

Cheers

49. Originally Posted by Markus Hanke
I think after all is said and done, and we end up with a complete model of gravity, we will find that it is merely an emergent phenomenon of something much more fundamental. That's my two cents' worth anyway. At present, all we have is a classical description ( GR ), the domain of which is limited, that can tell us how gravity acts, but not why it works the way it does. There is much speculation out there about what happens when we try to extend that domain into the quantum world, but few real answers - but even the answers we have so far seem to hint towards a far-ranging and fundamental paradigm shift, on par with the paradigm shift from Newtonian mechanics to relativistic physics a century ago. I certainly hope I will be able to witness ( and understand ) it in my lifetime !
The above, Markus, particularly the first sentence, is basically saying we don't really know what gravity is, or WHY two bodies move toward eash other. Which is basically what I asked in the first place.

This however, needs special emphasis ..
but even the answers we have so far seem to hint towards a far-ranging and fundamental paradigm shift, on par with the paradigm shift from Newtonian mechanics to relativistic physics a century ago.

Damn .. that's what I kinda just said, in my couple of posts, above. Go figure!

50. Marcbo, our most recent posts collided in cyberspace. My post 47 was responding to your post 45.

Yep, it was me referring to those shackles....and I agree that all knowledge is provisional. It would be a sad day if all was known.

51. Originally Posted by Implicate Order
Hi marcbo. Unlike most others here, I like you am on a steep learning curve and consequently am not in a confident position to respond to all your queries and therefore rely on the experts here to fill in any gaps. Please don't take this as avoiding questions. I really enjoy these discussions. As I mentioned above however I need to have a spell for a bit to get back to the pile of books sitting next to me and focus on the dreaded day to day necessities.

Cheers
Maybe the steep learning curve is the problem !

I like these several words of the poet, Ben Johnson ..

Deign on the passing world to turn your eyes
And pause a while from learning to be wise.

I'm outa here too :-)

Cheers my friend!

52. Originally Posted by marcbo
he above, Markus, particularly the first sentence, is basically saying we don't really know what gravity is, or WHY two bodies move toward eash other.
No, you misunderstood. We know what gravity is, but what we don't know yet is why it is that way, i.e. we don't know how gravity emerges on a quantum level. Why geodesic deviation, and not "just" mediation of vector bosons ? Why does gravity not work in the same way as all the other fundamental interactions ? Perhaps that is because gravity is not fundamental at all - that's all I was trying to hint at. In the classical domain, we can model and understand it perfectly well using standard GR, and we can test that model against experiment and observation - however, that is certainly not all there is to gravity. Interestingly, the exact same is true on the opposite end of the scale, being the Standard Model of Particle Physics - we can model the various particles, their properties and interactions reasonably well, but what we don't know and understand yet is why their properties and interactions are the way they are; we are missing the unifying theory that explains to us why the fundamental symmetries are broken, and just why that leads to the particle zoo with the exact properties and hierarchies which we can empirically observe, and some other construct. We are missing a grand unification, just as much as we are missing quantum gravity; and who knows, perhaps these will turn out to be the same thing.

If physics had all the answers, it would be an exceptionally boring discipline !

53. Originally Posted by Markus Hanke
Originally Posted by bangstrom
The energy lost as the two particles move closer becomes evident to us as an acceleration of the particles towards each other so energy is conserved.
The problem with this that you cannot easily define the concept of "conservation of energy" across curved regions of space-time. In flat space-time the situation is straightforward :

by virtue of the definition of the energy-momentum tensor of the system of particles, so in flat space-time the divergence vanishes and energy-momentum is automatically conserved. However, see what happens once we introduce curvature into the picture - we now have to replace the ordinary derivative with a covariant derivative, and get :

We get extra curvature-related terms in the divergence integral, so energy-momentum is not automatically conserved across extended regions of curved space-time. Fixing this is possible, but it requires introducing new objects ( such as the Landau-Lifschitz pseudotensor ) which are not covariant, and hence observer dependent - it makes the whole scenario very difficult to visualise and understand correctly.

This is just one example of the complications which arise if we try to understand GR in terms of Newtonian concepts such as energy and acceleration, hence my personal preference in trying to avoid this.
Thank you again for an excellent effort Markus!

So basically, if one uses the Euclidean metric (Newton) to measure distances between points in a non Euclidean space (Einstein) then they will get erroneous answers. And the general idea of energy is different between Newton and Einstein.

54. Originally Posted by GiantEvil
So basically, if one uses the Euclidean metric (Newton) to measure distances between points in a non Euclidean space (Einstein) then they will get erroneous answers.
Yes, absolutely. If you take a ball/globe/sphere and a flat sheet of paper, no matter how hard you try you will not be able to cover the surface of the ball with the paper unless you crumble it up really badly, or you do a cut-and-paste job. That's because the geometry of the sphere's surface is not the same as the geometry of a flat plane.

And the general idea of energy is different between Newton and Einstein.
Yes, correct.

