# Thread: Derivation of General Relativity

1. Method 1 : From Newton's Gravity

Start with Newtonian gravity field, the validity of which is locally verified for weak fields :

which is, expressed in terms of the energy-momentum tensor

Now attempt a first ansatz to formulate this in a Lorentz-invariant manner :

which further leads to a covariant formulation of the form

Our task will now be to determine the unknown tensor G. We impose the following conditions on that tensor :

1. G is a Riemann tensor
2. G is composed of the first and second derivatives of the metric tensor
3. The energy-momentum tensor obeys the usual symmetry and conservation laws and ; these properties then by default must also apply to our tensor G
4. The theory must reduce to Newton's gravity for weak fields

Using the above four points, the Bianchi identities, as well as the general ansatz

plus a little tensor algebra, one find that the easiest tensor which satisfies all of the above conditions is

2.

3. Method 2 : Via the Principle of Least Action

Start with the Hilbert action in the presence of matter fields

Now apply the principle of least action

which implies

We also know that

Putting this all together, and calculating the variations, one gets

4. Method 3 : From String Theory

Start with the first quantized action of String Theory on a curved space-time background :

The trace of energy-momentum tensor for this becomes

wherein the beta functions represents deviation from scale invariance. Using background field methods, one can explicitly calculate the beta function for the space-time metric :

Now perform a rescaling by

and we get

which are the Einstein equations. In other words, String theory in a curved space-time automatically yields General Relativity.

5. EDIT: Sorry, MW. You're right; changed and saved.

But I do have a question for Markus. So basically GR can be mathematically derived in multiple ways from different models?

6. Please let's keep this thread serious. It's a good reference and I'd like to pin it. I may move fluffy posts to clean it up.

7. Dear Markus Hanke!
I am sorry. I apologize for my poor English.

When negative energy(mass) and positive energy(mass) coexisted, can you make field equation?

----
Delete conditions
----

--- Icarus2

8. Not this crap again !
Why must you highjack my thread with your unadulterated nonsense ? If you have a point to make then open your own thread !

9. Originally Posted by Markus Hanke
Not this crap again !
Why must you highjack my thread with your unadulterated nonsense ? If you have a point to make then open your own thread !
I'm very sorry!, No highjack!, it is just question.

10. Originally Posted by icarus2
I'm very sorry!, No highjack!, it is just question.
Ok, apology accepted. Just open a new thread for your topic.

11. Originally Posted by epidecus
EDIT: Sorry, MW. You're right; changed and saved.

But I do have a question for Markus. So basically GR can be mathematically derived in multiple ways from different models?
Yes, absolutely. I have compiled three ways that I know of here on this thread. I find it fascinating that even completely unrelated models like String Theory automatically yield GR when closely examined - that is the main point of this thread.
Even method 1 is interesting in itself, because it shows that GR is a direct mathematical consequence if you try to formulate Newton's gravity in a covariant manner.

12. Originally Posted by Markus Hanke
Originally Posted by epidecus
EDIT: Sorry, MW. You're right; changed and saved.

But I do have a question for Markus. So basically GR can be mathematically derived in multiple ways from different models?
Yes, absolutely. I have compiled three ways that I know of here on this thread. I find it fascinating that even completely unrelated models like String Theory automatically yield GR when closely examined - that is the main point of this thread.
Even method 1 is interesting in itself, because it shows that GR is a direct mathematical consequence if you try to formulate Newton's gravity in a covariant manner.
Really interesting. That last fact must be considered a heavy add to GR's credibility, is it not?

13. Tomorrow, I will remove posts not strictly related to the topic.
ModeratorMW

14. Originally Posted by epidecus
Really interesting. That last fact must be considered a heavy add to GR's credibility, is it not?
Yeah, one could interpret it that way

15. That's pretty good. I see you know your math. I lift my hat to you on that.

Keeping serious and on the topic of general relativity, I am very familiar with it. I'd like to do a little test of seeing how you people see the derivation of general relativity from the actual observable behavior of the natural universe.

