Exploring the Role of Riemann Tensor in General Relativity

In summary, the reason that Einstein chose the Riemann tensor for his field equations is because it is a necessary component of the theory and it does a good job of conserving energy and momentum.
  • #1
eljose
492
0
A doubt..why einstein Chose Riemann Tensor for GR?..i know its covariant derivative is zero and all that..but Why Riemann tensor?...was not other tensor avaliable or simpler than that?..i studied that and found that for Geodesic deviation ( i didn,t understand that concept..sorry) the Riemann tensor was involved...:uhh: :uhh: :uhh:
 
Physics news on Phys.org
  • #2
eljose said:
A doubt..why einstein Chose Riemann Tensor for GR?..i know its covariant derivative is zero and all that..but Why Riemann tensor?...was not other tensor avaliable or simpler than that?..i studied that and found that for Geodesic deviation ( i didn,t understand that concept..sorry) the Riemann tensor was involved...:uhh: :uhh: :uhh:

We can break this down into parts

Part 1: Why use a tensor? The answer is that using tensors make the underlying theory independent of the coordinate system used. All tensors transform under a change of coordinates in a specific way (in the case of relativity and most of modern physics they transform via the Lorentz transform, the transform that Einstein originally derived in his first paper on special relativity). Because the transformation properties are known, a tensor equation is good in any coordinate system one chooses to use.

Part 2: What tensor(s) should one use? This is not an easy question, and it took Einstein a very long time to derive GR. The short version of the answer is that the choices Einstein made Einstein's field equations automatically conserve momentum and energy in almost the same way that Maxwell's equations automatically conserve charge.

The tensor that Einstein chose to represent matter was the stress-energy tensor, T_ij. This choice was made fairly early on, and can be regardes as "reasonably obvious" - at least, to someone familiar with tensors. The tensor that Einstein chose to represent "curvature" was not the Riemann, but a modified version of the Ricci tensor, a contraction of the Riemann.

The Riemann is R_abcd, the Ricci is formed by a process known as contraction which reduces the number of indices (the rank) of the tensor to two. The Ricci is usually written using the same symbol as the Riemann, but it can be distinguised by the fact that the Ricci only has two indices, i.e. R_ab.

Given that matter is represented by a rank-2 tensor, it is a necessity for a tensor equation that the left hand side of the equations also be a rank 2 tensor, so the Ricci is an obvious candidate.

Einstein originally tried using the Ricci on the left hand side, but he realized that the resulting theory had problems with the conservation of energy and momentum

I.e. R_ab = (some constant) T_ab does NOT work

Einstein eventually realiesed that what he needed on the left hand side was not the Ricci tensor, but a different tensor, known as the Einstein tensor, G_ab

where G_ab = R_ab - (1/2) g_ab R

and R is the contraction of the Ricci, which reduces the number of indices to zero, making it a scalar.

The result of this is Einsteins field equation:

G_ab = 8 pi T_ab does work properly.

Not only does it avoid the problems with momentum and energy conservation that the first formula had, it automatically implies the differential form of the conservation of energy and momentum of T_ab from the geometric properties of G_ab.

Note the form of the equation: the left-hand side contains information about the curvature of space-time, the right hand side contains information about matter - the density of energy and momentum.

This can be summed up neatly by the expression

"Space tells matter how to move, and matter tells space how to curve".

For more detailed online reading, I'd recommend

http://math.ucr.edu/home/baez/einstein/

Much of the information in my post is derived from MTW's textbook "Gravitation", but this is not a popular read, it's a graduate-level textbook. While large portions of the book are written in an informal style that might be accessible to the lay reader, equally large portions of the book are not so accessible.
 
  • #3
"Space tells matter how to move, and matter tells space how to curve".

I'd argue that the first part is given by the geodesic equation, and the second part by the Einstein Field equations (and not just the latter).

EDIT: even better. the first part by extremizing proper time with respect to neighbouring paths and the second part by extremizing the spacetime integral of the Ricci scalar with respect to the metric.
 
  • #4
I'm pretty sure that geodesic motion follows from Einstein's field equations alone - it does not need a separate postulate.