55. Originally Posted by Markus Hanke
Originally Posted by bangstrom
The energy lost as the two particles move closer becomes evident to us as an acceleration of the particles towards each other so energy is conserved.
The problem with this that you cannot easily define the concept of "conservation of energy" across curved regions of space-time. In flat space-time the situation is straightforward :

by virtue of the definition of the energy-momentum tensor of the system of particles, so in flat space-time the divergence vanishes and energy-momentum is automatically conserved. However, see what happens once we introduce curvature into the picture - we now have to replace the ordinary derivative with a covariant derivative, and get :

We get extra curvature-related terms in the divergence integral, so energy-momentum is not automatically conserved across extended regions of curved space-time. Fixing this is possible, but it requires introducing new objects ( such as the Landau-Lifschitz pseudotensor ) which are not covariant, and hence observer dependent - it makes the whole scenario very difficult to visualise and understand correctly.

This is just one example of the complications which arise if we try to understand GR in terms of Newtonian concepts such as energy and acceleration, hence my personal preference in trying to avoid this.
You make it sound as though it is impossible to determine if energy is conserved when an object falls. Even the crude experiments of Joules era gave us a clue.

56. Originally Posted by bangstrom
You make it sound as though it is impossible to determine if energy is conserved when an object falls. Even the crude experiments of Joules era gave us a clue.
Experimentation with small masses, at low velocities, in small gravitational fields over small distances does not a good test for Relativity make...

57. The conservation of energy is what is being tested, not relativity, so non-relativistic conditions work just fine.

58. Originally Posted by bangstrom
The conservation of energy is what is being tested, not relativity, so non-relativistic conditions work just fine.
You seem to have rather badly missed the whole point of Markus' post. The notion of conservation of energy does not apply over large scales. If you want to demonstrate what he is saying, you must go beyond what Joule's experiment showed.

59. Originally Posted by Markus Hanke

Yes, absolutely. If you take a ball/globe/sphere and a flat sheet of paper, no matter how hard you try you will not be able to cover the surface of the ball with the paper unless you crumble it up really badly,

Just a nit, Markus

"crumPle", not "crumBle" :-)

60. You are looking good in your new cloak Mr Roark :-)

61. Originally Posted by tk421
Originally Posted by bangstrom
The conservation of energy is what is being tested, not relativity, so non-relativistic conditions work just fine.
You seem to have rather badly missed the whole point of Markus' post. The notion of conservation of energy does not apply over large scales. If you want to demonstrate what he is saying, you must go beyond what Joule's experiment showed.
His comment was in reply to my observation in Post #39 which I will repeat below.

"The two test particles in the example are responding to their immediate space-time environment where time on the near side of the particles progresses at a slightly slower rate than time on the far side. The particles respond by moving through a time gradient from a point of high energy where time is fast to a point of lower energy where time is slow.

The energy lost as the two particles move closer becomes evident to us as an acceleration of the particles towards each other so energy is conserved."

My claim was that energy is conserved in a falling body ( In this case two particles moving together) and it had nothing to do with trying to observe the conservation of energy across many frames so that was irrelevant to the discussion.

My understanding of Marcus' post is that the conservation of energy is difficult to calculate at extremes. That makes sense and you could say the same for any physical measurement. But he did not go so far as to say that it does not apply over large scales although it could easily be interpreted that way. We just don't know for certain. And, since he mentioned nothing except how difficult it is measure the conservation of energy, he made it sound as if we don't know if energy is conserved or not- anywhere.

His statement was in reference to my comment that energy is conserved in falling bodies and this has been "law" for many years so his comment calls for clarification.

62. Originally Posted by bangstrom

The two test particles in the example are responding to their immediate space-time environment where time on the near side of the particles progresses at a slightly slower rate than time on the far side. The particles respond by moving through a time gradient from a point of high energy where time is fast to a point of lower energy where time is slow.

The energy lost as the two particles move closer becomes evident to us as an acceleration of the particles towards each other so energy is conserved.

First of all, the above is (yet) another word salad.
Second off, it is wrong, time passes at a slower rate in stronger gravitational fields, so you got the above backwards.
Third off, the time passage has nothing to do with energy conservation.
Fourth off, time slowdown happens only in the presence of gravitational fields (i.e. the spacetime is curved), so you HAVE to use GR, despite your protestations.
Fifth off, as explained by several of us already, energy is not conserved globally in curved spacetime, it is conserved only locally.

You got everything else correct :-)

63. "Second off, it is wrong, time passes at a slower rate in stronger gravitational fields, so you got the above backwards."

Is the gravitational field stronger between the two particles (their near sides) or away from each other (their far sides). Who has it backwards?

"Third off, the time passage has nothing to do with energy conservation."

I don't agree but that is another topic.

"Fourth off, time slowdown happens only in the presence of gravitational fields (i.e. the spacetime is curved), so you HAVE to use GR, despite your protestations."

I'm not disputing GR.

The discussion began with the question of why two planets move together. I changed the model to two particles to make it more general but you could go back to the two planet model where the presence of gravity is more clear. My example is about time slowing in a gravitational field.

"Fifth off, as explained by several of us already, energy is not conserved globally in curved spacetime, it is conserved only locally."

The test particles are responding to their local conditions, that is, their immediate environment. Marcus introduced the idea of global conservation of energy and my complaint is that global conservation is irrelevant to the model since the local energy environment is the is the ONLY consideration?