Are there people here who believe in "objects of different mass falling/accelerating at the same rate" relative to an stationary observer sitting on Earth and observing the fall? The old school way of testing gravity

Here's a thought experiment, let's say that two gravitational fall occurrences are observed:
1. an unmanned drone airplane (mass 2000 kg) is observed to free fall to the ground from the altitude of 8.0 kilometers
2. a UFO (the same mass as the moon) uses "anti-gravity tech" to hover at the altitude of 8.0 kilometers and the tech malfunctions and the craft free falls to Earth

The question now is that do you believe that in relativistic terms an observer on Earth will observe these two objects fall at the same rate of acceleration? Assume proportional net effect of air resistance to be equal for both.

16. Originally Posted by grandi
The question now is that do you believe that in relativistic terms an observer on Earth will observe these two objects fall at the same rate of acceleration?
Could you provide a quick comment on how this is connected to the topic of this thread, i.e. how to derive GR ?
Perhaps it might be better to put this into a new thread.

17. Originally Posted by Markus Hanke
Originally Posted by grandi
The question now is that do you believe that in relativistic terms an observer on Earth will observe these two objects fall at the same rate of acceleration?
Could you provide a quick comment on how this is connected to the topic of this thread, i.e. how to derive GR ?
Perhaps it might be better to put this into a new thread.
There are two distinct modes of derivation of scientific models:
1. how they are derived from other models (this alone is not complete natural science, but only hypothetical maneuvering)
2. how they are derived from observations of the natural universe (this is real hard core natural science)

Do you not want to include the real observable world into your discussions?

18. Originally Posted by grandi

There are two distinct modes of derivation of scientific models:
1. how they are derived from other models (this alone is not complete natural science, but only hypothetical maneuvering)
2. how they are derived from observations of the natural universe (this is real hard core natural science)

Do you not want to include the real observable world into your discussions?
I still don't get this, since no one has ever actually observed an object with a mass comparable to that of the moon free fall towards earth ( or else we wouldn't be here now to have this discussion ). The immediate problem here is that the moon has a mass of approximately 1/81 of that of the earth, so the gravitational interaction between the two cannot easily be ignored. What I mean by that is that the object with the mass of the moon falls towards earth, but at the same time earth also falls towards that heavy object, making this whole scenario a little difficult.

Anyway, to answer the question we shall assume that the earth is stationary - in that case both the drone and the UFO fall towards earth at the exact same rate of acceleration ( all other things being equal of course ). The underlying reason for that, in terms of general relativity is that free-falling objects in gravitational fields follow geodesics through space-time, and the length of this geodesic is

As you can see this length depends on the geometry of the gravitational field, but not on the mass of the test particle. The geodesic has the same length no matter how massive the test particle, all other things being equal.

19. Originally Posted by Markus Hanke
Originally Posted by grandi

There are two distinct modes of derivation of scientific models:
1. how they are derived from other models (this alone is not complete natural science, but only hypothetical maneuvering)
2. how they are derived from observations of the natural universe (this is real hard core natural science)

Do you not want to include the real observable world into your discussions?
I still don't get this, since no one has ever actually observed an object with a mass comparable to that of the moon free fall towards earth ( or else we wouldn't be here now to have this discussion ). The immediate problem here is that the moon has a mass of approximately 1/81 of that of the earth, so the gravitational interaction between the two cannot easily be ignored. What I mean by that is that the object with the mass of the moon falls towards earth, but at the same time earth also falls towards that heavy object, making this whole scenario a little difficult.

Anyway, to answer the question we shall assume that the earth is stationary - in that case both the drone and the UFO fall towards earth at the exact same rate of acceleration ( all other things being equal of course ). The underlying reason for that, in terms of general relativity is that free-falling objects in gravitational fields follow geodesics through space-time, and the length of this geodesic is

As you can see this length depends on the geometry of the gravitational field, but not on the mass of the test particle. The geodesic has the same length no matter how massive the test particle, all other things being equal.
We haven't observed that, it is true. However, we can discuss what GR predicts we should observe.

