Sources of Gravity: Exploring the Dependence of g_{\mu\nu}

[Rasalhague]

Is the following correct? In Newton's theory, gravity is a force (described in terms of a scalar potential field and its gradient), and the source of gravity is mass. In Einstein's theory, gravity is curvature of spacetime (described by the Riemann curvature tensor), $R_{\alpha\beta\gamma\delta}$, and the sources of gravity are (1) the stress-energy tensor, $T_{\mu\nu}$ (comprising energy density, momentum density and stress), representing the contribution to gravity of every physical thing except for gravity itself, and (2) something else, representing gravity's effect on itself. The stress-energy tensor is defined at each event except in a vacuum. The something else is in some sense nonlocal. (What sense?) Its influence is present at all events, even those in a vacuum.

In The Road to Reality Penrose appears to call the something else "gravitational field energy". I've also seen the term gravitational binding energy used; is that a synonym in the context of GR, or is it a purely Newtonian concept? In Essential Relativity (2nd ed., 1977), in the section on the Schwarzschild solution, Rindler makes use of a quantity he calls mass (which, setting c = 1, corresponds to what Taylor & Wheeler, in Spacetime Physics call energy). Is the mass that appears in Rindler's "Schwarzschild metric" (§ 8.3, p. 138) the same thing as Penrose's gravitational field energy (still taking c = 1), since it determines curvature in a vacuum. If so, is this the only other source of gravity in GR besides the stress-energy tensor (and thus the only source of gravity in a vacuum)? Is there no gravitational field momentum or force (nonlocal analogues to the momentum density and stress of the stress-energy tensor)?

Looking at what can be deduced from what...

g --> connection --> Riemann --> Ricci
g --> connection --> Riemann --> Weyl
g --> connection --> geodesic equation
g & T --> Ricci
g & Ricci --> T

...it seems that the metric tensor field, $g_{\mu\nu}$, can get us just about anywhere, but how does $g_{\mu\nu}$ depend on the sources of gravity?

 

[Dale]

(2) something else, representing gravity's effect on itself.

What is this (2)? Are you talking about the term with the cosmological constant? $$\Lambda g_{\mu \nu}$$

 

[Rasalhague]

I'd actually forgotten about the cosmological constant (dark energy). As it's a bit mysterious, maybe I should make that a separate category of source and call it (3).

By (2), I mean whatever the source of gravity in a vacuum is (apart from dark energy). Penrose talks about something called gravitational field energy. In the context of the Schwarzschild solution, Rindler talks about something he calls mass, which for him means relativistic mass, and which corresponds to what Taylor & Wheeler call energy. I'm wondering if Rindler's mass and Penrose's gravitational field energy are different names for the same thing.

Could we say there are three sources of gravity: (1) the stress-energy tensor field (nonzero only in continuous distributions of matter and nongravitational energy), (2) gravitational energy, (3) dark energy--the first source being defined locally, at each event, and the other two sharing some quality that Penrose calls a "nonlocal character"?

 

[Dale]

I'm sorry, but I am unclear about what terms you are talking about. The EFE is:
$$G_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu}$$
which can also be written
$$R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu}$$

There certainly would be nothing wrong with moving some of the terms on the left hand side over to the right hand side and calling them sources. One thing that is occasionally done is to express the R term as a T term instead.
$$R_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa (T_{\mu \nu} - \frac{1}{2} T g_{\mu \nu})$$

So, which of these terms are you interested in?

 

[Rasalhague]

Well, which terms express the contribution to curvature of whatever sources (apart from dark energy) aren't described by the stress-energy tensor? And what is the nature of those sources: are they limited to the energy of the gravitational field, or do they include other properties of the gravitational field (perhaps analogous in some way to the other components of the stress-energy tensor besides energy density)?

If we leave out the cosmological constant, for the moment, and set the stress-energy tensor to zero, as in a vacuum, the EFE tells us that Ricci = 0. What information do we need, in that case, to determine curvature? I think we can get Riemann from g, but how does the the value of g at a point depend on matter elsewhere (i.e. how is it calculated), and which properties of that distant matter are relevant?