64. Originally Posted by bangstrom
"Second off, it is wrong, time passes at a slower rate in stronger gravitational fields, so you got the above backwards."

Is the gravitational field stronger between the two particles (their near sides) or away from each other (their far sides). Who has it backwards?
You still have it backwards and you don't know it. The effect on the passage of time has nothing to do with the two test particles.

"Third off, the time passage has nothing to do with energy conservation."
I don't agree but that is another topic.
Doesn't matter what you agree or disagree, you have amply demonstrated that you are pushing anti-mainstream ideas.

"Fourth off, time slowdown happens only in the presence of gravitational fields (i.e. the spacetime is curved), so you HAVE to use GR, despite your protestations."

I'm not disputing GR.
You don't understand GR, you don't understand your own scenario. So, you cannot make any claims whatsoever.

"Fifth off, as explained by several of us already, energy is not conserved globally in curved spacetime, it is conserved only locally."

The test particles are responding to their local conditions, that is, their immediate environment. Marcus introduced the idea of global conservation of energy and my complaint is that global conservation is irrelevant to the model since the local energy environment is the is the ONLY consideration?
Good, if you are talking locally, then energy is conserved , so you have nothing to complain about.

Oh, by the way, you did not address my "first off". Yes, your post is still a word salad.

65. Originally Posted by Howard Roark
Originally Posted by bangstrom

The two test particles in the example are responding to their immediate space-time environment where time on the near side of the particles progresses at a slightly slower rate than time on the far side. The particles respond by moving through a time gradient from a point of high energy where time is fast to a point of lower energy where time is slow.

The energy lost as the two particles move closer becomes evident to us as an acceleration of the particles towards each other so energy is conserved.

First of all, the above is (yet) another word salad.
Second off, it is wrong, time passes at a slower rate in stronger gravitational fields, so you got the above backwards.
Third off, the time passage has nothing to do with energy conservation.
Fourth off, time slowdown happens only in the presence of gravitational fields (i.e. the spacetime is curved), so you HAVE to use GR, despite your protestations.
Fifth off, as explained by several of us already, energy is not conserved globally in curved spacetime, it is conserved only locally.

You got everything else correct :-)
I think Bangstrom has got it right for an an object at at infinite distance has zero potential energy but when it is in a gravitational well it has negative potential energy so it is lower (more negative than zero) and time is slower in the stronger gravitational field.

66. Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by bangstrom

The two test particles in the example are responding to their immediate space-time environment where time on the near side of the particles progresses at a slightly slower rate than time on the far side. The particles respond by moving through a time gradient from a point of high energy where time is fast to a point of lower energy where time is slow.

The energy lost as the two particles move closer becomes evident to us as an acceleration of the particles towards each other so energy is conserved.

First of all, the above is (yet) another word salad.
Second off, it is wrong, time passes at a slower rate in stronger gravitational fields, so you got the above backwards.
Third off, the time passage has nothing to do with energy conservation.
Fourth off, time slowdown happens only in the presence of gravitational fields (i.e. the spacetime is curved), so you HAVE to use GR, despite your protestations.
Fifth off, as explained by several of us already, energy is not conserved globally in curved spacetime, it is conserved only locally.

You got everything else correct :-)
I think Bangstrom has got it right for an an object at at infinite distance has zero potential energy
Err, basic physics says that an object at infinity has infinite potential energy.

but when it is in a gravitational well it has negative potential energy so it is lower (more negative than zero) and time is slower in the stronger gravitational field.
You guys tend to support each other when it comes to fringe, incorrect stuff. Do you really think that the people who know physics don't see through this?

67. Originally Posted by Howard Roark
You guys tend to support each other when it comes to fringe, incorrect stuff. Do you really think that the people who know physics don't see through this?
Ahem,
Gravitational Potential Energy
Gravitational Potential Energy

The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. Because of the inverse square nature of the gravity force, the force approaches zero for large distances, and it makes sense to choose thezero of gravitational potential energy at an infinite distance away. The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches. This negative potential is indicative of a "bound state"; once a mass is near a large body, it is trapped until something can provide enough energy to allow it to escape. The general form of the gravitational potential energy of mass m is:
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers.

This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
note the negative sign

68. Originally Posted by dan hunter
Originally Posted by Howard Roark
You guys tend to support each other when it comes to fringe, incorrect stuff. Do you really think that the people who know physics don't see through this?
Ahem,
Gravitational Potential Energy
Gravitational Potential Energy

The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. Because of the inverse square nature of the gravity force, the force approaches zero for large distances, and it makes sense to choose thezero of gravitational potential energy at an infinite distance away. The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches. This negative potential is indicative of a "bound state"; once a mass is near a large body, it is trapped until something can provide enough energy to allow it to escape. The general form of the gravitational potential energy of mass m is:
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers.

This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
Give it a rest, basic physics teaches you that [utl=http://en.wikipedia.org/wiki/Gravitational_potential#Potential_energy]] gravitational energy[/URL] is , where h is the altitude.
is the gravitational potential.
Obviously you do not understand the difference between the two. It is also refreshing to see you aligning yourself with Robittybob1. Like I said, you guys tend to support each other.

note the negative sign
Yes, I note it. I also note that you don't know what you are talking about.

