The question is: Can GR be derived from the Einstein-Hilbert action?

[friend]

Prof Hagen Kleinert suggested that the action (3)

$$\[A = \int_{{t_a}}^{{t_b}} {dt\frac{M}{2}{g_{\mu \nu }}(q){{\dot q}^\mu }{{\dot q}^\nu }} \]$$

can lead to GR. He writes, "Einstein's equivalence principle amounts to the postulate that the transformed action (3) describes directly the motion of the particle in the presence of a gravitational field caused by other masses." See the Webpage at:

http://users.physik.fu-berlin.de/~kleinert/kleiner_re252/node3.html#SECTION00021000000000000000

But he does not explicitly derive GR from this action. And I don't understand how he can make this claim. Can GR be derived from the above, and if so how? Thanks.

 

[JustinLevy]

I think it is plausible if you take "in the presence of a gravitational field caused by other masses" to mean the particle in this Lagrangian is moving in the curved spacetime caused by an external mass. Sort of like solving for the motion of Mercury using the Schwarzschild metric for the effect of the sun, but we don't consider the smaller effects of the gravitation caused by Mercury.

You seem to be asking if one can derive the GR field equations for the metric from that Lagrangian. In that case I agree, it is not possible.

 

[tom.stoer]

The above action describes how a test particle moves on curved spacetime. This is not special for GR but applies to all curved manifolds; a similar equation holds for a particle moving on a sphere.

The question in GR is not how to derive what happens to a test particle on spacetime, but what happens to spacetime itself. This dynamics is encoded in the so-called Einstein-Hibert action

$$S \sim \int \sqrt{-g}R$$

where g means "determinant of the metric tensor" and R is the curvature scalar, calculated from contracting indices of the Riemann curvature tensor.

 

[atyy]

"Einstein's equivalence principle amounts to the postulate that the transformed action (3) describes directly the motion of the particle in the presence of a gravitational field caused by other masses." is not the same as "leading to GR". It just means that it leads to postulating the geodesic motion of a free falling test particle.

A list of actions for GR are found under the heading of "General Relativity" at http://www.phy.olemiss.edu/~luca/Topics/list.html

 

[friend]

A list of actions ...?

This makes me wonder: is there a way to start with the equations of motion and work backward to find the general form for all possible actions? Is there an inverse variation operation, maybe functional integration?

 

[atyy]

This makes me wonder: is there a way to start with the equations of motion and work backward to find the general form for all possible actions? Is there a inverse variation operation, maybe functional integration?

http://arxiv.org/abs/1008.3177
"It is worth noting that any system of equations can be derived from a variational principle: Simply multiply each equation by an undetermined multiplier, add them together, and integrate over spacetime (for PDE’s) or time (for ordinary differential equations). Such an action principle does not add any insights, and probably has no practical benefit. What we want in an action principle is an encoding of the equations of motion without the addition of any extra variables. Not all systems of equations can be derived from a variational principle using only the variables that appear in the original PDE’s."

A somewhat related point is that although different actions may lead to the same classical solution, the actions may differ when interpreted quantum mechanically. Eg. http://arxiv.org/abs/1012.4280 , which is speculative, but suggests that the application of asymptotic safety ideas leads to different theories depending on whether a metric action or the Holst action is used.

 

[friend]

...
The question in GR is not how to derive what happens to a test particle on spacetime, but what happens to spacetime itself. This dynamics is encoded in the so-called Einstein-Hibert action

$$S \sim \int \sqrt{-g}R$$

where g means "determinant of the metric tensor" and R is the curvature scalar, calculated from contracting indices of the Riemann curvature tensor.

It's interesting that you did not include the differential element for your integral. Wikipedia shows a more complete version of the action as

$$\[S = - \frac{{{c^4}}}{{16\pi G}}\int {R\sqrt { - g} d{x^4}} \]$$

See:http://en.wikipedia.org/wiki/Einstein-Hilbert_action

I'm accustomed to seeing the action integral with respect to the differential dt, where t parameterizes paths of particles x(t), with start and ending points specified.