Google finds, for instance

http://www.mathpages.com/home/kmath588/kmath588.htm

Einstein’s claim that general relativity dispenses with the principle of inertia might seem to be supported by the fact that, uniquely among field theories, the equations of motion in general relativity need not be postulated separately from the field equations. In other words, we don’t need to assume what amounts to the principle of inertia, i.e., that material particles follow geodesic paths in spacetime. The field equations themselves imply that free particles follow geodesic paths. This can easily be overlooked when dealing only with test particles of negligible mass moving in a vacuum, because we typically don’t treat such object as sources, we just apply the vacuum field equations Rmn = 0, augmented with the assumption that the test particles follow geodesic paths. But the need to make this separate assumption arises only because we have neglected the role of the particles as sources. If we treat the particles correctly as contributors to the source term, we must apply the full field equations, and those equations do indeed imply that the particles must follow geodesic paths (in the absence of non-gravitational forces).

The author goes on to point out that the differential conservation laws "built into" GR imply that particles follow geodesics in the lack of any external forces.

Unfortunately, I'm not quite sure who the author of the above actually is, which makes it a less than perfect source to resolve a debate, but it does agree with my recollection that we don't need to postulate geodesic motion separately.
 
  • #5
That is a very appealing argument. I wonder if anyone has worked through the problem and derived the geodesic equation...
 
  • #6
Possibly relevant reading

http://links.jstor.org/sici?sici=0003-486X(193801)2%3A39%3A1%3C65%3ATGEATP%3E2.0.CO%3B2-P
The Gravitational Equations and the Problem of Motion
A. Einstein, L. Infeld, B. Hoffmann
Annals of Mathematics, 2nd Ser., Vol. 39, No. 1 (Jan., 1938) , pp. 65-100
"In this paper we investigate the fundamentally simple question of the extent to which the relativistic equations of gravitation determine the motion of ponderable bodies..."

http://prola.aps.org/abstract/RMP/v21/i3/p408_1
On the Motion of Test Particles in General Relativity
L. Infeld and A. Schild
Rev. Mod. Phys. 21, 408–413 (1949)
"In this paper we give a simple derivation of the geodesic motion of test particles from Einstein's gravitational equations for empty space. The history of this problem is connected with the development of basic physical concepts..."

http://www.artsci.wustl.edu/~jashiffl/einstein_infeld.html [Broken]
 
Last edited by a moderator:
  • #8
masudr said:
That is a very appealing argument. I wonder if anyone has worked through the problem and derived the geodesic equation...

Yes, to most peoples' satisfaction, this is has been done. The simplest derivations are present in most textbooks, actually. These have a rather limited scope, though, which led a number of (picky) people to call the principle that objects move along geodesics the "geodesic hypothesis."

Some particularly picky people still call it this, but its precise applicability is much better understood now. Some work to this effect is discussed in http://www.arxiv.org/abs/gr-qc/0309074" [Broken]. The two authors (Ehlers and Geroch) have made many important contributions to the rigorous understanding of relativistic mechanics.

Anyway, the fact the Einstein's equation determines mechanics (to a large extent) was historically considered one of GR's most attractive features.
 
Last edited by a moderator:
  • #9
masudr said:
That is a very appealing argument. I wonder if anyone has worked through the problem and derived the geodesic equation...

See Dirac's 70-page textbook.

Daniel.
 
  • #10
eljose said:
A doubt..why einstein Chose Riemann Tensor for GR?..i know its covariant derivative is zero and all that..but Why Riemann tensor?...was not other tensor avaliable or simpler than that?..i studied that and found that for Geodesic deviation ( i didn,t understand that concept..sorry) the Riemann tensor was involved...:uhh: :uhh: :uhh:

Hi,

The following does not undermine answers given by others.

1) The importance of the Riemann tensor for general relativity can be understood solely from differential geometry (tensor analysis) which shows that a space is flat if and only if the riemann tensor vanishes.

