# Thread: Gravity, space, space-time, mass, how do these things work?

1. I understand that gravity is the attracting force between two bodies of mass, for instance, the earth is pulling the moon towards it and is simply in free fall towards earth, i read that a objects orbit is credited to mass bending the space around it. this sounds great and all but I cant seem to find anything explaining how does mass actually bend space-time, what is it about the properties of mass that causes space to warp?

it seems that most scientist believe that mass is condensed energy, they also believe that the faster a object travels=the more mass/energy that object has. does this mean that a small object can become equivalent to a much larger mass if it were to be accelerated to a sufficient velocity? would that object bend the space around in scale with its velocity? wouldnt this mean that could actually bend space and create gravitational waves by simply making a chunk of mass spin fast enough?

speed is measured by how much distance a object travels within a given time frame right? imagine you have a spinning wheel with a pipe attached horizontally at the wheels center, spin it and you will see that the pipe is actually traveling a greater distance of space the further the pipe extends from the wheel, this means that the end of the pipe is traveling faster then other end that is attached to the wheels center. now imagine you create a pipe that is trillions of miles long, you bring this pipe into space and begin to swing it in a circle.....so if the pipes speed is increased relative to the observer that is swinging it, then what happens if the pipe was so long that it would have to break the speed of light?? would the end of the pipe be moving so fast that it gains enough mass to bend the space-time around itself which would cause time to dilate and slow down on order to keep the speed of light constant?

2.

3. There are a lot of different questions in your post, the answers of which are not necessarily easy to explain.

First of all, General Relativity is a descriptive model of spacetime and gravity - this means it tells us very well how gravity behaves, but it has nothing to say as to why it does so. Using this model, we can take a given distribution of energy-momentum, and calculate from it how the trajectories of objects in and around it would behave, as well as the evolution of that distribution itself. That's all it does. GR makes no attempt to provide an explanation as to why energy-momentum influences the geometry of spacetime in the particular way it does; part of the reason seem to be fundamental principles of topology, but that's likely not the full story. This is currently an area of ongoing and very active research.

1. The source of gravity is energy-momentum, not just mass. The key here is to understand that the mathematical object that describes energy-momentum is not just a single number, but a more complicated object called a tensor; you can envision this as a matrix of 16 real numbers, each of which describes a certain aspect of the energy-momentum distribution.

2. When a mass is in relative motion with respect to an observer, then that observer will say that the object's relativistic mass has increased. It is important to understand that 'relativistic mass' can be seen as a measure of an object's resistance to further acceleration - hence, the faster something moves with respect to yourself, the harder it becomes to accelerate that object even more. However, relativistic mass - despite what its rather unfortunate name would imply - is not in isolation a source of gravity. That means that the gravity an object in relative motion exerts on itself and its surrounds is not "increased" just by being in relative motion; this would obviously create paradoxes. Mathematically, what happens is that - when you put something in relative motion - some components of the energy-momentum tensor will increase, while others will decrease in magnitude, leaving the overall object unchanged. All of this is not to say that momentum has no effect on gravity - it does -, but only that relativistic mass on its own isn't a source of spacetime distortions.

3. Yes, you can generate gravitational waves by making a chunk of mass spin fast enough, so long as that chunk isn't a perfect sphere ( in which case there will be no gravitational waves ). However, the underlying mechanism is very much more complicated than the simple notion of relativistic mass. A spinning chunk of mass will emit gravitational waves, and it will also distort the background spacetime through which those waves propagate in complicated ways, overall giving quite a complex system.

4. If you spin a long pipe at high velocity, then its two ends no longer share the same notion of simultaneity. Measured from any point on the pipe, the outermost end will start to "lag behind" your own clock, so it will never exceed the speed of light as measured by you. An observer looking on such a spinning setup will see the pipe ceasing to be straight, and instead "bend" backwards with respect to the direction of rotation. There will be a gravitational field associated with such a setup, but again, it will be quite complicated, and not due to 'relativistic mass'. Mathematically, this is quite a difficult scenario to treat.

In general terms, the main message is - when talking about gravity, it is best not to think in terms of 'relativistic mass', since that quantity taken in isolation isn't a source of gravity. It is necessary to consider all forms of energy - energy density itself, momentum, pressure, flux etc etc. Gravity cannot be reduced down to a single number; even the concept of "strength of gravity" is problematic, and can't be easily defined.