But I'm not sure what dx⁴ is supposed to mean, and I wonder how or if there is a starting and ending "volumes" as the limits of the integration in x⁴ space. I've read that the action integral integrates wrt volume when one is considering "mechanics of continua", where bulk properties are being considered. Is that what's going on with GR? Is GR a consideration of bulk volumes of spacetime?

Parameterized curves x(t) seem natural when considering path integrals of QM. But if one wanted to put this 4D Hilbert-Einstein action in the path integral to quantize gravity, how would one justify a path integaral when integrating with respect to x⁴ spacetime? Where is the "path"? Is the 3D volume being parameterized by t, from an initial to a final volume?

 

[fzero]

But I'm not sure what dx⁴ is supposed to mean, and I wonder how or if there is a starting and ending "volumes" as the limits of the integration in x⁴ space. I've read that the action integral integrates wrt volume when one is considering "mechanics of continua", where bulk properties are being considered. Is that what's going on with GR? Is GR a consideration of bulk volumes of spacetime?

Yes, field theories (not necessarily quantum) have actions that are expressed in terms of Lagrangian densities:

$$S = \int_\Sigma d^4x ~\mathcal{L}.$$

The Lagrangian is more properly defined as

$$S = \int dt~ L, ~~~ L = \int_X d^3x ~\mathcal{L},$$

whenever the spacetime $$\Sigma$$ admits nice enough splitting as $$\mathbb{R}\times X$$.

For instance, the Maxwell equations are derivable from the action

$$S_{\text{EM}} = \int d^4x \left( -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right),$$

where $$F_{\mu\nu}$$ is a tensor constructed from the electric and magnetic fields.

Parameterized curves x(t) seem natural when considering path integrals of QM. But if one wanted to put this 4D Hilbert-Einstein action in the path integral to quantize gravity, how would one justify a path integaral when integrating with respect to x⁴ spacetime? Where is the "path"? Is the 3D volume being parameterized by t, from an initial to a final volume?

The "paths" in a path integral formulation are essentially Riemannian metrics, so we'd write

$$Z = \int \mathcal{D} g e^{-S(g)/\hbar}.$$

A complete definition of the measure $$\mathcal{D}g$$ is not known, but loop quantum gravity involves various methods to try to formalize the definition.

 

[tom.stoer]

A complete definition of the measure $$\mathcal{D}g$$ is not known, but loop quantum gravity involves various methods to try to formalize the definition.

But it does not use the metric g as dynamical variable

 

[fzero]

But it does not use the metric g as dynamical variable

The Ashtekar variables are supposed to be equivalent variables that replace the metric. I'm not very familiar with the details, but that is the general idea.

 

[atyy]

This makes me wonder: is there a way to start with the equations of motion and work backward to find the general form for all possible actions? Is there an inverse variation operation, maybe functional integration?

AEM gave a good set of references in https://www.physicsforums.com/showpost.php?p=2474351&postcount=16 . Tonti's papers are downloadable from http://www.dic.univ.trieste.it/perspage/tonti/papers.htm .

 

[tom.stoer]

The Ashtekar variables are supposed to be equivalent variables that replace the metric. I'm not very familiar with the details, but that is the general idea.

We had (and still have) long discussions in the "beyond forum" regarding LQG. The basic idea of LQG is that the Ashtekar variables are classical equivalent, but lead to an inequivalent quantum theory. That was one reason why Ashtekar started the business.

 

[fzero]

We had (and still have) long discussions in the "beyond forum" regarding LQG. The basic idea of LQG is that the Ashtekar variables are classical equivalent, but lead to an inequivalent quantum theory. That was one reason why Ashtekar started the business.

