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Einstein Field Equations


Farsight
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I was kicking something around with a few guys a couple of weeks back, and something came up. I wonder if anybody here could confirm or or counter what I was told?

 

In The Foundation of the General Theory of Relativity on page 185 of Doc 30, 3.6 Mbytes, Einstein says "the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy".

 

The people I was talking to said that this energy is not included in the Einstein Field Equations. They seemed to know what they were talking about, but I thought surely not?

 

I'd appreciate any information you can offer.

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The energy-momentum tensor on the RHS of the field equations is that of the matter and non-gravitational fields on the space-time. It does not contain any contribution from gravity directly. (The energy-momentum is a function of the metric though).

 

This does at first seem strange, but physically this is due to the equivalence principle.

 

Mathematically is it due to the fact that there is no unique well-defined notion of energy tensor in general relativity. People have defined various pseudo tensors that can be useful.

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Thanks ajb, but I confess I'm still not clear on this. For reference:

 

[math]G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}[/math]

 

Can I can try and ask the question another way? Imagine you measure gravitational time dilation at locations in the space around a planet, and at locations within the planet. All these locations lie on a circular horizontal plane. When you then plot the time dilation, you find you're effectively plotting a "Newtonian" gravitational potential:

 

300px-GravityPotential.jpg

http://en.wikipedia.org/wiki/Gravitational_potential

 

If you now turn this upside down, surely what you have is a plot of spatial energy density, what Einstein called "the energy of the gravitational field". His following words "shall act gravitatively in the same way as any other kind of energy" are a reminder that a non-uniform energy distribution causes gravity, and that matter only causes gravity because of the "energy content". But this spatial energy isn't contained within in the matter. If we could somehow lose the matter whilst somehow retaining this non-uniform spatial energy distribution, the latter would cause a gravitational field in its own right.

The guys I was talking to said this spatial energy isn't included in the EFEs, and I note on the last page of document 30, Einstein talks about a first approximation, but the document is truncated at that point, and I'm a bit stuck. Any further assistance you can offer would be appreciated.

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How is gravitational energy not involved in Einstein Field Equation?!

It is totally built upon the effect of Mass on the spacetime curviture, which in turns define gravity. Gravitational energy is implied in the Einstein Tensor in the Equation.

But, don't forget. There is another part that is remaining which is "An object in a gravitational field follows the shortest distance between two points". It is completing the description of the gravitaional effect on masses.

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How is gravitational energy not involved in Einstein Field Equation?!

It is totally built upon the effect of Mass on the spacetime curviture, which in turns define gravity. Gravitational energy is implied in the Einstein Tensor in the Equation

 

I am not sure you can really interpret it that way.

 

The Einstein tensor is a purely geometric thing, i.e. it is built directly from the metric. The gravitational constant you can interpret as an energy-density of space-time.

 

Defining energy in general relativity is involved and generally there is no real good single notion of energy of a space-time.

 

The important thing is that in the field equations the term [math]T_{\mu \nu}[/math] has no direct contribution from gravity. This is the point that Farsight's friends were making.

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I don't know of any sensible decomposition of the field equations that allows us to really think like that.

 

It is a fact that the gravitational energy-density is non-localisable. Weinberg's pesudotensor is probably the best from our point of view in that it is a source of gravity.

Edited by ajb
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But, as you said, Einstein Tensor depends upon the Metric Tensor, therefore the metric (which is sufficient to describe the spacetime curvature) is defined in The Einstein Field Equation. Then, the basic background of gravity is defined in " The Einstein Field Equation".

 

It is, then, what is remaining, is the way that spacetime curvature is acting upon the mass situated at certain position.

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But is this the "gravitational energy"?

 

The self-interactions of gravity are included in the field equations viz the non-linearity. That is true, but I am not aware of any general reformulation that states this as the energy of the gravitational field.

 

Most of the work I am aware of, but I am not very familiar is devoted to the issue of defining an energy-momentum for the graviton field. My understanding is that most people think that gravitational waves do carry energy, but this has not been completely formulated.

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Grab a modern reference is my advice. I'd suggest Carroll's lecture notes. They are a great place to start to learn general relativity.
I'm not a novice, but thanks anyway. On page 111 he says this is an equation relating derivatives of the metric to the energy density so maybe there'll be something in there that will help.


