Questions and Some Attempted Answers on General Relativity

I’ve been reading A. Einstein’s popular book “Relativity, the Special and the General Theory”, translated by R. Lawson. Reading this has reminded me of how much I enjoy this area of physics and also how rusty is my understanding of it. I had some questions that I could not immediately answer so, at the risk of exposing my ignorance, I thought that I’d write them down here, possibly together with some of my own or Einstein’s attempted explanations. I’ve also included some verbatim quotes (italicised) that I particularly liked. Corrections are welcomed.

First of all, the following comment puzzled me and I still have no explanation:

“The general theory of relativity renders it likely that the electrical masses of an electron are held together by gravitational forces.”

I suspect this is related to some (now) outdated attempt at unification by Einstein and/or some old model of the electron.

“…the law of the constancy of the velocity of light in vacuo…cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position…We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena.”

This is not a point that I’ve considered much before: light rays curve in the presence of a gravitational field and therefore the velocity of light is not constant with respect to a non-inertial system of coordinates. The principle of the constancy of the speed of light is replaced by the more general law that light rays follow null geodesics in spacetime–of course the latter can be suitably interpreted as saying that light rays follow “straight lines” and have a constant speed c (with respect to a local system of inertial coordinates).

Question: I know that the general principle of relativity puts all coordinate systems on an equal footing, but what does that have to do with gravity?

Answer: The fundamental idea, which is explained elegantly in the book, follows from the equivalence of inertial and gravitational mass–the fact that gravity acts on masses irrespective of construction or composition. The idea is that a reference frame which is accelerated with respect to an inertial reference frame can legitimately be regarded, by a comoving observer, as being “at rest” since any forces on some object due to its inertial mass (as viewed from the inertial reference frame) can be attributed, by a comoving observer, to the action of a (possibly time-varying) gravitational field on that object’s gravitational mass. This is only possible thanks to the equivalence of inertial and gravitational mass.

This next one is just a quote that I quite liked.

“No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.”

I wonder when general relativity will rightly inherit its fair destiny?

Question: How does the mathematical formulation of general relativity incorporate the general principle of relativity?

Short answer: By formulating the laws of nature in terms of tensor equalities.

Longer answer: We want all coordinate systems to be equivalent, insofar as any observer can deduce the same general forms for the laws of nature. Thus, the relevant conceptual framework should represent physical quantities by coordinate independent geometrical objects. It turns out that tensors are just such suitable geometrical objects: they are mathematical objects defined on the tensor bundle of a differentiable manifold and have sufficient complexity to represent things of interest to physicists, such as energy-momentum and curvature. So, if we write down our physics equations in terms of tensors, then we automatically satisfy the general principle of relativity.

Caveat: From a quick google search, it looks as if the matter is much more complicated than my attempted answers make out. There’re all sorts of relativity principles and they aren’t all necessarily realised in general relativity. This free online book dedicates a large amount of space to the issue. Possibly a future blog post will try to sort this out properly.

Question: Are the laws of special relativity tensorial?

In special relativity there are a special class of observers, the inertial observers, who observe the laws of nature in their simplest form. Different coordinate systems corresponding to different inertial observers are related by the Lorentz transformations. It is not true in special relativity, unlike in general relativity, that observers in coordinate systems related by arbitrary transformations will derive the same form for the laws of nature. That is, the laws of nature are not required to be generally covariant. Since it is general covariance that tensorial laws embody, the laws of special relativity are not necessarily tensorial. We do, however, require the laws to be invariant under arbitrary Lorentz transformations, which we do by expressing physical quantities as Lorentz invariant scalars.

Question: In general relativity coordinates are unphysical; why is this? Does this hold in special relativity, or is some intrinsic physical meaning then attached to coordinates?

Question: Is a spacetime devoid of matter and fields the only possible physical realisation of a strict Euclidean (Minkowski) space?

Question: What is the nature of time in general relativity?

Answer: “Clocks, for which the law of motion is of any kind, however irregular, serve for the definition of time. We have to imagine each of these clocks fixed at a point on the non-rigid reference-body. These clocks satisfy only the one condition, that the ‘readings’ which are observed simultaneously on adjacent clocks (in space) differ from each other by an indefinitely small amount.”

Question: General relativity is a special example of a gauge field theory. In what exact mathematical sense is this true and what, if any, consequences does this have in quantum gravity?

Partial Answer: A gauge theory means that the Lagrangian is invariant under a group of local transformations. Another way of looking at gauge invariance is to say that there is some redundancy in the mathematical description of a physical system i.e. the map from conceptual framework to physical reality is not injective, although this redundancy does have physically implications. In general relativity the redundancy is that many coordinate systems describe the same physical system–the descriptions of laws are generally covariant. My understanding is that the gauge, or symmetry, group is special because it is infinite dimensional, whereas, for example, the standard model gauge group U(1)xSU(2)xSU(3) is finite dimensional.

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