69. Originally Posted by bangstrom
But he did not go so far as to say that it does not apply over large scales
What I was trying to say is that there is no global law of energy conservation across regions of space-time where curvature cannot be neglected - it is simply not definable, and it does not make any physical sense. That's because the energy-momentum tensor is the conserved Noether current of space-time translation invariance, but in a curved space-time notions of "space" and "time" and purely local ( so in general there is no such invariance ), so this tensor is quite simply not defined for an extended region, hence it makes no sense to talk about its conservation either. The entire concept is not physically meaningful. Energy-momentum is rigorously defined and conserved everywhere locally, but not defined globally.

And, since he mentioned nothing except how difficult it is measure the conservation of energy, he made it sound as if we don't know if energy is conserved or not- anywhere.
While the notion of "conservation of energy-momentum" makes no physical sense globally in a curved space-time, one can define another object which is a linear combination of the energy-momentum tensor and a mathematical entity called the Landau-Lifschitz pseudotensor. This latter entity can intuitively be understood as gravitational self-energy, i.e. it is a measure of how much energy gravity itself "carries" with it. As being a pseudotensor, it is an observer-dependent quantity ( as it must be of course ); however, the divergence of its combination with the energy-momentum is again is fully covariant tensor.

To cut a long ( and mathematically complicated ) story short - conservation of energy-momentum is not defined globally, however, conservation of a combination of energy-momentum and gravitational self-energy is in fact globally conserved. In other words, to generalise the Newtonian energy conservation law to curved space-times, you need to account for both sources of energy-momentum and gravitational self-energy; neither one in isolation is conserved, but the combination of the two is, and forms a conserved current. See here for more details :

Stress

70. Originally Posted by Markus Hanke
Originally Posted by bangstrom
But he did not go so far as to say that it does not apply over large scales
What I was trying to say is that there is no global law of energy conservation across regions of space-time where curvature cannot be neglected - it is simply not definable, and it does not make any physical sense. That's because the energy-momentum tensor is the conserved Noether current of space-time translation invariance, but in a curved space-time notions of "space" and "time" and purely local ( so in general there is no such invariance ), so this tensor is quite simply not defined for an extended region, hence it makes no sense to talk about its conservation either. The entire concept is not physically meaningful. Energy-momentum is rigorously defined and conserved everywhere locally, but not defined globally.

And, since he mentioned nothing except how difficult it is measure the conservation of energy, he made it sound as if we don't know if energy is conserved or not- anywhere.
While the notion of "conservation of energy-momentum" makes no physical sense globally in a curved space-time, one can define another object which is a linear combination of the energy-momentum tensor and a mathematical entity called the Landau-Lifschitz pseudotensor. This latter entity can intuitively be understood as gravitational self-energy, i.e. it is a measure of how much energy gravity itself "carries" with it. As being a pseudotensor, it is an observer-dependent quantity ( as it must be of course ); however, the divergence of its combination with the energy-momentum is again is fully covariant tensor.

To cut a long ( and mathematically complicated ) story short - conservation of energy-momentum is not defined globally, however, conservation of a combination of energy-momentum and gravitational self-energy is in fact globally conserved. In other words, to generalise the Newtonian energy conservation law to curved space-times, you need to account for both sources of energy-momentum and gravitational self-energy; neither one in isolation is conserved, but the combination of the two is, and forms a conserved current. See here for more details :

Stress
The above is a beautiful post. Congratulations, Markus!

71. Originally Posted by dan hunter
I should perhaps mention here that the notion of "gravitational potential energy" is definable only for space-times with very specific symmetries; it is not a generally valid concept in curved space-times. More specifically, it can only be meaningfully defined in space-times that possess a time-like Killing vector field. The usual textbook Schwarzschild metric is an example of this.

72. Originally Posted by Markus Hanke
Originally Posted by bangstrom
But he did not go so far as to say that it does not apply over large scales
What I was trying to say is that there is no global law of energy conservation across regions of space-time where curvature cannot be neglected - it is simply not definable, and it does not make any physical sense. That's because the energy-momentum tensor is the conserved Noether current of space-time translation invariance, but in a curved space-time notions of "space" and "time" and purely local ( so in general there is no such invariance ), so this tensor is quite simply not defined for an extended region, hence it makes no sense to talk about its conservation either. The entire concept is not physically meaningful. Energy-momentum is rigorously defined and conserved everywhere locally, but not defined globally.

And, since he mentioned nothing except how difficult it is measure the conservation of energy, he made it sound as if we don't know if energy is conserved or not- anywhere.
While the notion of "conservation of energy-momentum" makes no physical sense globally in a curved space-time, one can define another object which is a linear combination of the energy-momentum tensor and a mathematical entity called the Landau-Lifschitz pseudotensor. This latter entity can intuitively be understood as gravitational self-energy, i.e. it is a measure of how much energy gravity itself "carries" with it. As being a pseudotensor, it is an observer-dependent quantity ( as it must be of course ); however, the divergence of its combination with the energy-momentum is again is fully covariant tensor.