4. Originally Posted by Markus Hanke
3. Yes, you can generate gravitational waves by making a chunk of mass spin fast enough, so long as that chunk isn't a perfect sphere ( in which case there will be no gravitational waves ). However, the underlying mechanism is very much more complicated than the simple notion of relativistic mass. A spinning chunk of mass will emit gravitational waves, and it will also distort the background spacetime through which those waves propagate in complicated ways, overall giving quite a complex system.
Although it is commonly believed that a rotating rigid dumbbell emits gravitational radiation, there is good reason to believe that it doesn't, though I can't provide a rigorous proof. The thing to note is that a spacetime containing a solitary spinning rigid object is a stationary spacetime, and there is no gravitational radiation in a stationary spacetime. The stationary nature of the spacetime is evident in the spinning frame of reference of the object: The energy-momentum of the source of the gravitation is stationary and therefore so is the surrounding gravitation of the rest of the spacetime. Furthermore, conservation of angular momentum guaranteed by Noether's theorem ensures that gravitational radiation isn't produced by decreasing the spin of the object.

In the case of the decay of orbits, gravitational radiation is emitted due to the decreasing distance between the objects, not the orbital motion itself, which would seem to be a red herring. As the distance between the orbiting objects decrease, the gravitation around the objects changes, leading to a small part of this gravitation being emitted away as radiation.

It should also be noted that although the gravitation around a rotating rigid dumbbell cyclically changes in the non-rotating frame of reference, this change is not the same as gravitational radiation as it doesn't actually radiate away from the rotating source.

5. Interesting point, KJW
Your argument does make sense. However, I distinctly remember that in the very first GR textbook I ever read ( a German language university textbook, probably no point referencing it here ), the calculation was explicitly performed for a spinning rigid dumbbell, and it was found that it does indeed emit gravitational radiation, and the amplitudes and frequencies were calculated. It was always my understanding that any setup that has a non-zero quadrupole or higher multipole moment will emit such radiation - independent of the actual nature of the source. This is also what MTW seems to say ( page 977, bottom ) - and it then goes on to explicitly perform the calculation for a rotating steel beam ( §36.3, page 979 ).

I'm not saying your argument is wrong, I'm just trying to point out that it seems to contradict standard literature, unless I'm missing something.

6. Not sure if this is a contribution to the thread but I actually sent a question (back in 2014) to the NASA website as to whether spinning neutron stars had extra mass on account of their rotation . They said yes.(although it was small)
This is the exchange :

Topic: Neutron Stars
Level: I am an adult who knows high school physics and have a casual interest in astronomy.

Hello
I have a specific question regarding pulsars.

Is their gravitational field (I mean the way they distort SpaceTime) stronger
than that of a normal ,non-rotating neutron star of equivalent rest mass ?

If this is not the case do they affect Space Time differently at all from
this hypothetical neutron star?

I understand rotation increases relativistic mass but am unclear as to the
implications of this.

By the way I assume that all neutron stars and all pulsars must be practically
homogeneous .

-------------------------------------------------------------------------------

There is energy in the movement of rotation, and that energy has a mass
equivalent. Thus a rotating neutron star does have a slightly higher mass,
but the difference is insignificant, even for a very rapidly rotating neutron
star.

So the gravitational field of a rotating neutron star is not significantly
stronger than that of a non-rotating neutron star. However, it is not quite
the same. Far away there is no difference, but there are some differences
nearby. A satellite above the equator will orbit slightly faster in the
same direction as the rotation than it would in the opposite direction. A
rotating satellite will experience a torque (twisting force) due to the
rotation of the neutron star.

These effects are present for any rotating object, such as the Earth, but
very weak except near to a compact object.

"Compact" in this case means more dense than ordinary matter. Neutron stars
and black holes are what an astronomer calls "compact objects". The
standard model is that a neutron star is a solid ball of neutrons all the
way through, but there may be some exotic physics going on in the center.
Naturally it is difficult to investigate the state of matter, since we can
compress only very small amounts of matter to such high density in our
laboratories. There is probably a thin crust of mostly iron isotopes on
the surface.

By the way, a non-rotating neutron star would not be "normal". Any object
that has not interacted strongly with another would be expected to have a
random amount of angular momentum. Many moons do not rotate relative to
their primaries, but that is because of strong tidal interactions. It
would be very unexpected, not normal, to find a neutron star that had
exactly zero angular momentum out of all the possible values of angular
momentum.