Why would we assume that the Earth is stationary when we know that it isn't? That is inadequate for modeling the observational reality in relativistic terms. The actual relativistic answer to the question is trivial. Do you see the answer?

20. Why would we assume that the Earth is stationary when we know that it isn't?
I meant stationary with respect to the falling UFO. If we do away with this, we are faced with a relativistic Kepler problem, which is not at all trivial.

The actual relativistic answer to the question is trivial. Do you see the answer?
So long as we ensure that all things are equal apart from the mass of the falling objects then the answer is indeed trivial, and given by

which is invariant for all observers. Both objects will fall at the same rate of acceleration as measured by an observer on the surface.

21. I would assume that in each case, both the UFO and drone would fall at the same rate relative to their starting points, while the earth will accelerate away faster from it's starting position towards the UFO than to the drone. That would be the Newtonian answer, which would be closely mirrored by the relativistic answer I presume?

22. Originally Posted by KALSTER
I would assume that in each case, both the UFO and drone would fall at the same rate relative to their starting points, while the earth will move away faster from it's starting position towards the UFO than to the drone. That would be the Newtonian answer, which would be closely mirrored by the relativistic answer I presume?
The original question was as to the rate of acceleration from the perspective of an observer on the earth's surface. The assumption that the earth remains stationary and does not move towards the UFO is necessary to obtain a closed solution under General Relativity. If we abandon this assumption we are suddenly faced with the full relativistic two-body problem, which is highly non-trivial and has only approximate numerical solutions :

Two-body problem in general relativity - Wikipedia, the free encyclopedia

23. Interesting, but it should still closely mirror the Newtonian answer, no? At least in this scenario?

24. Originally Posted by KALSTER
Interesting, but it should still closely mirror the Newtonian answer, no? At least in this scenario?
Probably in this scenario it would be very close to the Newtonian answer, since we are talking weak fields here. For strong fields, e.g. two neutron stars orbiting each other, you would likely get very different results.

25. Originally Posted by KALSTER
I would assume that in each case, both the UFO and drone would fall at the same rate relative to their starting points, while the earth will accelerate away faster from it's starting position towards the UFO than to the drone. That would be the Newtonian answer, which would be closely mirrored by the relativistic answer I presume?
Yes. This is the correct answer derived from the Newtonian view point. What this shows us is that while from the initial spatial positions the two objects (the drone and the massive UFO) will be accelerating at the same rate towards Earth, Earth will also be accelerating towards these objects and faster in the gravitational field of the UFO. Demonstrating that mathematically the models dictate that the myth of observers seeing objects fall at the same rate on Earth is false. We just can not measure the difference of acceleration rates between small objects, it is so minuscule.

The relativistic notion is that the observer will *observe* the drone accelerate at approximately 9-10 m/s² (it is varying, compensated with the distance). The UFO will be *observed* to accelerate relative to the observer at approx. 11-12 m/s² (also increasing as the distance shortens).

This is how the mathematics of the gravity model dictates the observations.

26. Originally Posted by grandi

Yes. This is the correct answer derived from the Newtonian view point. What this shows us is that while from the initial spatial positions the two objects (the drone and the massive UFO) will be accelerating at the same rate towards Earth, Earth will also be accelerating towards these objects and faster in the gravitational field of the UFO. Demonstrating that mathematically the models dictate that the myth of observers seeing objects fall at the same rate on Earth is false. We just can not measure the difference of acceleration rates between small objects, it is so minuscule.
The relativistic notion is that the observer will *observe* the drone accelerate at approximately 9-10 m/s² (it is varying, compensated with the distance). The UFO will be *observed* to accelerate relative to the observer at approx. 11-12 m/s² (also increasing as the distance shortens).
This is how the mathematics of the gravity model dictates the observations.
So then I must repeat my question - what does this have to do with the topic of this thread, i.e. how to derive General Relativity ? I still don't get that.

