Re: Symmetries of general relativity

On 26 Nov 2003, Serenus Zeitblom wrote:

> Where in Steve Carlip's message did he declare this? He didn't.

Of course that he *did* implicitly declare (the wrong claim) that the
metric was a variable in special relativity (as opposed to a part of the
definition of the theory) - simply because he allows its value to *change*
and still he wants to claim that the operation was a symmetry. To make my
point totally clear - and to avoid a new discussion without consequences
for physics, I should also add a comment that the word "variable" means
"something that can change and/or does change".

The reality is very different. While the metric is a dynamical variable in
general relativity, it is *not* a dynamical variable in special
relativity. Two backgrounds with non-equivalent metrics define *different*
non-gravitational theories. Because the metric is non-dynamical in special
relativity, it is not allowed to change it. If you're changing it, you're
changing the form of the equations. The equations are not invariant under
a generic diffeomorphism!

Of course, it is always possible to say that a theory has any symmetry you
want, as long as you artificially add new degrees of freedom - but it does
not increase the physical, "pragmatic" amount of symmetry that the system
actually has.

> He was talking about SR, where it is declared from the outset,
> as a postulate of the theory, that the curvature tensor is zero.

You are confusing several different things. There are *two* postulates of
special relativity. The phrase "postulate of special relativity" is
something that *everyone else* except for you agrees with, and they
certainly don't say anything about "curvature" because "curvature" is not
a notion that appears in special relativity at all. Let me remind everyone
about the postulates of special relativity:

1. The laws of physics should be the same for all observers in uniform
   motion.

2. The speed of light is the same for all observers in uniform motion.

If anyone needs references that explain the postulates of special
relativity, I will be happy to give him or her some. ;-)

You see, there is no comment about *curvature* simply because *curvature*
is not a construct that one should operate with in special relativity; of
course, no one was talking about spacetime curvature until general
relativity was found.  One can talk about *curvature* only if she
describes a limit of general-relativistic theory that *looks*
non-gravitational. But "curvature" does not belong to special relativity
just like the "spin of a point-like particle" does not belong to classical
physics.

> Diffeomorphism invariance doesn't imply that
> the metric has to be dynamical.

Of course that it does (the topological theories are exceptions). Without
the metric - or another variable that compensates the effects of a
diffeomorphism - the diffeomorphism *changes* the physical laws, the local
speed of light etc., and because it *changes* something (an equation) that
defines the theory, it is *not* an invariance. If you allowed the
equations to change, then *everything* would be an "invariance" and the
word would be meaningless. One is only allowed to say that something is a
symmetry/invariance of a theory if the corresponding transformation only
changes the things that are allowed to change, i.e. the dynamical
variables that define the particular configuration of the theory.

Just like the gauge (Yang-Mills) symmetry implies the existence of the
(dynamical) gauge field, the diffeomorphism invariance implies the
existence of dynamical metric.

> This obvious fact was pointed out decades ago.

Many undefendable things have been said in the history of humankind.
But the opposite statement is not only true, but it is very important
because it is essentially the key argument that leads to general
relativity: the diffeomorphism invariance requires the metric to be
dynamical. Once Einstein understood that *all* observers should be treated
on equal footing, he had all the tools to realize that the spacetime is
curved.

> > condition "the curvature must be zero" clearly has a unique solution -
> > up to a trivial reparameterization of "x" - namely the flat space.
>
> Well, it's nice to see you acknowledging that diffeomorphism
> invariance is a "trivial reparameterization"!

More or less everything in this sentence is incorrect again. First of all,
there is *no* diffeomorphism invariance in special relativity. Second of
all, the choice of different coordinates in special relativity - which is
what you *incorrectly* call "diffeomorphism invariance" - is trivial in
special relativity, indeed. It is so trivial that I can tell you how to
solve it universally: the resulting metric after the appropriate change of
coordinates is ds^2 = -dt^2 + dx^2 + dy^2 + dz^2. ;-) This contrasts with
general relativity where there are infinitely many non-trivial solutions
for the metric - for example the Kerr-Newman black hole.

> What are you talking about? Steve just emphasised what everyone
> has been trying to tell you for weeks: special relativity is
> diffeomorphism invariant.

I don't know whether *everyone*, but let me admit that the number of the
people on this newsgroup who misunderstand the symmetry structure of
special and general relativity at a pretty basic level is just
overwhelming.

Someone expressed doubts whether the Harvard High Energy Theory Group
agreed about these issues. Now I can tell you that we *do* agree. ;-) The
last person whom I asked was Prof. Nima Arkani-Hamed, a leading
phenomenologist - one of the top five cited particle physicists during the
last 5 years - and a renowned co-author of the large extra dimensional
models. His comment about the participants of this newsgroup who claim
that the symmetry of special relativity is the whole diffeomorphism group
was too strong, and the rules of this newsgroup do not allow me to
reproduce his opinion here.