--
Jay and Alaina

Here is a good site for anyone interested in astronomy:
http://antwrp.gsfc.nasa.gov/apod/astropix.html

7. Originally Posted by Markus Hanke
There are a lot of different questions in your post, the answers of which are not necessarily easy to explain.

First of all, General Relativity is a descriptive model of spacetime and gravity - this means it tells us very well how gravity behaves, but it has nothing to say as to why it does so. Using this model, we can take a given distribution of energy-momentum, and calculate from it how the trajectories of objects in and around it would behave, as well as the evolution of that distribution itself. That's all it does. GR makes no attempt to provide an explanation as to why energy-momentum influences the geometry of spacetime in the particular way it does; part of the reason seem to be fundamental principles of topology, but that's likely not the full story. This is currently an area of ongoing and very active research.

1. The source of gravity is energy-momentum, not just mass. The key here is to understand that the mathematical object that describes energy-momentum is not just a single number, but a more complicated object called a tensor; you can envision this as a matrix of 16 real numbers, each of which describes a certain aspect of the energy-momentum distribution.

2. When a mass is in relative motion with respect to an observer, then that observer will say that the object's relativistic mass has increased. It is important to understand that 'relativistic mass' can be seen as a measure of an object's resistance to further acceleration - hence, the faster something moves with respect to yourself, the harder it becomes to accelerate that object even more. However, relativistic mass - despite what its rather unfortunate name would imply - is not in isolation a source of gravity. That means that the gravity an object in relative motion exerts on itself and its surrounds is not "increased" just by being in relative motion; this would obviously create paradoxes. Mathematically, what happens is that - when you put something in relative motion - some components of the energy-momentum tensor will increase, while others will decrease in magnitude, leaving the overall object unchanged. All of this is not to say that momentum has no effect on gravity - it does -, but only that relativistic mass on its own isn't a source of spacetime distortions.

3. Yes, you can generate gravitational waves by making a chunk of mass spin fast enough, so long as that chunk isn't a perfect sphere ( in which case there will be no gravitational waves ). However, the underlying mechanism is very much more complicated than the simple notion of relativistic mass. A spinning chunk of mass will emit gravitational waves, and it will also distort the background spacetime through which those waves propagate in complicated ways, overall giving quite a complex system.

4. If you spin a long pipe at high velocity, then its two ends no longer share the same notion of simultaneity. Measured from any point on the pipe, the outermost end will start to "lag behind" your own clock, so it will never exceed the speed of light as measured by you. An observer looking on such a spinning setup will see the pipe ceasing to be straight, and instead "bend" backwards with respect to the direction of rotation. There will be a gravitational field associated with such a setup, but again, it will be quite complicated, and not due to 'relativistic mass'. Mathematically, this is quite a difficult scenario to treat.

In general terms, the main message is - when talking about gravity, it is best not to think in terms of 'relativistic mass', since that quantity taken in isolation isn't a source of gravity. It is necessary to consider all forms of energy - energy density itself, momentum, pressure, flux etc etc. Gravity cannot be reduced down to a single number; even the concept of "strength of gravity" is problematic, and can't be easily defined.
I find it so un-fair that there is so much stuff left to learn about the universe we live in.

8. Since we're here at relativistic mass, is it the same difference as proper time vs. coordinate time? Or maybe not?

9. Originally Posted by xxsolarplexusxx
I find it so un-fair that there is so much stuff left to learn about the universe we live in.
Or exciting?

10. Originally Posted by GiantEvil
Since we're here at relativistic mass, is it the same difference as proper time vs. coordinate time? Or maybe not?
There are similarities, in that relativistic mass and coordinate time are observer dependent. But I think coordinate time is more "real" or fundamental. You can avoid using relativistic mass as a concept (a lot of people think it causes confusion).

11. I find it so un-fair that there is so much stuff left to learn about the universe we live in.
Hey, I think it's actually great What this does is foster a sense of curiosity, mystery, awe and striving within us - how boring would life be, if we already knew everything there is to know about the universe ?

Our knowledge is both incomplete, and yet almost too extensive for any one person to learn within a single lifetime - and you know what ? I wouldn't want to have it any other way

12. Originally Posted by GiantEvil
Since we're here at relativistic mass, is it the same difference as proper time vs. coordinate time? Or maybe not?
Yes, it's very similar - coordinate measurements are valid only in a single frame, and no one else will agree on them. The same is true for relativistic mass - it's essentially a frame-dependent quantity.