Merged post follows:

Consecutive posts merged

Amr: thanks for chipping in too. This seems to be a trickier problem than I thought.

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I'm not a novice, but thanks anyway. On page 111 he says this is an equation relating derivatives of the metric to the energy density so maybe there'll be something in there that will help.

 

All Carroll has said here is that in the weak field limit we have

 

[math]R_{00} \propto T_{00}[/math].

 

The rhs is the energy density of some "stuff". It is not the energy density of the gravitational field.

 

You can then use this to get back to the Poisson equation for Newtonian gravity.

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You are welcome, Farsight. It is a good question.

 

I think that we must differentiate between Field Energy (Energy per unit volume) and Potential Energy (Energy per unit mass). The later is not included in the Einstein Field Equation, I admit that. But, the former can be derived from the spacetime curvature (the metric tensor).

 

We can look to electromagnetism to have an idea. Maxwell's Equations, which describes the EM field, can define EM Energy density totally away from Lorentz Force Equation. However, the Potential Energy is not included. I think the situation is similar here.

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The rhs is the energy density of some "stuff". It is not the energy density of the gravitational field.
I should read it all I suppose. Apologies for not doing so. But did you mean energy density of the gravitational field? Surely a gravitational field is only there because the energy density is not uniform. And your response perhaps hints at what the underlying issue is, and it's a big one, to do with cause and effect: what stuff?

 

You're doing geometry, so you might have a head start on why this is so important. Think back to that Einstein quote: "the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy". Forget about the planet and look just at the surrounding space. There's no "stuff" there. Just curved spacetime, which one can think of as a geometrical distortion. And there's no "stuff" in a gravitational wave either. Take a look at LIGO re length change in the interferometer arms, and imagine a gravity wave passing through a cubic lattice that represents flat space. Take a snapshot, and the presence of the gravity wave is betrayed by those lattice lines that are now curved instead of straight. The curvature is the presence of energy-momentum.

 

I think that we must differentiate between Field Energy (Energy per unit volume) and Potential Energy (Energy per unit mass). The later is not included in the Einstein Field Equation, I admit that. But, the former can be derived from the spacetime curvature (the metric tensor).
Yes, no problem re potential energy. This whole issue revolves around the fact that mass per se isn't the cause of a gravitational field, energy is. Or more properly, a non-uniform energy density. And that's what a gravitational field is. I read this 1920 Einstein quote:

 

"According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)..."

 

...and I can't reconcile it with the FLRW metric which says The FLRW metric starts with the assumption of homogeneity and isotropy of space. It also says where Σ ranges over a 3-dimensional space of uniform curvature. If there's curvature, the space can't be homogeneous.

 

We can look to electromagnetism to have an idea. Maxwell's Equations, which describes the EM field, can define EM Energy density totally away from Lorentz Force Equation. However, the Potential Energy is not included. I think the situation is similar here.
Me too. What I was saying to ajb about a gravitational wave maybe applies to a photon too.

 

Any chance either of you guys could ask around for an answer to this one?

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I am not sure you can really interpret it that way.

 

 

The important thing is that in the field equations the term [math]T_{\mu \nu}[/math] has no direct contribution from gravity. This is the point that Farsight's friends were making.

 

Wrong. True only in a mass free zone. Otherwise, mass does contribute to T.

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... what stuff?

 

Any matter or fields that are not the gravitational field (the metric or other related formulations).

 

Think back to that Einstein quote: "the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy".

 

I must say I do not understand Einstein's quote.

 

The best I can say is that Einstein is saying that gravity includes self-interactions. This makes us think that gravitational radiation should have a well defined energy-momentum tensor. Even without gravity there is a lot of freedom in defining such tensors, but I beleive there is no formuation of a true tensor for gravity.

 

 

The curvature is the presence of energy-momentum.

 

You can have non-trivial geometry in vacua. Only the scalar curvature is zero.

 

One could wish to think of the "energy of gravity" as the origin of the curvature, but I have no idea how you would do this.

 

Yes, no problem re potential energy. This whole issue revolves around the fact that mass per se isn't the cause of a gravitational field, energy is. Or more properly, a non-uniform energy density.

 

More primative is the energy-momentum. This is the sourse of gravity.

 

And that's what a gravitational field is. I read this 1920 Einstein quote:

 

"According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)..."