2) Einstein was looking for generally covariant (i.e tensorial) geometrodynamical field theory (i.e the dynamical variables are intrinsic geometrical objects) which has causal solutions(i.e the field equations are 2nd-order differential equations).
To arrive at such theory, one needs a scalar density(Lagrangian) that satisfies;

(i) it is constructed out of the geometry of spacetime alone (i.e the metric tensor)
(ii) it contains no higher derivatives (of the metric tensor) than the second.
(iii) it is linear in the 2nd derivatives of the metric so that it can be put in the form;

[tex] (g)^{1/2}\mathcal{L} = (g)^{1/2} G(g^{\mu\nu},\partial^{\rho}g^{\mu\nu}) + \partial_{\mu}[(g)^{1/2} J^{\mu}(g^{\mu\nu}, \partial^{\rho}g^{\mu\nu})][/tex]

(iv) it vanishes in flat spacetime (the geometry of flat spacetime is not dynamical)

3) From the symmetries and the antisymmetries of [itex]R^{\mu\nu\rho\sigma}[/itex] one can show that the scalar curvature is the only scalar that exists that satisfies (i)-(iv).

4) If one takes

[tex]\mathcal{L}=R=g^{\mu\nu}R_{\mu\nu}=g^{\mu\nu}R^{\rho} {}_{\mu\rho\nu}[/tex]

one arrives (through action principle) at Einstein equation in free space;

[tex]R^{\mu\nu} - 1/2 g^{\mu\nu} R = 0 [/tex]

regards

sam
 
Last edited:
  • #11
pervect said:
I'm pretty sure that geodesic motion follows from Einstein's field equations alone - it does not need a separate postulate.

Google finds, for instance

http://www.mathpages.com/home/kmath588/kmath588.htm
The author goes on to point out that the differential conservation laws "built into" GR imply that particles follow geodesics in the lack of any external forces.

Unfortunately, I'm not quite sure who the author of the above actually is, which makes it a less than perfect source to resolve a debate, but it does agree with my recollection that we don't need to postulate geodesic motion separately.
He is professor Ken Brown from Cornell. See here:

http://www.math.cornell.edu/People/Faculty/brownk.html
.
 
Last edited:

1. What is the Riemann tensor and why is it important in general relativity?

The Riemann tensor is a mathematical object that describes the curvature of space-time in general relativity. It is important because it is used to calculate the effects of gravity on the motion of objects and the bending of light.

2. How does the Riemann tensor relate to the theory of general relativity?

The Riemann tensor is a fundamental part of the mathematical framework of general relativity. It is used in the Einstein field equations, which describe the relationship between the curvature of space-time and the distribution of matter and energy.

3. Can you explain the mathematical equations used to calculate the Riemann tensor?

The Riemann tensor is calculated using a combination of the metric tensor, which describes the geometry of space-time, and the Christoffel symbols, which represent the connection between space-time points. The specific equations used can vary depending on the coordinate system being used.

4. How does the Riemann tensor help us understand the behavior of black holes?

The Riemann tensor is crucial in understanding the behavior of black holes as it allows us to calculate the curvature of space-time around them. This helps us understand how black holes warp space and time, causing the extreme gravitational effects we observe.

5. What are some current research areas exploring the role of the Riemann tensor in general relativity?

Current research areas include using the Riemann tensor to study the properties of gravitational waves, investigating the behavior of black holes in different environments, and exploring alternative theories of gravity that involve modifications to the Riemann tensor. Additionally, the Riemann tensor is also being used in cosmology to study the large-scale structure of the universe and the effects of dark energy and dark matter.

Similar threads

  • Special and General Relativity
Replies
4
Views
922
Replies
5
Views
1K
  • Special and General Relativity
Replies
20
Views
1K
  • Special and General Relativity
Replies
8
Views
2K
  • Special and General Relativity
Replies
9
Views
879
  • Special and General Relativity
Replies
17
Views
1K
  • Special and General Relativity
Replies
1
Views
1K
  • Special and General Relativity
Replies
8
Views
2K
  • Special and General Relativity
Replies
5
Views
161
  • Special and General Relativity
Replies
6
Views
3K
Back
Top