To cut a long ( and mathematically complicated ) story short - conservation of energy-momentum is not defined globally, however, conservation of a combination of energy-momentum and gravitational self-energy is in fact globally conserved. In other words, to generalise the Newtonian energy conservation law to curved space-times, you need to account for both sources of energy-momentum and gravitational self-energy; neither one in isolation is conserved, but the combination of the two is, and forms a conserved current. See here for more details :

Stress
Thank you for your efforts Markus!
I find these general descriptions of the mathematical modeling of GR to be of the most help to my own understandings. I tend to find scenario descriptions to be tedious and confusing.

73. Originally Posted by GiantEvil
I tend to find scenario descriptions to be tedious and confusing.
I feel exactly the same, which is why I attempt to very much focus on the bigger picture, instead of getting lost in specific scenarios.

74. ^From my own perspective you do an excellent job of describing the abstract modeling going on in whatever subject matter. I think a common problem of many is a lack of experience in "abstraction". They just haven't managed to expand the idea of the "x" from algebra beyond equations to whole models.

75. Originally Posted by Howard Roark

Err, basic physics says that an object at infinity has infinite potential energy.
Prove that or admit you are wrong.

76. Originally Posted by Robittybob1
Originally Posted by Howard Roark

Err, basic physics says that an object at infinity has infinite potential energy.
Prove that or admit you are wrong.
You need to learn how to click on links. Or you could go back to school and learn that potential gravitational energy is defined as . They teach that in 9-th grade.
You also need to learn how to stop trolling. Makes you look silly.

77. Originally Posted by Howard Roark
Originally Posted by Robittybob1
Originally Posted by Howard Roark

Err, basic physics says that an object at infinity has infinite potential energy.
Prove that or admit you are wrong.
You need to learn how to click on links. Or you could go back to school and learn that potential gravitational energy is defined as . They teach that in 9-th grade.
You also need to learn how to stop trolling. Makes you look silly.
What links? I want you to prove that "an object at infinity has infinite potential energy" for that is what you claimed. "Potential gravitational energy is defined as " is only an approximation close to Earth's surface, isn't it?
You can't just say that since is infinite, that will be infinite, for is not going to be constant over that distance!
I'm surprised at how stupid you seem.

78. Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by Robittybob1
Originally Posted by Howard Roark

Err, basic physics says that an object at infinity has infinite potential energy.
Prove that or admit you are wrong.
You need to learn how to click on links. Or you could go back to school and learn that potential gravitational energy is defined as . They teach that in 9-th grade.
You also need to learn how to stop trolling. Makes you look silly.
What links? I want you to prove that "an object at infinity has infinite potential energy" for that is what you claimed. "Potential gravitational energy is defined as " is only an approximation close to Earth's surface, isn't it?
See if you can stop trolling for a while just to make in .

79. Originally Posted by Howard Roark
Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by Robittybob1
Originally Posted by Howard Roark

Err, basic physics says that an object at infinity has infinite potential energy.
Prove that or admit you are wrong.
You need to learn how to click on links. Or you could go back to school and learn that potential gravitational energy is defined as . They teach that in 9-th grade.
You also need to learn how to stop trolling. Makes you look silly.
What links? I want you to prove that "an object at infinity has infinite potential energy" for that is what you claimed. "Potential gravitational energy is defined as " is only an approximation close to Earth's surface, isn't it?
See if you can stop trolling for a while just to make in .
As I say above "You can't just say that since is infinite, that will be infinite, for is not going to be constant over that distance!" Who's trolling?

80. Originally Posted by Robittybob1
Who's trolling?
You are, as always. Give it a rest.

81. Originally Posted by Howard Roark
Originally Posted by Robittybob1
Who's trolling?
You are, as always. Give it a rest.
I gave you a challenge and I'd say you lost it.

82. Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by Robittybob1
Who's trolling?
You are, as always. Give it a rest.
I gave you a challenge and I'd say you lost it.

83. Originally Posted by Howard Roark
Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by Robittybob1
Who's trolling?
You are, as always. Give it a rest.
I gave you a challenge and I'd say you lost it.
You are behaving like a troll.

84. Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by Robittybob1
Who's trolling?
You are, as always. Give it a rest.
I gave you a challenge and I'd say you lost it.
You are behaving like a troll.
In thic case Robbity is correct.
Even the Wiki page Roark gave contradicts what Roark stated and Roark has simply been making personal attacks since he was told he was making a mistake.

At this moment he is the one acting delusional instead of Robbity.

85. You only have to give an object sufficient energy to reach escape velocity to send it to infinity, and that definitely isn't an infinite amount of energy.
Thanks Dan.

86. Technically in an imaginary universe where there are only two objects, it will require infinite energy to go infinite distance as even though g may become VERY small it is still not 0, and once you really grasp infinity it's kind of cool

87. Originally Posted by dan hunter
Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by Robittybob1
Who's trolling?
You are, as always. Give it a rest.
I gave you a challenge and I'd say you lost it.
You are behaving like a troll.
In thic case Robbity is correct.
Even the Wiki page Roark gave contradicts what Roark stated and Roark has simply been making personal attacks since he was told he was making a mistake.

At this moment he is the one acting delusional instead of Robbity.
You mean that you STILL don't understand he difference between gravitational potential energy and gravitational potential. It is amnusing to see you and Robittybob1 on the same side of mistake.

