No Symmetry Spacetimes/Metrics in Theory of Relativity

[Suekdccia]

TL;DR Summary

In the context of the Theory of Relativity are there any spacetimes or metrics with a complete absence of symmetries?

In the context of the Theory of Relativity are there any spacetimes or metrics with a complete absence of symmetries?

I mean, consider a type of space or metric where no symmetries would hold (at least not exactly, but approximately). A space or metric where the Poincaré invariance (including the Lorentz symmetry), diffeomorphism invariance, CPT symmetry, time and spatial translational symmetries...and even internal symmetries (like gauge invariances) would not hold (again, at least not exactly). Is there anything like this in the context of the Theory of Relativity? Or would this necessarily be "outside" of it?

 

[vanhees71]

I think then we'd have no regularities and couldn't observe "natural laws". First of all there are physical "global" symmetries, which are treated in Noether's 1st theorem, according to which for any one-parameter symmetry Lie group there's a conserved quantity and if there's a conserved quantity there's a corresponding one-parameter symmetry Lie group. A symmetry is defined as a transformation of the coordinates that leaves the 1st variation of the action invariant.

Second there are also "local gauge symmetries", which occur if the physical situation is described by the theory with more degrees of freedom than determined by the equations of motion, i.e., there are indetermined but unobservable quantities in the the theory. The paradigmatic example is Maxwell's electrodynamics when formulated in terms of the potentials (scalar and vector potential or, in relativistic notation, the four-potential). The potentials are determined only up to an arbitrary gauge transformation, i.e., if $A_{\mu}$ describes a certain physical situation, the same situation is equivalently described also by $A_{\mu}'=A_{\mu}+\partial_{\mu} \chi$, where $\chi$ is an arbitrary scalar field. Formally this is described by a "local symmetry", i.e., transformations, where the group parameters are arbitrary functions of the spacetime arguments.

GR is also a gauge theory. It makes the global Poincare symmetry of SR local, leading to full diffeomorphism invariance, i.e., the gauge group is GL(4), i.e., all local diffeomorphisms between arbitrary spacetime coordinates. That's in the way the mathematical version of the Einstein equivalence principle: Locally (i.e., small regions of spacetime around any "event") everything looks as in special relativity, i.e., it's described by Minkowski space, and you can, at any spacetime point, introduce a local inertial frame of reference, where the physical laws hold as in SR. The gravitational interaction is in this way reinterpreted as "geometrodynamics", i.e., as a pseudo-Riemannian spacetime manifold. Locally you can always transform the gravitational interactions away by using a free falling local reference frame, and that's what makes inertia and sources of gravitation equivalent and thus described by the energy-momentum-stress tensor of matter and radiation. What cannot be "gauged away" of true gravitational fields are the "tidal forces", being closely related to the curvature of the spacetime manifold.

 

[haushofer]

TL;DR Summary: In the context of the Theory of Relativity are there any spacetimes or metrics with a complete absence of symmetries?

In the context of the Theory of Relativity are there any spacetimes or metrics with a complete absence of symmetries?

I mean, consider a type of space or metric where no symmetries would hold (at least not exactly, but approximately). A space or metric where the Poincaré invariance (including the Lorentz symmetry), diffeomorphism invariance, CPT symmetry, time and spatial translational symmetries...and even internal symmetries (like gauge invariances) would not hold (again, at least not exactly). Is there anything like this in the context of the Theory of Relativity? Or would this necessarily be "outside" of it?

No, there isn't. Diffeomorphism invariance is a property of the field equations. Of course, solutions to them can break these symmetries, like e.g.
planetary orbits as solutions don't possess the spherical symmetry of the underlying "field equation" (Poisson equation) of Newtonian gravity. In GR this means an absence of Killing vectors and this tells you something about the spacetime geometry. But you always have the freedom to choose coordinates/observers as you like; this tells you something about the mathematical structure in which the field equations are formulated (general covariance).

So I'd answer your last question with a "yes".

Interestingly, you can reformulate Newtonian gravity in a general covariant way, but you can't formulate GR in a "less than general covariant way"; at least, afaik.

 

[PeterDonis]

In the context of the Theory of Relativity are there any spacetimes or metrics with a complete absence of symmetries?

You need to be clear about what "symmetries" means. In GR, it means Killing vector fields. With that said, the answer is that yes, there are solutions to the Einstein Field Equation that do not have any Killing vector fields.

A space or metric where the Poincaré invariance (including the Lorentz symmetry), diffeomorphism invariance

GR does not include any such cases; GR as a theory is built on the assumption that spacetime is locally Lorentz invariant and that solutions which are diffeomorphic to each other are physically equivalent.

CPT symmetry

GR says nothing about this. You have to look at quantum field theory.

time and spatial translational symmetries

These are Killing vector fields in spacetimes that have them, but not all spacetimes do.

and even internal symmetries (like gauge invariances)

Like CPT symmetry, GR says nothing about these (except to the extent that diffeomorphism invariance can be considered a kind of gauge invariance). You have to look at quantum field theory.

 

[PeterDonis]

I think then we'd have no regularities and couldn't observe "natural laws".

That depends on what kind of "symmetry" you say is absent. It's perfectly possible to have a spacetime with no Killing vector fields that still has "natural laws"--it's still a solution of the Einstein Field Equation and still has local Lorentz invariance and diffeomorphism invariance, and it's perfectly possible to have quantum fields in such a spacetime that are CPT symmetric and have gauge invariance.

 

[vanhees71]

Exactly that's my argument. If you'd have no general symmetries at all, there'd be no clue, which mathematical structures should be used to describe the phenomena, starting from the space-time structure, and there's be also no regularities which could serve as "natural laws". The local Poincare invariance determines the description of the GR spacetime as a pseudo-Riemannian manifold (or rather an Einstein-Cartan manifold to enable the presence of fields with spin). Of course, you can have a completely unsymmetric distribution of energy, momentum, and stress, leading to a solution of the Einstein equations, which has no symmetries/Killing vectors.

 

[PeterDonis]

If you'd have no general symmetries at all, there'd be no clue, which mathematical structures should be used to describe the phenomena, starting from the space-time structure, and there's be also no regularities which could serve as "natural laws".

This seems like a tautology: if we had no natural laws, we'd have no natural laws.

 

[vanhees71]

Indeed. That's why without any symmetries there'd be no physics or other natural sciences.