Proper measurements are different, because all observers agree on them. In terms of mass, it is invariant mass ( also called rest mass ) that plays this role.

13. Originally Posted by Markus Hanke
I'm not saying your argument is wrong, I'm just trying to point out that it seems to contradict standard literature, unless I'm missing something.
No, you will find it stated in various texts that rotating objects of particular shapes emit gravitational radiation (including Wikipedia). And I haven't come across anything in the literature that says that such objects don't emit gravitational radiation, so my statement that a rotating rigid object of any shape doesn't emit gravitational radiation is a reasoned statement rather than something I have read.

Originally Posted by Markus Hanke
It was always my understanding that any setup that has a non-zero quadrupole or higher multipole moment will emit such radiation
An alternative way to interpret this is as a statement of what does not emit gravitational radiation (i.e. zero quadrupole and higher multipole moments).

Originally Posted by Markus Hanke
I distinctly remember that in the very first GR textbook I ever read, the calculation was explicitly performed for a spinning rigid dumbbell, and it was found that it does indeed emit gravitational radiation, and the amplitudes and frequencies were calculated ... This is also what MTW seems to say ( page 977, bottom ) - and it then goes on to explicitly perform the calculation for a rotating steel beam ( §36.3, page 979 ).
As for the explicit calculations, all I can suggest is that they made a subtle error. Bear in mind though that all the different instances of the calculations could be based on the same subtle error, so that the number of such calculations shouldn't add more weight to the argument that such objects emit gravitational radiation. Perhaps one should consider whether the calculations have properly accounted for what may be called "Mach's principle", as it is this principle that is central to my argument that rotating rigid objects (of any shape) do not emit gravitational radiation.

It seems to me that it is easy to rely on an intuitive viewpoint when considering the gravitation from a rotating rigid non-axisymmetric object. However, it also seems obvious that if the same object were not rotating, then it would not emit gravitational radiation. So the question becomes: What is it about rotation that causes an object to emit gravitational radiation? Note that we are considering this from a general relativistic point of view where we are able to consider this in the frame of the rotating object itself and see that it is just as stationary as if the object were not rotating.

Also, note that a rotating rigid object is different to two objects orbiting each other. In the case of two objects orbiting each other, there is freedom for the two objects to change the distance between them, and in doing so, change the gravitational field around them. By contrast, a rotating rigid object can only change the rate of rotation, which doesn't actually change the gravitational field around the object (when considered in the rotating frame of the object).

14. Originally Posted by KJW
No, you will find it stated in various texts that rotating objects of particular shapes emit gravitational radiation (including Wikipedia). And I haven't come across anything in the literature that says that such objects don't emit gravitational radiation, so my statement that a rotating rigid object of any shape doesn't emit gravitational radiation is a reasoned statement rather than something I have read.

An alternative way to interpret this is as a statement of what does not emit gravitational radiation (i.e. zero quadrupole and higher multipole moments).

It seems to me that it is easy to rely on an intuitive viewpoint when considering the gravitation from a rotating rigid non-axisymmetric object. However, it also seems obvious that if the same object were not rotating, then it would not emit gravitational radiation. So the question becomes: What is it about rotation that causes an object to emit gravitational radiation? Note that we are considering this from a general relativistic point of view where we are able to consider this in the frame of the rotating object itself and see that it is just as stationary as if the object were not rotating.

Also, note that a rotating rigid object is different to two objects orbiting each other. In the case of two objects orbiting each other, there is freedom for the two objects to change the distance between them, and in doing so, change the gravitational field around them. By contrast, a rotating rigid object can only change the rate of rotation, which doesn't actually change the gravitational field around the object (when considered in the rotating frame of the object).
What is a "rotating rigid body" ? (surely not the neutron star in post#5,is it?)

Doesn't any rotation cause eddies internally ?

15. Originally Posted by geordief
What is a "rotating rigid body"?
A rigid object is an object that doesn't change shape, and if it is rotating, it doesn't change shape as a result of the centrifugal forces. It doesn't exist in the real world but is an idealisation that is the limit of increasing stiffness to infinite stiffness. If one considers the gravitational radiation from a rotating object that does change shape slightly, then considering the effect of decreasing the change in shape on the gravitational radiation, what is the gravitational radiation in the limit of no change in the shape of the object?

Originally Posted by geordief
Doesn't any rotation cause eddies internally?
By the definition of a rigid object, no. (Note what I said above about a rigid object being an idealisation).