88. Originally Posted by dan hunter
Originally Posted by Howard Roark
You guys tend to support each other when it comes to fringe, incorrect stuff. Do you really think that the people who know physics don't see through this?
Ahem,
Gravitational Potential Energy
Gravitational Potential Energy

The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. Because of the inverse square nature of the gravity force, the force approaches zero for large distances, and it makes sense to choose thezero of gravitational potential energy at an infinite distance away. The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches. This negative potential is indicative of a "bound state"; once a mass is near a large body, it is trapped until something can provide enough energy to allow it to escape. The general form of the gravitational potential energy of mass m is:
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers.

This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
note the negative sign

dan hunter,

is the gravitational potential, NOT the gravitational energy.

This explains why, the gravitational force , ok?
Then, gravitational energy is equal to the work expanded to move a test probe frome to :

When you copy stuff off the internet, you need to be aware that it isn't always correct, doing physics by selective copying doesn't always work.

You need to learn, rather than persisting in your basic errors. OK?

89. Originally Posted by Howard Roark
You mean that you STILL don't understand he difference between gravitational potential energy and gravitational potential. It is amnusing to see you and Robittybob1 on the same side of mistake.

This problem could be solved if you would enlighten us with your wisdom rather than amusing yourself with everyone's ignorance. So what is the difference between your gravitational potential and gravitational potential energy that others have been discussing?

Originally Posted by Howard Roark
Originally Posted by dan hunter
Originally Posted by Howard Roark
You guys tend to support each other when it comes to fringe, incorrect stuff. Do you really think that the people who know physics don't see through this?
Ahem,
Gravitational Potential Energy
Gravitational Potential Energy

The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. Because of the inverse square nature of the gravity force, the force approaches zero for large distances, and it makes sense to choose thezero of gravitational potential energy at an infinite distance away. The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches. This negative potential is indicative of a "bound state"; once a mass is near a large body, it is trapped until something can provide enough energy to allow it to escape. The general form of the gravitational potential energy of mass m is:
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers.

This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
note the negative sign

dan hunter,

is the gravitational potential, NOT the gravitational energy.

This explains why, the gravitational force , ok?
Then, gravitational energy is equal to the work expanded to move a test probe frome to :

When you copy stuff off the internet, you need to be aware that it isn't always correct, doing physics by selective copying doesn't always work.

You need to learn, rather than persisting in your basic errors. OK?
Now you have introduced a new phrase "gravitational energy" to the mix without yet explaining how gravitational potential is different from gravitational potential energy.

Your explanation of gravitational energy appears sufficient but could you clarify how gravitational potential is different from gravitational potential energy since this is essential for our understanding.

90. Originally Posted by ScienceNoob
Technically in an imaginary universe where there are only two objects, it will require infinite energy to go infinite distance as even though g may become VERY small it is still not 0, and once you really grasp infinity it's kind of cool
If those two objects were just two hydrogen atoms it wouldn't take an infinite energy to separate them. Scaling the problem up doesn't make it that much harder.

91. Originally Posted by Robittybob1
Originally Posted by ScienceNoob
Technically in an imaginary universe where there are only two objects, it will require infinite energy to go infinite distance as even though g may become VERY small it is still not 0, and once you really grasp infinity it's kind of cool
If those two objects were just two hydrogen atoms it wouldn't take an infinite energy to separate them. Scaling the problem up doesn't make it that much harder.
Actually Sciencenoob is right Robbity.

An infinity multiplied by anything is still an infinity.
(Multiplying infinity by zero doesn't count because zero is a nothing.
According to Paul Dirac the answer would be one but he was applying it in a special way.
In other areas of math it might be considered zero or it might be considered undefined.)

An infinity plus one is still an infinity, an infinity minus one is an infinity, an infinity times an infinity is still an infinity, and oddly enough an infinity divided by an infinity is undefined instead of being one.

92. Originally Posted by dan hunter
Originally Posted by Robittybob1
Originally Posted by ScienceNoob
Technically in an imaginary universe where there are only two objects, it will require infinite energy to go infinite distance as even though g may become VERY small it is still not 0, and once you really grasp infinity it's kind of cool
If those two objects were just two hydrogen atoms it wouldn't take an infinite energy to separate them. Scaling the problem up doesn't make it that much harder.
Actually Sciencenoob is right Robbity.

An infinity multiplied by anything is still an infinity.
(Multiplying infinity by zero doesn't count because zero is a nothing.
According to Paul Dirac the answer would be one but he was applying it in a special way.
In other areas of math it might be considered zero or it might be considered undefined.)

An infinity plus one is still an infinity, an infinity minus one is an infinity, an infinity times an infinity is still an infinity, and oddly enough an infinity divided by an infinity is undefined instead of being one.
I don't give up that easy. I have infinite energy. The idea that one atom of hydrogen forever and a day can influence another hydrogen atom at an infinite distance is bizarre. I'd rather accept the concept of God than to accept these two atoms having God-like properties. Energy in the end comes in quantum units. (does it not? A type of (virtual) photon.) There is no way of dividing the energy required into ever smaller quantum bits.