I suppose I should be more honest. When I say I'm not familiar with the details, I really mean that there are obvious inconsistencies that I've never had adequately explained to me. I thought it was worth mentioning LQG in the context, but that going into more detail than I did (none at all) wouldn't be helpful to the OP anyway.

I suppose that it's a good thing that the Ashtekar formalism is not quantum equivalent to GR, since the latter has incurable divergences. However, I am not entirely convinced that the classical theory is equivalent either. Since the diffeomorphism group is not completely generated by its algebra (at least in dimension 4), any attempt to account for all of the redundancies by gauging a spin connection is unlikely to result in the correct degrees of freedom. This is not strictly a classical problem, since the point is that the correct classical gauge symmetries are not present even in the classical prescription to translate between the metric and new variables.

The definition of the variables also involves introducing the Immirzi parameter, which is misleadingly claimed to to be fixed by black hole entropy calculations. I haven't followed every paper, but I believe that one requires different values for the Schwarzschild and Kerr-Newman BHs. I've also seen a recent paper on a quantum group calculation where the authors claimed that the BH entropy was completely independent of the Immirzi parameter. Whether the ambiguity is due to an inconsistency in the LQG prescription for computing the entropy or an even deeper problem.

I suppose I will find some of the older discussions you're referring to at some point, though I'm not sure that any of these topics would be addressed.

 

[tom.stoer]

Your problem that the two formailsms are not equivalent classically is not due to the Ashtekar variables but due to the global foliation R*M³ which is possible with canonical gravity using the metric, too. So it's nozt on the level of variables but already on the level of the geometry. I would say that both formalisms are exactly equivalent in a certain subsector (e.g. excluding the Goedel cosmos with closed, timelike curves).

The Immirzi parameter is a quantization ambiguity that seems to be closed to the theta-angle in QCD and is reated to a topological term in the action. But there are also some differences and I agree that the issue has not yet been fully clarified.

But I agree that this is not relevant for the OP. Considering classical GR the Einstein-Hilbert action plus mathematical different formulation (like Palatini) are equivalent.

 

[friend]

Yes, field theories (not necessarily quantum) have actions that are expressed in terms of Lagrangian densities:

$$S = \int_\Sigma d^4x ~\mathcal{L}.$$

The Lagrangian is more properly defined as

$$S = \int dt~ L, ~~~ L = \int_X d^3x ~\mathcal{L},$$

whenever the spacetime $$\Sigma$$ admits nice enough splitting as $$\mathbb{R}\times X$$.

I guess I was asking about the necessity of the volume integral in the action. Is the curvature scalar, R, necessarily a bulk property? Could R also be calculated by, say, a line integral of some function over a closed loop? Or perhaps the ratio of a open surface to its closed boundary?

 

[fzero]

I guess I was asking about the necessity of the volume integral in the action. Is the curvature scalar, R, necessarily a bulk property? Could R also be calculated by, say, a line integral of some function over a closed loop? Or perhaps the ratio of a open surface to its closed boundary?

You could try to do those things, but then you should have a physical explanation of why the action depends on the choice of some submanifold. You would also be introducing background dependence, which means that we have to modify the definition of the action depending on what sources are present. By integrating over the entire volume, we avoid many potential problems.

 

[atyy]

I guess I was asking about the necessity of the volume integral in the action. Is the curvature scalar, R, necessarily a bulk property? Could R also be calculated by, say, a line integral of some function over a closed loop? Or perhaps the ratio of a open surface to its closed boundary?

In the case of a particle, the path integral is a weighted sum over all possible paths x(t).

In the case of a field, the path integral is then a weighted sum over all possible field configurations A(t,x).

In both cases, we are summing over all possible answers, ie. in the classical case, reality consists of one particular answer, which is x(t) for a particle and A(x,t) for a field.

The details on how the boundary is handled for the Einstein-Hilbert action can be found at http://www.physics.uoguelph.ca/poisson/research/agr.pdf