27. Originally Posted by Markus Hanke
Originally Posted by grandi

Yes. This is the correct answer derived from the Newtonian view point. What this shows us is that while from the initial spatial positions the two objects (the drone and the massive UFO) will be accelerating at the same rate towards Earth, Earth will also be accelerating towards these objects and faster in the gravitational field of the UFO. Demonstrating that mathematically the models dictate that the myth of observers seeing objects fall at the same rate on Earth is false. We just can not measure the difference of acceleration rates between small objects, it is so minuscule.
The relativistic notion is that the observer will *observe* the drone accelerate at approximately 9-10 m/s² (it is varying, compensated with the distance). The UFO will be *observed* to accelerate relative to the observer at approx. 11-12 m/s² (also increasing as the distance shortens).
This is how the mathematics of the gravity model dictates the observations.
So then I must repeat my question - what does this have to do with the topic of this thread, i.e. how to derive General Relativity ? I still don't get that.
In terms of natural science this relates the derivation of the relativistic model from actual observations. I gave identical mass to the moon for the UFO because we can observe these accelerations from the Earth-moon system. The barycenter of that system is slightly below Earth's surface.

28. Originally Posted by grandi
In terms of natural science this relates the derivation of the relativistic model from actual observations.
And how, exactly, would you do that ? The mere fact that two bodies attract each other is already well known from Newtonian mechanics - I do not see the connection to GR. So far as I can tell the difference between GR and Newton in this particular scenario would be so small as to be unobservable. So how do you get GR from this, mathematically ?

29. Originally Posted by Markus Hanke
Originally Posted by grandi
In terms of natural science this relates the derivation of the relativistic model from actual observations.
And how, exactly, would you do that ? The mere fact that two bodies attract each other is already well known from Newtonian mechanics - I do not see the connection to GR. So far as I can tell the difference between GR and Newton in this particular scenario would be so small as to be unobservable. So how do you get GR from this, mathematically ?
You are correct in that the difference between the two models (GR and Newton) in this scenario is unobservable and the distinction of between the two models can not be mathematically derived from the observational data of this scenario. But that was not the point, the point was simply to make the notion of how GR as a whole can also be understood to fit the observational data.

30. Originally Posted by grandi
how GR as a whole can also be understood to fit the observational data.
All tests of GR have shown a very close fit between theory and observation.

31. Originally Posted by grandi
But that was not the point, the point was simply to make the notion of how GR as a whole can also be understood to fit the observational data.
Well, yes. All theories must fit observational data, or else they are obviously useless. GR owes its success to the fact that it fits observational data to a very high degree - refer also here :

Modern Tests of Relativity

32. We have the precession of Mercury for that, a perfect example of observed facts validating GR, as well as the wealth of other experimental verifications given as a sticky in this section.

By the way, the myth of falling bodies is only a myth among those who do not understand the physics involved.

I will separate all of this out a bit later.

33. By the way, the myth of falling bodies is only a myth among those who do not understand the physics involved.
That's true, but it is actually a really good approximation so long as the mass of one of the bodies is very much smaller than the mass of the other, like in the drone and the earth.

34. So basically if I've undershood this correctly GR would be either the end (classical limit) or the beginning, or at least Maxwell's equations would, for trying to understand quantum electrodynamics.

35. Originally Posted by Chrisgorlitz
So basically if I've undershood this correctly GR would be either the end (classical limit) or the beginning, or at least Maxwell's equations would, for trying to understand quantum electrodynamics.
I think GR is a low-energy approximation for a more comprehensive theory of quantum gravity, whatever this will turn out to be.

36. Originally Posted by Markus Hanke
Method 1 : From Newton's Gravity

Start with Newtonian gravity field, the validity of which is locally verified for weak fields :

which is, expressed in terms of the energy-momentum tensor

Now attempt a first ansatz to formulate this in a Lorentz-invariant manner :

which further leads to a covariant formulation of the form

Our task will now be to determine the unknown tensor G. We impose the following conditions on that tensor :

1. G is a Riemann tensor
2. G is composed of the first and second derivatives of the metric tensor
3. The energy-momentum tensor obeys the usual symmetry and conservation laws and ; these properties then by default must also apply to our tensor G
4. The theory must reduce to Newton's gravity for weak fields

Using the above four points, the Bianchi identities, as well as the general ansatz

plus a little tensor algebra, one find that the easiest tensor which satisfies all of the above conditions is

Not correct. See Hynecek, Physics Essays 22, 4, (2009).
I would upload this file if I knew how.

