General Covariance in GR

[Ben]

So, I apologize in advance for the length ( and content, even!) of this post, but I have been reading A. Pais's excellent biog of Einstein.

 

It seems that for a couple of years prior to publishing his famous field equations E. was convinced that there could be no generally covariant field theory for gravitation.

 

Bearing in mind I am reading a biog not a text, I have tried as best I can to re-create his argument.

 

Suppose, says E. a region [math]D[/math] of spacetime endowed with a gravitational source, given by the energy-momentum tensor [math]T_{ij}[/math]. Assume that there is a metric tensor field [math]g_{ij}[/math] which is everywhere completely and uniquely determined by this source. It seems this assumption is wrong, but for historical interest let's leave that aside.

 

Suppose that in [math]D[/math] there is a coordinate system [math]x^h[/math]. Insist that [math]D[/math] can be partitioned such that in [math]D_1[/math] we may not have that [math]T_{ij} \ne 0[/math] whereas in [math]D_2[/math] this may be the case.

 

Now apply a generally covariant transformation [math]x^h \to \overline{x}^h[/math] such that [math]x^h = \overline{x}^h \in D_1[/math] whereas [math]x^h \ne \overline{x}^h \in D_2[/math] (This is allowed by "general covariance").

 

Then in [math]D_1[/math] we have that [math]T_{ij} = \overline{T}_{ij}[/math] and therefore, by hypothesis [math]g_{ij} = \overline{g}_{ij}[/math] there.

 

Consider the field elements [math]g_{ij}[/math] as being the 10 functions in the 4 variables [math]x^h[/math] (10, because due to skew symmetry there are not 4 x 4 functions, rather there are [math]\frac{1}{2}n(n+1) = 10[/math]. Then, in [math]D_1[/math] one has that [math]\overline{x}^h = x^h,\,\,\overline{T}_{ij} = T_{ij} \Rightarrow \overline{g}_{ij}(\overline{x}^h) = g_{ij}(x^h)[/math]

 

Now turn to [math]D_2[/math] and, using the usual benign abuse of notation write [math]\overline{x}^h = \overline{x}^h(x^k)[/math] and then define [math]\overline{g}_{ij}(\overline{x}^h) = \overline{g}_{ij}(\overline{x}^h(x^h))\equiv h_{ij}(x^h)[/math] and have that [math]h_{ij}[/math] as a new metric field in [math]D_2[/math].

 

But, says E. (I am putting words in his mouth here!) we have that [math]\overline{T}_{ij} = T_{ij} = 0 \in D_2[/math], and two distinct metric fields over the same coordinates which are seemingly determined by the same source.

 

This cannot be so for a theory to be generally covariant. Apparently he called it a "failure of causality"

 

Discuss.

 

PS. I believe Einstein's 1914 paper is availbale online, but I speak no word of German

[modest]

I'm aware of a wiki article addressing this, and offer it if it may be of use,

 

Einstein's hole argument

 

In a usual field equation, knowing the source of the field determines the field everywhere. For example, if we are given the current and charge density and appropriate boundary conditions, Maxwell's equations determine the electric and magnetic fields. They do not determine the vector potential though, because the vector potential depends on an arbitrary choice of gauge.

 

Einstein noticed that if the equations of gravity are generally covariant, then the metric cannot be determined uniquely by its sources as a function of the coordinates of spacetime. The argument is obvious: consider a gravitational source, such as the sun. Then there is some gravitational field described by a metric g®. Now perform a coordinate transformation r to r' where r' is the same as r for points which are inside the sun but r' is different from r outside the sun. The coordinate description of the interior of the sun is unaffected by the transformation, but the functional form of the metric for coordinate values outside the sun is changed.

 

This means that one source, the sun, can be the source of many seemingly different metrics. The resolution is immediate: any two fields which only differ by a coordinate transformation are physically equivalent, just as two different vector potentials which differ by a gauge transformation are equivalent. Then all these different fields are not different at all.

 

There are many variations on this apparent paradox. In one version, you consider an initial value surface with some data and find the metric as a function of time. Then you perform a coordinate transformation which moves points around in the future of the initial value surface, but which doesn't affect the initial surface or any points at infinity. Then you can conclude that the generally covariant field equations don't determine the future uniquely, since this new coordinate transformed metric is an equally valid solution. So the initial value problem is unsolvable in general relativity. This is also true in electrodynamics--- since you can do a gauge transformation which will only affect the vector potential tomorrow. The resolution in both cases is to use extra conditions to fix a gauge.

 

<...>

 

Einstein's resolution

 

In 1915, Einstein realized that the hole argument makes an assumption about the nature of spacetime, it presumes that the gravitational field as a function of the coordinate labels is physically meaningful by itself. By dropping this assumption general covariance became compatible with determinism, but now the gravitational field is only physically meaningful to the extent that it alters the trajectories of material particles. While two fields that differ by a coordinate transformation look different mathematically, after the trajectories of all the particles are relabeled in the new coordinates, their interactions are manifestly unchanged. This was the first clear statement of the principle of gauge invariance in physical law.

Hole argument - Wikipedia, the free encyclopedia

 

And, a paper with more particulars, http://arxiv.org/PS_cache/gr-qc/pdf/0512/0512021v2.pdf

 

~modest

[Ben]

Thanks modest, I hadn't realized the argument was so well-known. Though I DID realize I was flogging a horse that has been dead for 95 years!

[Qfwfq]

Sure, as soon as Albert realized that what he was attempting to make is a gauge theory, his problem was solved.

[Ben]

Well, I am not completely sure he DID realize this, at least in the sense that it is meant now.

 

But I found this as a footnote in one of my Lie Theory texts:

 

The theory of gravitation can be ...considered as a gauge theory, where the gauge group is the Lorentz group..............Both theories are the result of applying the dynamical principle of gauge invariance (called general covarinance the case of gravitation).

 

......The Yang-Mills gauge group, the Yang-Mills potentials and the field strength have their analogues in the group of general coordinate transformations, the Christoffel symbols and the spacetime curvature {respectively}. However, in GR, ......it is the metric (and not the connection) that primarily defines the the gravitational field. In contrast, the Yang-Mills fields are themselves the connection

 

Do I understand this statement? No, not really, but some of it.

 

Ho Hum

[Qfwfq]

Well, I am not completely sure he DID realize this, at least in the sense that it is meant now.

And that must be why he got so confused! :P

 

But, surely Ricci and Levi-Civita would have already been able to clear up his confusion, as much as Minkowski had already cleared up Special Relativity...

 

Ho Hum

Given the differential relations between metric and connection in the GR case, I don't agree with their words "and not the connection" in there. :shrug:

 

What they could have better dwelt on is that, being GR the special case of the gauge group coinciding with the coordinate transformations, there isn't the same ambivalence in the gauge theories for the other interactions. Oh, well, I guess GR is the trickiest gauge theory around. I was gonna say the stickiest, but it hit me that QCD deserves that description.