93. Originally Posted by Robittybob1
Originally Posted by dan hunter
Originally Posted by Robittybob1
Originally Posted by ScienceNoob
Technically in an imaginary universe where there are only two objects, it will require infinite energy to go infinite distance as even though g may become VERY small it is still not 0, and once you really grasp infinity it's kind of cool
If those two objects were just two hydrogen atoms it wouldn't take an infinite energy to separate them. Scaling the problem up doesn't make it that much harder.
Actually Sciencenoob is right Robbity.

An infinity multiplied by anything is still an infinity.
(Multiplying infinity by zero doesn't count because zero is a nothing.
According to Paul Dirac the answer would be one but he was applying it in a special way.
In other areas of math it might be considered zero or it might be considered undefined.)

An infinity plus one is still an infinity, an infinity minus one is an infinity, an infinity times an infinity is still an infinity, and oddly enough an infinity divided by an infinity is undefined instead of being one.
I don't give up that easy. I have infinite energy. The idea that one atom of hydrogen forever and a day can influence another hydrogen atom at an infinite distance is bizarre. I'd rather accept the concept of God than to accept these two atoms having God-like properties. Energy in the end comes in quantum units. (does it not? A type of (virtual) photon.) There is no way of dividing the energy required into ever smaller quantum bits.
Anything divided by an infinity becomes effectively zero, except that it is really undefined because you can't actually divide anything by infinity. Infinity is not actually a number.
In set theory they use the idea of infinite sets to define limits as included. they want the sets to sum up to real numbers and avoid infintesimal numbers. Sciencenoob is using the idea of infinitesimals, the idea you can approach a limit but never actually reach it.

But this isn't why gravitational potential is set by convention to be zero at an infinite distance. The reason gravitational potentail is negative is that a falling object is losing potential energy and converting it into kinetic energy as it falls. If gravitational potential was positive the signs would be changed and a falling object would gain potential energy as it fell and lose kinetic energy. You know from dropping things that such is not the case.
If you were to set the gravitational zero point at the center of earth you would still have to label it as a negative force when dealing with more massive objects, like black holes, relative to the earth. You would also still have a serious math problem when looking at objects falling towards the earth.
So gravitational potential was set to be zero at an infinite distance.

Be careful about confusing gravitational potential with gravitational force too, they are not the same thing.
Consider an infinitely massive object and it is at the bottom of an infinitely deep gravity well. The gravitaional potential would be negative infinity, but the gravitational force would be positive infinity instead.
An object falling towards it from an infinite distance would also have infinite velocity and thus infinite kinetic energy.

Of course sombody else may want to correct my obvious misunderstanding here.

94. Originally Posted by dan hunter
....
Anything divided by an infinity becomes effectively zero, except that it is really undefined because you can't actually divide anything by infinity. Infinity is not actually a number.
In set theory they use the idea of infinite sets to define limits as included. they want the sets to sum up to real numbers and avoid infinitesimal numbers. Sciencenoob is using the idea of infinitesimals, the idea you can approach a limit but never actually reach it. ....
Can anything other than numbers be divided by infinity? Not as far as I know.
I get tired even thinking the problem through in my mind.

95. Originally Posted by bangstrom
Originally Posted by Howard Roark
You mean that you STILL don't understand he difference between gravitational potential energy and gravitational potential. It is amnusing to see you and Robittybob1 on the same side of mistake.

This problem could be solved if you would enlighten us with your wisdom rather than amusing yourself with everyone's ignorance. So what is the difference between your gravitational potential and gravitational potential energy that others have been discussing?

Originally Posted by Howard Roark
Originally Posted by dan hunter
Originally Posted by Howard Roark
You guys tend to support each other when it comes to fringe, incorrect stuff. Do you really think that the people who know physics don't see through this?
Ahem,
Gravitational Potential Energy
Gravitational Potential Energy

The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. Because of the inverse square nature of the gravity force, the force approaches zero for large distances, and it makes sense to choose thezero of gravitational potential energy at an infinite distance away. The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches. This negative potential is indicative of a "bound state"; once a mass is near a large body, it is trapped until something can provide enough energy to allow it to escape. The general form of the gravitational potential energy of mass m is:
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers.

This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
note the negative sign

dan hunter,

is the gravitational potential, NOT the gravitational energy.

This explains why, the gravitational force , ok?
Then, gravitational energy is equal to the work expanded to move a test probe frome to :

When you copy stuff off the internet, you need to be aware that it isn't always correct, doing physics by selective copying doesn't always work.

You need to learn, rather than persisting in your basic errors. OK?
Now you have introduced a new phrase "gravitational energy" to the mix without yet explaining how gravitational potential is different from gravitational potential energy.
I did. Twice.

96. Originally Posted by Robittybob1
Can anything other than numbers be divided by infinity? Not as far as I know.
I get tired even thinking the problem through in my mind.
Just tired, or infinitely tired?

Nothing in the real world can be multiplied or divided infinitely because nobody has enough time to do it.

97. Originally Posted by dan hunter
Originally Posted by Robittybob1
Originally Posted by ScienceNoob
Technically in an imaginary universe where there are only two objects, it will require infinite energy to go infinite distance as even though g may become VERY small it is still not 0, and once you really grasp infinity it's kind of cool
If those two objects were just two hydrogen atoms it wouldn't take an infinite energy to separate them. Scaling the problem up doesn't make it that much harder.
Actually Sciencenoob is right Robbity.