37. Originally Posted by Markus Hanke
Method 1 : From Newton's Gravity

Start with Newtonian gravity field, the validity of which is locally verified for weak fields :

which is, expressed in terms of the energy-momentum tensor

Now attempt a first ansatz to formulate this in a Lorentz-invariant manner :

which further leads to a covariant formulation of the form

Our task will now be to determine the unknown tensor G. We impose the following conditions on that tensor :

1. G is a Riemann tensor
2. G is composed of the first and second derivatives of the metric tensor
3. The energy-momentum tensor obeys the usual symmetry and conservation laws and ; these properties then by default must also apply to our tensor G
4. The theory must reduce to Newton's gravity for weak fields

Using the above four points, the Bianchi identities, as well as the general ansatz

plus a little tensor algebra, one find that the easiest tensor which satisfies all of the above conditions is

I would upload the file, but it is a bit too large.
Physics Essays,22,4,2009

38. Originally Posted by Sanford
Not correct. See Hynecek, Physics Essays 22, 4, (2009).
I would upload this file if I knew how.
Perhaps you should explain, rather than simply say "not correct". Do you understand the contents of the paper and the theory it argues against?

Anyway, I'm not sure why you don't know how to go to the website of said journal and find the article:

http://physicsessays.org/doi/pdf/10.4006/1.3239584

Incidentally, I see that Stephen J Crothers is also a prolific contributor to that journal. I can't see many respected contributors there, and above paper has citations from nobody except the author.

Just adding to the fluff.

EDIT: Oh, the article is behind a pay-wall. Well in that case we cannot read it, and nor should you upload it.

You'll have to explain it then. I would recommend you start your own thread to do that, rather than post it here.

39. I believe Mr Crothers is als an editor of the "journal" I've gone a few rounds with him before.

40. Originally Posted by SpeedFreek
Originally Posted by Sanford
Not correct. See Hynecek, Physics Essays 22, 4, (2009).
I would upload this file if I knew how.
Perhaps you should explain, rather than simply say "not correct". Do you understand the contents of the paper and the theory it argues against?

Anyway, I'm not sure why you don't know how to go to the website of said journal and find the article:

An Error Occurred Setting Your User Cookie

Incidentally, I see that Stephen J Crothers is also a prolific contributor to that journal. I can't see many respected contributors there, and above paper has citations from nobody except the author.

Just adding to the fluff.

EDIT: Oh, the article is behind a pay-wall. Well in that case we cannot read it, and nor should you upload it.

You'll have to explain it then. I would recommend you start your own thread to do that, rather than post it here.
Can't explain it. Too complicated. Write the author.

41. Originally Posted by Sanford
Write the author.
Or not.

42. Originally Posted by Sanford
I would upload the file, but it is a bit too large.
Physics Essays,22,4,2009
It is very much correct, unless you can show us exactly where the error lies. Are you a troll of some kind, or what is your agenda ? This is the standard derivation for the GR field equations.
The exact same derivation can be found in the following university textbook on General Relativity, which is my primary source :

Fliessbach, Prof Torsten: Allgemeine Relativitätstheorie, Mannheim : Wien : Zürich : BI-Wiss.-Verl. 1990, ISBN 3-411-14331-2

Torsten Fliessbach is a professor of theoretical physics at the university of Siegen in Germany, and has published several well known textbooks in the areas of quantum mechanics, QFT and General Relativity. You will find all his credentials and publications here :

Homepage von Torsten Fließbach

Can't explain it. Too complicated. Write the author.
In other words - you really don't know what you are talking about.

43. Originally Posted by Sanford
Can't explain it. Too complicated. Write the author.
Is this the most feeble excuse since Fermat?

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