 

[Dale]

Well, which terms express the contribution to curvature of whatever sources (apart from dark energy) aren't described by the stress-energy tensor? And what is the nature of those sources: are they limited to the energy of the gravitational field, or do they include other properties of the gravitational field (perhaps analogous in some way to the other components of the stress-energy tensor besides energy density)?

Well, if you expand out the curvature tensors you will get the metric and derivatives of the metric which are proportional to the stress energy tensor. There is nothing else in the EFE.

If we leave out the cosmological constant, for the moment, and set the stress-energy tensor to zero, as in a vacuum, the EFE tells us that Ricci = 0. What information do we need, in that case, to determine curvature? I think we can get Riemann from g, but how does the the value of g at a point depend on matter elsewhere (i.e. how is it calculated), and which properties of that distant matter are relevant?

Only the stress-energy tensor of matter fields is important.

 

[Rasalhague]

Well, if you expand out the curvature tensors you will get the metric and derivatives of the metric which are proportional to the stress energy tensor. There is nothing else in the EFE.

By "the curvature tensors", do you mean only those that appear in the EFE: Ricci and the scalar curvature? Or do you mean we can derive g from the Riemann curvature (which doesn't appear in the EFE)?

Is the EFE sufficient to determine everything about curvature from any relavant properties of matter (taken in the broad sense of whatever the sources of curvature are)? (There's nothing else, and we don't need anything else.) Or does it only tell half the story of GR?

Only the stress-energy tensor of matter fields is important.

There are multiple vacuum solutions to the EFE. What makes these solutions different when the stress-energy tensor field is the same (having the value zero everywhere) for all of them. It seems like there's something other than the stress-energy tensor which determines the important difference between Minkowski and Schwarzschild spacetimes. What is it, and how is its effect (on g and hence on the Riemann curvature) calculated?

 

[Russell E]

Which terms express the contribution to curvature of whatever sources (apart from dark energy) aren't described by the stress-energy tensor?

The gravitation of gravity itself, in general relativity, is not represented by specific terms in the field equations, it is a consequence of the non-linearity of those equations. It is possible to write down a pseudo-tensor that can be said to represent the gravitational stress-energy, but it is not a true tensor, i.e., it's not covariant under coordinate transformations, and in fact we can make it vanish at any given point (in vacuum) by a suitable choice of coordinates. Nevertheless, it gives a reasonable expression for the stress-energy of gravity, and it has measurable consequences, e.g., the gravitational effects of astronomical bodies include the contribution of the stress-energy of the gravitational fields themselves (as confirmed by the absence of any "Nordtvedt effect").

If we leave out the cosmological constant, for the moment, and set the stress-energy tensor to zero, as in a vacuum, the EFE tells us that Ricci = 0. What information do we need, in that case, to determine curvature?

The field equations are differential equations, so you need a complete set of initial/boundary conditions to determine the field at every point, as in a Cauchy initial-value problem.

It seems like there's something other than the stress-energy tensor which determines the important difference between Minkowski and Schwarzschild spacetimes. What is it, and how is its effect (on g and hence on the Riemann curvature) calculated?

Did you mean DeSitter instead of Schwarzschild? Your question would make sense for De-Sitter spacetime, because it is a vacuum solution as is Minkowski spacetime. The difference between them is the boundary conditions at infinity. If you meant Schwarzschild, then your question doesn't make sense, because the Schwarzschild spacetime is based on the presence of a quantity of ordinary mass-energy which represents one of the boundary conditions that are imposed on the field equations, distinguishing it from Minkowski spacetime. If you're just trying to understand how a differential equation can give unique values for the solution at points other than at the boundary conditions, that's a question of understanding the basics of algebra and differential equations, not uniquely related to understanding general relativity.

 

[Dale]

By "the curvature tensors", do you mean only those that appear in the EFE: Ricci and the scalar curvature? Or do you mean we can derive g from the Riemann curvature (which doesn't appear in the EFE)?