An infinity multiplied by anything is still an infinity.
(Multiplying infinity by zero doesn't count because zero is a nothing.
Actually, this is wrong. Basic calculus shows how to treat the case as a common case.

98. Originally Posted by Howard Roark
Originally Posted by bangstrom
Originally Posted by Howard Roark
You mean that you STILL don't understand he difference between gravitational potential energy and gravitational potential. It is amnusing to see you and Robittybob1 on the same side of mistake.

This problem could be solved if you would enlighten us with your wisdom rather than amusing yourself with everyone's ignorance. So what is the difference between your gravitational potential and gravitational potential energy that others have been discussing?

Originally Posted by Howard Roark
Originally Posted by dan hunter
Originally Posted by Howard Roark
You guys tend to support each other when it comes to fringe, incorrect stuff. Do you really think that the people who know physics don't see through this?
Ahem,
Gravitational Potential Energy
Gravitational Potential Energy

The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. Because of the inverse square nature of the gravity force, the force approaches zero for large distances, and it makes sense to choose thezero of gravitational potential energy at an infinite distance away. The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches. This negative potential is indicative of a "bound state"; once a mass is near a large body, it is trapped until something can provide enough energy to allow it to escape. The general form of the gravitational potential energy of mass m is:
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers.

This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
note the negative sign

dan hunter,

is the gravitational potential, NOT the gravitational energy.

This explains why, the gravitational force , ok?
Then, gravitational energy is equal to the work expanded to move a test probe frome to :

When you copy stuff off the internet, you need to be aware that it isn't always correct, doing physics by selective copying doesn't always work.

You need to learn, rather than persisting in your basic errors. OK?
Now you have introduced a new phrase "gravitational energy" to the mix without yet explaining how gravitational potential is different from gravitational potential energy.
I did. Twice.
Actually you didn't. But who cares, you are still wrong.

99. Originally Posted by Howard Roark
Originally Posted by dan hunter
Originally Posted by Robittybob1
Originally Posted by ScienceNoob
Technically in an imaginary universe where there are only two objects, it will require infinite energy to go infinite distance as even though g may become VERY small it is still not 0, and once you really grasp infinity it's kind of cool
If those two objects were just two hydrogen atoms it wouldn't take an infinite energy to separate them. Scaling the problem up doesn't make it that much harder.
Actually Sciencenoob is right Robbity.

An infinity multiplied by anything is still an infinity.
(Multiplying infinity by zero doesn't count because zero is a nothing.
Actually, this is wrong. Basic calculus shows how to treat the case as a common case.
could you give us a reference please?

100. Originally Posted by Robittybob1
Originally Posted by Howard Roark
Originally Posted by dan hunter
Originally Posted by Robittybob1
Originally Posted by ScienceNoob
Technically in an imaginary universe where there are only two objects, it will require infinite energy to go infinite distance as even though g may become VERY small it is still not 0, and once you really grasp infinity it's kind of cool
If those two objects were just two hydrogen atoms it wouldn't take an infinite energy to separate them. Scaling the problem up doesn't make it that much harder.
Actually Sciencenoob is right Robbity.

An infinity multiplied by anything is still an infinity.
(Multiplying infinity by zero doesn't count because zero is a nothing.
Actually, this is wrong. Basic calculus shows how to treat the case as a common case.
could you give us a reference please?

L'Hospital rule. Did you take calculus in high school?

101. Originally Posted by dan hunter
Originally Posted by Howard Roark
Originally Posted by bangstrom
Originally Posted by Howard Roark
You mean that you STILL don't understand he difference between gravitational potential energy and gravitational potential. It is amnusing to see you and Robittybob1 on the same side of mistake.

This problem could be solved if you would enlighten us with your wisdom rather than amusing yourself with everyone's ignorance. So what is the difference between your gravitational potential and gravitational potential energy that others have been discussing?

Originally Posted by Howard Roark
Originally Posted by dan hunter
Originally Posted by Howard Roark
You guys tend to support each other when it comes to fringe, incorrect stuff. Do you really think that the people who know physics don't see through this?
Ahem,
Gravitational Potential Energy
Gravitational Potential Energy

The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. Because of the inverse square nature of the gravity force, the force approaches zero for large distances, and it makes sense to choose thezero of gravitational potential energy at an infinite distance away. The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches. This negative potential is indicative of a "bound state"; once a mass is near a large body, it is trapped until something can provide enough energy to allow it to escape. The general form of the gravitational potential energy of mass m is:
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers.

This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
note the negative sign

dan hunter,

is the gravitational potential, NOT the gravitational energy.

This explains why, the gravitational force , ok?
Then, gravitational energy is equal to the work expanded to move a test probe frome to :

When you copy stuff off the internet, you need to be aware that it isn't always correct, doing physics by selective copying doesn't always work.

You need to learn, rather than persisting in your basic errors. OK?
Now you have introduced a new phrase "gravitational energy" to the mix without yet explaining how gravitational potential is different from gravitational potential energy.
I did. Twice.
Actually you didn't. But who cares, you are still wrong.
It is amusing to see you persisting in your crankery. You have no clue , yet you persist, in classical crank fashion.

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