Diffemorphism invariance is not a symmetry (or invariance) of special
relativity because a general diffeomorphism does *not* keep the equations
of special relativity - the equations the govern the (dynamical) variables
- invariant.

If you define the word "invariance", "variable" or another word
differently, be sure that your definition is different from the definition
used in physics. The ability to write something in different coordinates
is not a *symmetry* in the physical sense of this word.

> What is the relevance of your remark to this fact?

The relevance of my remark is that it is the only meaningful and true
remark that can be said about this discussion - a discussion about
the physics questions that are being taught at high schools.

The only meaningful comment among hundreds of others - and hundreds of
less meaningful ones will probably follow. ;-) I believe that even most
high-school seniors just know that the Lorentz invariance and translations
are symmetries of special relativity, but a general coordinate
transformation is *not*.

Every individual web page about the problem that you can find on the web -
as well as every article in physics - agrees that it is only the
Lorentz/Poincare group that is *the* symmetry of special relativity. With
the hope that this un-constructive discussion won't continue, let me list
the first ten web pages that Google returns for the query "symmetry of
special relativity":

        http://www.emmynoether.com/intro.htm

"Special relativity emphasizes, in fact is built on, Lorentz symmetry or
Lorentz invariance, which is one of the most crucial concepts in 20th
Century Physics."       C. N. Yang (Nobel Laureate in Physics)

To give up the notion that the speed of light is a fundamental limitation
on the propagation of signals would be to give up the fundamental symmetry
of Special Relativity, i.e., Lorentz invariance, and its consequences,
such as the equivalence of matter and anti-matter, etc. Symmetry controls
physics in a most profound way, and this was the ultimate lesson of the
20th century.

        http://www.emmynoether.com/rel.htm

This formula produces the same value of  in the primed coordinate system
as in the unprimed coordinate system. This is the defining symmetry
principle of Special Relativity (It's actually called Lorentz Invariance).
Let us now note some very important implications of our new symmetry, the
symmetry of Lorentz Invariance.

        http://www.navi.net/~rsc/physics/B3/evans/evans08.txt

If the underlying symmetry of special relativity is represented by the
Poincare group, it follows that the Maxwell equations ...

        http://www2.berry.edu/academics/science/physics/astro/research.asp

Recently his work has focused on tests of Lorentz symmetry, which is the
fundamental symmetry of Special Relativity and one of the basic tenets of
the Standard Model of Particle Physics.

        http://mathquest.com/discuss/sci.math/a/t/68598

... obey the Lorentz symmetry of special relativity.

        http://www.physicsforums.com/archive/topic/5588-1.html

First, it took me a while to accept that the acceleration breaks the
symmetry of special relativity and accounts for the difference in aging of
the twins when one returns to the other.

        http://www.fandm.edu/Departments/Physics/force/1996.html

That is, expressions in Coulomb gauge do not display the Lorentz symmetry
of special relativity, and so are harder to calculate.

        http://www.stardrive.org/Sarmail4-10-01.shtml

One can use Lorentz symmetry of special relativity with the 4-dim
2-form...

        http://superstringtheory.com/forum/stringboard/messages2/268.html

In general relativity there is a generalization of Lorentz symmetry of
special relativity, which ensures that different (not necessarily
inertial) observers...

        http://www.mcgoodwin.net/pages/elegantuniverse.html

The equivalence principle of general relativity extended the symmetry of
special relativity to make the laws of physics identical for observers
moving and accelerating with respect to each other (motion symmetry). ...

> Wow, so now diffeomorphisms are *not* necessarily symmetries after all!

Diffeomorphisms are the symmetries that underlie *general* relativity, but
they are not symmetries of special relativity. Is it really so difficult
to understand the difference between these two structures?

I hope that this will be the last text that I post on this thread.

> That's very convincing. Still, I think it would be good if you could
> tell us where Misner Thorne and Wheeler made their mistake.

They did not make any conceptual mistake in this discussion, but the goal
of their explanation of general covariance was exactly as mine: to show
how empty the notion of "general covariance" becomes if we allow ourselves
to add new degrees of freedom and unnatural conditions. Yes, if it is
allowed to add new degrees of freedom, we can easily write Newton's
equations in a generally covariant form. The same is true about loop
quantum gravity. Loop quantum gravity, just like Newton's equations, does
not respect the physical rules of 20th century physics - but it is always
easy to claim that a theory is, in fact, generally covariant.

But the price of this sort of "general covariance" is zero.

Let me conclude with a variation of the famous quote due to Daniele Amati.
Loop quantum gravity is part of 17th century physics that fell by chance
into the 20th century. Well, let's try to appreciate that we already live
in the 21st century and let's get rid of the children's diseases at last!

Best
Lubos
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