Sorry, I meant the other way around. We can derive the Riemann, Ricci, and scalar curvature from the metric. The curvature tensors are nothing more than combinations of derivatives of the metric.

Is the EFE sufficient to determine everything about curvature from any relavant properties of matter (taken in the broad sense of whatever the sources of curvature are)? (There's nothing else, and we don't need anything else.) Or does it only tell half the story of GR?

The EFE is sufficient, plus, of course, the boundary conditions.

There are multiple vacuum solutions to the EFE. What makes these solutions different when the stress-energy tensor field is the same (having the value zero everywhere) for all of them. It seems like there's something other than the stress-energy tensor which determines the important difference between Minkowski and Schwarzschild spacetimes. What is it, and how is its effect (on g and hence on the Riemann curvature) calculated?

The boundary conditions. You can say the same thing about pretty much any physics equation. E.g. there are multiple vacuum solutions to Maxwells equations, what makes these solutions different are the boundary conditions.

 

[Rasalhague]

Sorry, I meant the other way around. We can derive the Riemann, Ricci, and scalar curvature from the metric. The curvature tensors are nothing more than combinations of derivatives of the metric. [...]

Okay, good. I've seen Riemann defined in terms of g, and how Ricci, Weyl and scalar curvature come from Riemann.

The gravitation of gravity itself, in general relativity, is not represented by specific terms in the field equations, it is a consequence of the non-linearity of those equations. It is possible to write down a pseudo-tensor that can be said to represent the gravitational stress-energy, [...]

So g can be found from the EFE if we know three things: the stress-energy tensor, the cosmological constant, and boundary conditions. The "gravitation of gravity itself"--curvature due to (other components of?) curvature--is among the boundary conditions. Are there other possible boundary conditions?

I think you're saying gravitational stress-energy is the source of the gravitation of gravity: that part of curvature at a given event which isn't due to the value of the stress-energy tensor at that event, or to the cosmological constant. (Source in the sense that mass is the source of Newtonian gravity.) Is GSE what Penrose calls gravitational field energy, and what Rindler, in the case of the Schwarzschild solution, calls mass? Is the contribution of GSE to curvature at one event in a vacuum (partly or wholly) determined by the values of the stress-energy tensor at other, distant events, or is it entirely independent of the stress-energy tensor field?

When the word energy refers to the 00 component of the stress-energy tensor, it really means energy density, but when Penrose says gravitational field energy is, in some sense, nonlocal, does he mean that gravitational field energy is literally energy (a property of a whole system) rather than energy density; or is GSE actually a quantity that needs to be represented as a nontensorial matrix exactly analogous to the stress-energy tensor, with components of energy density (perhaps Penrose's gravitational field energy), momentum density and stress?

Did you mean DeSitter instead of Schwarzschild?

No, I did mean Schwarzschild. If I seem to be talking nonsense here, that's probably because I am ;-)

I don't have a very sophisticated understanding of differential equations, but I can appreciate that different boundary conditions may result in a different function (i.e. one having different values anywhere, not just at the boundary). I'm just trying to get an overview of what the geometry of spacetime depends on, what we need to know, to calculate a metric tensor field, and hence curvature tensors: what sort of boundary conditions might be involved here.

 

[Russell E]

So g can be found from the EFE if we know three things: the stress-energy tensor, the cosmological constant, and boundary conditions.

Well, the cosmological constant is just part of your postulated field equations, and the stress-energy tensor is part of the boundary conditions (with the understanding that "boundary conditions" include the initial conditions). But it's not so simple. For example, prior to knowing the metric of spacetime, how do you propose to specify the spacetime distribution of stress-energy? Also, there's a related subtlety, because we aren't free to arbitrarily select boundary/initial conditions (as we are with, for example, Maxwell's equations, using electrically neutral manipulation), because the field equations of general relativity are over-specified in the sense that the boundary conditions are constrained by the requirement to satisfy the Bianchi identities. On the other hand, once we have specified initial/boundary conditions that satisfy these requirements, the field equations are under-specified, in the sense that there are four arbitrary degrees of freedom in the solution. But these degrees of freedom correspond to our free choice of coordinates.