 

...and I can't reconcile it with the FLRW metric which says The FLRW metric starts with the assumption of homogeneity and isotropy of space. It also says where Σ ranges over a 3-dimensional space of uniform curvature. If there's curvature, the space can't be homogeneous.

 

Well, what Einstein is saying is that in general the space-time near a massive object is effected by the presence of the object. That is the vacuum arround an object is geometrically non-trivial.

 

The FLRW metric is based on the assumption that on large enough scales the universe looks homogeneous and isotropic. The presence of curvature does not nessisarily mean non-homogeneous.

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I finished it last year, yourdadonapogostick. This is just another piece of verification.

 

Are we ever going to see the final version? Do you still think it is correct, or have you moved on to something else?

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Any matter or fields that are not the gravitational field (the metric or other related formulations).
Sorry not to have responded sooner. This gets quite interesting. An electron's electric field comprises energy, so there's a mass equivalence, so there's "stuff" in the surrounding space. When you look at the Aharanov-Bohm experiment, there's no detectable electromagnetic field outside the solenoid, but there's electromagnetic potential there such that the interference pattern is deflected when you turn the solenoid on. So there's something there. But it's just empty space. Then when you look at real "empty space", it has vacuum energy. Take a block of space and call it a system, and the energy content implies a mass equivalence, so we're back to "stuff" again, even though there's no "stuff" there.

 

I must say I do not understand Einstein's quote.
I think I do. If you take a region of empty space, there's a uniform energy density. If you stick a planet in the middle then mask it off and just look at the space, the energy density isn't uniform any more. The energy density is higher near the planet, and reduces with distance. You've added energy to the space, and this is the energy of the gravitational field. A non-uniform energy density causes gravity, so this causes gravity too.

 

The best I can say is that Einstein is saying that gravity includes self-interactions. This makes us think that gravitational radiation should have a well defined energy-momentum tensor.
Sounds good to me. After all, electromagnetic radiation involves a non-uniform energy density moving from a to b.

 

Even without gravity there is a lot of freedom in defining such tensors, but I beleive there is no formulation of a true tensor for gravity.
Noted.

 

You can have non-trivial geometry in vacua. Only the scalar curvature is zero.
I was rather suggesting it isn't.

 

One could wish to think of the "energy of gravity" as the origin of the curvature, but I have no idea how you would do this.

 

More primitive is the energy-momentum. This is the sourse of gravity.

Again sounds good. For an analogy, imagine a cube of jelly marked out with lattice lines. Insert a hypodermic needle into the middle, and inject more jelly. The lattice lines around this area are now curved. However this represents curved space, not curved spacetime, and this is a snapshot of a photon.

 

Well, what Einstein is saying is that in general the space-time near a massive object is effected by the presence of the object. That is the vacuum arround an object is geometrically non-trivial.
No problem with that.

 

The FLRW metric is based on the assumption that on large enough scales the universe looks homogeneous and isotropic. The presence of curvature does not nessisarily mean non-homogeneous.
This I can't see. Einstein talks about curvilinear motion being the result of inhomogeneous space. That curvilinear motion marks out curved spacetime, so I just can't see how we can have curvature and homogeneity.


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Consecutive posts merged
Are we ever going to see the final version? Do you still think it is correct, or have you moved on to something else?
It's generally not done to plug your own work, and people can be hostile on forums like this, but yes, there is a final version. I'll PM you. And yes, I do think it's correct. There's a couple of little things I could have said better, like I talk about gravity as a "negative tension gradient" rather than a "pressure gradient" or a "stress-energy density gradient", and there's a typo that messes up the distinction between a tidal gradient and the gμv gradient. But it's been a year now, and it still holds up. Indeed, like I said, it seems to be getting verified, and I like to think that more professionals are thinking along these lines. Having said that, I imagine there will turn out to be something wrong in there, because nothing is ever perfect. And it is only an outline sketch, so it definitely isn't perfect.
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Enough with this theoretical nonsense, like medieval scholars debating the number of angels on a pin. Please clarify the basic assumptions of GR, and how we can devise experiments to verify this. I do not mean experiments that verify various conclusions, such as the orbit or Mercury, but experiments that verify the two initial assumptions as Weinberg presented it in his 1972 book.

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