16. Originally Posted by KJW

A rigid object is an object that doesn't change shape, and if it is rotating, it doesn't change shape as a result of the centrifugal forces. It doesn't exist in the real world but is an idealisation that is the limit of increasing stiffness to infinite stiffness. If one considers the gravitational radiation from a rotating object that does change shape slightly, then considering the effect of decreasing the change in shape on the gravitational radiation, what is the gravitational radiation in the limit of no change in the shape of the object?

I don't know but I can see that it is mathematically a limit. You seem to be saying that it is the changes in internal gravitational fields of a rotating body that cause changes to external spacetime curvature and that ,in an idealized rigid rotating body this external change would not occur.

Have I got it?

17. Originally Posted by geordief
I don't know but I can see that it is mathematically a limit. You seem to be saying that it is the changes in internal gravitational fields of a rotating body that cause changes to external spacetime curvature and that ,in an idealized rigid rotating body this external change would not occur.

Have I got it?
Instead of "internal gravitational fields", that should be "energy-momentum field" as this is the source of the gravitational field (both internal and external). But yes, if you keep the source energy-momentum the same, then the resulting gravitational field will also remain the same. The only difference between the rotating and non-rotating case is the auxiliary condition that determines the background spacetime (rotating or non-rotating metric).

18. The crux of the argument is as follows:

Consider the Einstein equation (ignoring the proportionality constant):

where:

is the Einstein tensor, a 4x4 matrix of differential expressions that are second-order in the solution metric tensor, and:

is a 4x4 matrix of functions of the coordinates

There are two things to note:

1: As a second-order system of differential equations, a particular solution requires auxiliary conditions to be specified as the equation itself is not sufficient to determine the metric.

2: As a tensor equation, the coordinates have no meaning until the equation is solved. Thus, for a given specification of functions:

the general solution will be the same regardless of the meaning of the coordinates .

Thus, if the functions are the same in the frame of the rotating object as they are in the frame of the corresponding non-rotating object, then the general solution will be the same, and the specification of the rotation of the object will be part of the auxiliary conditions. Any gravitational radiation will be part of the general solution and not part of the auxiliary conditions which specify the flat background spacetime.

19. Thus, if the functions are the same in the frame of the rotating object as they are in the frame of the corresponding non-rotating object, then the general solution will be the same, and the specification of the rotation of the object will be part of the auxiliary conditions. Any gravitational radiation will be part of the general solution and not part of the auxiliary conditions which specify the flat background spacetime.
I understand your argument, but I see an issue here - when we speak about "general solution", then what we mean is a metric that covers both the interior as well as the exterior vacuum region, while remaining everywhere smooth and differentiable. So you start with a field , which, as you rightly point out, is physically the same regardless of our choice of coordinate basis. The question is now this - what are appropriate boundary conditions, in order for us to obtain a solution to the field equations under these circumstances ?

One boundary condition is imposed on us right away, because we must demand that the general solution - the one covering both the interior and exterior region - remains smooth and everywhere differentiable at the boundary of the rotating body. If, in addition to this, we also demand that the exterior region is everywhere both Ricci flat ( vacuum ) and Weyl flat ( no gravitational radiation ), are we still going to be able to obtain a general solution to the problem, as you suggest ? This means, can we match the interior region of such a body to the exterior vacuum in such a way as to keep the metric smooth at the boundary, while at the same time guaranteeing a stationary spacetime ?

I don't know the answer to this, but I have a strong intuition that it is the physically necessary condition to keep spacetime smooth and differentiable at the surface boundary of the body, that might throw a spanner into these works. Even in the much simpler case of the Kerr spacetime, no one has as of yet been able to find an interior solution that smoothly connects to the exterior Kerr metric; on the other hand though, Wikipedia mentions that the outer parts of the Kerr interior can be modelled as a colliding plane wave spacetime (!), which again does not seem to match any asymptotically flat vacuum solution on the exterior. How much more complicated would this be if the body is not axially symmetric !

20. Originally Posted by Markus Hanke
when we speak about "general solution", then what we mean is a metric that covers both the interior as well as the exterior vacuum region, while remaining everywhere smooth and differentiable.
No. What is meant by "general solution" is a metric tensor field that contains arbitrary functions that are eliminated when the Einstein tensor is formed. For example, if we consider the equation:

then the general solution is:

where is any flat metric and are arbitrary functions (for -dimensional space). However, if we consider the equations:

or

then the solutions are no longer limited to flat spaces, and there are more arbitrary functions to cover the expanded set of solutions (compared to )

However, these three equations have one thing in common... the source term is zero. This means that the equation has no explicit dependence on the coordinates. The consequence of this is that given any solution metric, a coordinate transformation yields another solution. In other words, the arbitrary functions contains functions of the form:

(though only in the case of is that form complete.)