The "gravitation of gravity itself"--curvature due to (other components of?) curvature--is among the boundary conditions.

No, not at all. The gravitation of gravity is essentially a consequence of the non-linearity of the field equations themselves, not a boundary condition imposed on the field equations. The field equations impose this non-linearity on the solutions.

Are there other possible boundary conditions?

Other? Boundary/initial conditions can be specified and imposed in many different ways, such as by requiring spherical symmetry (as we do in deriving the Schwarzschild metric), or specifying Minkowski metric "at infinity" relative to some frame of reference, or by imposing "conformal boundary conditions", etc. And of course the most common way of specifying boundary conditions for practical problems is to specify where all the mass is and where it's going (initially).

I think you're saying gravitational stress-energy is the source of the gravitation of gravity: that part of curvature at a given event which isn't due to the value of the stress-energy tensor at that event, or to the cosmological constant.

I think you're operating under a basic misconception. The curvature is NEVER just due to the value of the stress-energy tensor *at that event*. Remember that the Einstein tensor "G" is a function of the metric *and its derivatives*, so by equating it at each event to the stress energy tensor T at that event, we are NOT saying the metric g at that event is due to T at that event. The entire solution, everywhere, is constrained by the boundary and initial conditions.

Also, don't confuse this issue with the issue of gravitational energy and the non-linearity of the field equations. You're really mixing up two very different issues. One is how differential equations subject to boundary/initial conditions work. The other involves gravitational energy and how gravity gravitates according to the non-linear field equations of general relativity. The first question is trivial freshman calculus. The second question is very deep and difficult. I suggest you focus on understanding the first question first, because without that, I don't see how you can even begin to understand the second.

Is GSE what Penrose calls gravitational field energy, and what Rindler, in the case of the Schwarzschild solution, calls mass?

Yes and no.

Is the contribution of GSE to curvature at one event in a vacuum (partly or wholly) determined by the values of the stress-energy tensor at other, distant events, or is it entirely independent of the stress-energy tensor field?

Again, this is hopelessly confused. You need to first have at least some concept of how a differential equation works. Specifically, you need to understand how boundary/initial conditions determine a solution everywhere. This seems to be your main interest, and it really has nothing to do with the issue of gravitational energy. (The same issue arises for Maxwell's equations, and yet the photon has no electric charge.) Also, your attempt to establish "cause and effect" for the non-linear aspects of solutions of the field equations is misguided... but I don't think you can begin to understand this without knowing how a differential equation works (at least qualitatively).

When the word energy refers to the 00 component of the stress-energy tensor, it really means energy density, but when Penrose says gravitational field energy is, in some sense, nonlocal, does he mean that gravitational field energy is literally energy (a property of a whole system) rather than energy density; or is GSE actually a quantity that needs to be represented as a nontensorial matrix exactly analogous to the stress-energy tensor, with components of energy density (perhaps Penrose's gravitational field energy), momentum density and stress?

You're waaaay off base with all this.

No, I did mean Schwarzschild. If I seem to be talking nonsense here, that's probably because I am ;-)

Yes, it was nonsense, because you said Schwarzschild was a vacuum solution, whereas of course it isn't, since it has a central mass. Your confusion - again - is due to the fact that you imagine the solution of the field equations at each event is independent of the conditions at other events. This is not true at all. The equation G=T is not an algebra equation, it is a differential equation. This issue has nothing to do with subtle issues of gravitational energy and non-linearity of the Einstein field equations. The same is applies to linear equations like Maxwell's. You just need to understand how differential equations work, and how they are constrained by boundary/initial conditions.

 

[Rasalhague]

Also, don't confuse this issue with the issue of gravitational energy and the non-linearity of the field equations. You're really mixing up two very different issues. One is how differential equations subject to boundary/initial conditions work. The other involves gravitational energy and how gravity gravitates according to the non-linear field equations of general relativity. The first question is trivial freshman calculus. The second question is very deep and difficult. I suggest you focus on understanding the first question first, because without that, I don't see how you can even begin to understand the second.