For equations in which the source term explicitly depends on the coordinates, it is no longer true that a coordinate transformation of a solution is a solution, though presumably (this is something I'm not too sure about) the set of solutions that form the general solution is the same "size" as the set of solutions of the corresponding equations with zero source terms. Nevertheless, even in the cases in which the source term explicitly depends on the coordinates, the arbitrary functions are eliminated by forming the equation. Thus, the auxiliary conditions that reduce the general solution to a particular solution are separate from the equation, and in the case of the Einstein equation, separate from the energy-momentum distribution of the spacetime. And given that we are only interested in the gravitation from the energy-momentum (as distinct from source-free gravitation), the auxiliary conditions are separate from the gravitation in which we are interested.

Originally Posted by Markus Hanke
and Weyl flat ( no gravitational radiation )
No, not Weyl flat. It is essential to distinguish between gravitational radiation and the ordinary gravitational field that surrounds an energy-momentum distribution. For both a non-rotating and rotating rigid object, there will be a gravitational field in the surrounding spacetime and thus the Weyl tensor will not be zero in either case. The distinction between a gravitational field and gravitational radiation is analogous to the distinction between an electromagnetic field and electromagnetic radiation (one does not see light emitted from a magnet).

Originally Posted by Markus Hanke
I have a strong intuition that it is the physically necessary condition to keep spacetime smooth and differentiable
I feel that you may be over-emphasising the importance of smoothness, whereas the true issue is integrability, which impose constraints on the source terms that yield any solutions at all. The conservation laws are a consequence of integrability conditions, and in the case of the Einstein equation, whether an energy-momentum distribution for a non-rotating object can possibly be an energy-momentum distribution for a rotating object, or vice versa. However, although I'm assuming that it can, even if it can't, I don't think the internal stress due to centrifugal force would be sufficient to invalidate my argument that rotating rigid objects do not emit gravitational radiation.

21. For both a non-rotating and rotating rigid object, there will be a gravitational field in the surrounding spacetime and thus the Weyl tensor will not be zero in either case.
Apologies, you are right of course - I got this badly wrong. Both Ricci flatness and Weyl flatness simultaneously would imply a vanishing Riemann tensor, hence no gravitation at all.

22. And given that we are only interested in the gravitation from the energy-momentum (as distinct from source-free gravitation), the auxiliary conditions are separate from the gravitation in which we are interested.
Yes, it seems obvious indeed that the auxiliary conditions are separate; they are what you need to impose in order to obtain specific solutions to the system of differential equations that are the Einstein equations.

However, I am unsure about the other point you bring up - why are we only interested in the gravitation within the region of non-vanishing energy-momentum ? Since the original question is whether or not there exists gravitational radiation external to the rotating body, is the aim not to find a metric that covers the entire spacetime, i.e. interior and exterior region ?

Another question - if it turns out that the interior spacetime is non-stationary ( as seems to be the case for an axialsymmetric rotating ideal fluid ), what would that imply with regards to the exterior vacuum region ? Is it - at least in principle - even possible to have a non-stationary interior, but a stationary exterior region ?

23. Originally Posted by Markus Hanke
And given that we are only interested in the gravitation from the energy-momentum (as distinct from source-free gravitation), the auxiliary conditions are separate from the gravitation in which we are interested.
However, I am unsure about the other point you bring up - why are we only interested in the gravitation within the region of non-vanishing energy-momentum ? Since the original question is whether or not there exists gravitational radiation external to the rotating body, is the aim not to find a metric that covers the entire spacetime, i.e. interior and exterior region ?
I think you misunderstood, or I chose my words poorly. By "gravitation from the energy-momentum", I meant "gravitation from the energy-momentum as source". This includes the gravitation external to the energy-momentum, as well as gravitation internal to the energy-momentum (Weyl tensor only). However, I said "gravitation from the energy-momentum" so as to distinguish this gravitation from any gravitation that may exist that does not have any source, the gravitation that would remain if the energy-momentum were removed from the spacetime. The point is that we are only interested in the gravitation that is a direct consequence of the specified energy-momentum distribution and not any background curvature that is independent of the specified energy-momentum distribution (though because curvature isn't linear, it isn't entirely clear what this actually means).

Originally Posted by Markus Hanke
Another question - if it turns out that the interior spacetime is non-stationary ( as seems to be the case for an axialsymmetric rotating ideal fluid ), what would that imply with regards to the exterior vacuum region ? Is it - at least in principle - even possible to have a non-stationary interior, but a stationary exterior region ?
It can be proven that if an equation has a unique solution, then that solution has all the symmetries of the equation. However, if an equation has multiple solutions, then those solutions do not necessarily possess the symmetries of the equation. This is called "symmetry breaking". But, even if there are multiple solutions, the set of solutions is itself unique, and therefore must possess all the symmetries of the equation.

To answer your question, can one derive a unique interior from the exterior region? Normally, in solving the Einstein equation, one is obtaining an exterior spacetime metric from an interior energy-momentum distribution. Since the exterior spacetime metric is a unique solution (given the auxiliary conditions), then a stationary energy-momentum distribution along with stationary auxiliary conditions implies a stationary exterior spacetime metric. But you're asking if the absence of a given symmetry in the interior energy-momentum distribution can produce that symmetry in the exterior spacetime metric. The answer is - at least in principle - yes, provided that there isn't an equivalence between the interior and exterior regions. This means that it can only occur if multiple distinct interior regions lead to the same exterior region. But it should be noted that if the metric tensor is stationary everywhere¹, then so is the Einstein tensor (as well as all the other curvature tensors).

¹ In the case of the interior regions of blackholes, while it is technically true that these are not stationary, the corresponding symmetry still exists, which may be considered as what is being referred to as "stationary".

24. Originally Posted by KJW

then the solutions are no longer limited to flat spaces, and there are more arbitrary functions to cover the expanded set of solutions (compared to )
Actually, for this equation, I can state the general solution:

where is another arbitrary function in addition to the arbitrary functions from the general solution of the equation

PS: The equation is an identity for .

25. Right, understood. So, given all of this, how could one investigate the original question mathematically, in a manner that is as general as possible, and yet still analytically doable ? What I mean is - in what way could one rigorously prove/disprove the existence of gravitational radiation in these cases, at least in principle ?

Or asked differently - what subtle error do you think the textbooks were making here ?

26. Recently I 've reread some of Dan Brown's novels . In Angels and Demons, he recommended some theory about the space, (its creation, expansion, explosion, so on) and some facts about some famous people like Isaac Newton, Einstein. The point here is, when I first read them (at high school), I believed them immediately, but now, I raise some suspicion, whether they really exist or not. Anyone here is fan of Dan? Can you answer my question? Link deleted.

27. Dan Brown is not trustworthy as a source (for science, history or much else).

28. Originally Posted by Markus Hanke
Right, understood. So, given all of this, how could one investigate the original question mathematically, in a manner that is as general as possible, and yet still analytically doable ? What I mean is - in what way could one rigorously prove/disprove the existence of gravitational radiation in these cases, at least in principle ?

Or asked differently - what subtle error do you think the textbooks were making here ?
As I said in my initial post, I don't have a proof. However, it is a topic in which I have a particular interest. I see two different approaches to resolving the issue: (1) Construct an exact solution to the Einstein equation for a rod, a dumbbell, or other suitably shaped object. The solution would initially be for the non-rotating object, then modified to determine the effect of rotation on the object while staying in a frame that is fixed to the object by rotating the background spacetime. The goal would be to determine if the rotating object admits a stationary solution. (2) Derive a manifestly covariant expression for the source of gravitational radiation. The difficulty is to distinguish gravitational radiation from other forms of gravitation. For example, the covariant divergence of the Weyl tensor is easily derived, but the Weyl tensor itself does not define gravitational radiation.

I also have an interest in a covariant expression for electromagnetic radiation. The expression I have seen is for the Liénard-Wiechert potential, where electromagnetic radiation manifests itself in the equations for the electromagnetic field as the term (the non-radiation field is the term). The problem is that these expressions, in particular the distinction between the radiation and non-radiation terms, are not covariant. However, it does occur to me that the term represents a conserved electromagnetic field, whereas the term represents a conserved energy-momentum field that is the "square" of the electromagnetic field. Thus, it is the "square" of the electromagnetic field that best represents electromagnetic radiation, including its source.

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