Okay, thanks for your advice.

Yes and no.

Yes to the former, and no to the latter; or "yes and no" in the sense of partly (or conditionally) yes to both, perhaps because the terms overlap to some extent but not perfectly?

Yes, it was nonsense, because you said Schwarzschild was a vacuum solution, whereas of course it isn't, since it has a central mass.

Perhaps, as often, there are different dialects:

The fact that the Schwarzschild metric is not just a good solution, but is the unique spherically symmetric vacuum solution, [...]
http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html

According to Birkhoff's theorem, the Schwarzschild solution is the most general spherically symmetric, vacuum solution of the Einstein field equations.
http://en.wikipedia.org/wiki/Schwarzschild_solution

The solution is assumed to be spherically symmetric, static and vacuum.
http://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

 

[Russell E]

Yes to the former, and no to the latter; or "yes and no" in the sense of partly (or conditionally) yes to both, perhaps because the terms overlap to some extent but not perfectly?

Yes to the former and no to the latter. (Penrose and Rindler are not talking about the same things.)

Perhaps, as often, there are different dialects:

It isn't a matter of dialects, it's just you not understanding how differential equations work and what they mean. The Schwarzschild solution is derived by determining the solution of the vacuum field equations under the condition of spherical symmetry, and the result is singular at the origin and has one free parameter. This is taken as the global solution with the boundary condition consisting of a single mass at the origin, and the free parameter represents the mass-energy. For any non-zero value of the mass parameter, this is obviously not a global vacuum solution, it is the solution corresponding to a spherically symmetrical mass.

Of course, it's a vacuum solution *locally* at every point outside the source mass, but the point is, it has a source mass, whereas Minkowski spacetime does not. That's why it made no sense, in the context of your question, for you to ask why these two solutions were different. You thought they should be the same because "they are both vacuum solutions", but they aren't. One has a source mass, the other doesn't. That's exactly why I've suggested that you try to learn the basics of the subject you're asking about. All your questions seem to be motivated by not understanding how mass over there affects the locally vacuum solution over here. Again, this has nothing to do with gravitational energy and non-linearity of the field equations.

 

[Rasalhague]

Thanks for explaining, and taking the trouble to unpick my muddled questions. I certainly intend to follow your advice and read up on differential equations.

I gather there are several quantities called mass in GR, none of which correspond to rest mass in SR. When you say "source mass" and "a single mass at the origin", is this mass synonymous (setting c = 1) with your mass-energy, and with what Rindler calls mass (in Essential Relativity: "the metric around a spherically symmetric mass"), and with what Taylor & Wheeler call energy (in Spacetime Physics, the time component of the energy-momentum vector in SR)?

By dialects, I just meant that some writers apply the name "vacuum solution" to the Schwarzschild solution (because it defines a metric tensor field in a vacuum), whereas you don't (because the region where the metric tensor field is defined has a boundary which is not a vacuum).

 

[TrickyDicky]

Rassalhague, have you considered the Weyl (conformal) tensor? it deals with the part of the Riemannian curvature that acts when the Ricci tensor vanishes. So it fits with some of the features you were asking about and it is implicit in the EFE.

 

[Rasalhague]

Thanks, yes, I'll bear Weyl in mind. I mentioned it earlier, but I think at the moment my priority is to follow Russell's suggestion and get some more familiarity with differential equations. Having read Wheeler's famous motto "spacetime tells matter how to move; matter tells spacetime how to curve" I was expecting there to be a function "curvature(some qualities of matter expressed as tensor fields including a stress-energy tensor field)", but it seems, from what I've been told here, that the reality isn't so simple, and I may need more conceptual tools to make sense of it.

 

[Mentz114]

I notice that there's no link in this thread to an excellent Wiki article. It is made clear what role the EMT plays in the variation of the action wrt the inverse metric.

http://en.wikipedia.org/wiki/Einstein–Hilbert_action