Formalizing Special and General Relativity: Axioms and Conservation Laws

[jdstokes]

Hi,

If one were to formalize SR as an axiomatic theory, what axioms would be required other than Minkowski space?

It seems like conservation of 4-momentum is the only other independent axiom.

What about general relativity? Is it sufficient to simply replace Minkowski space by a semi-Riemannian manifold and keep conservation of 4-momentum?

 

[Fredrik]

That's a good question. I've been thinking about this too. I believe that the only axiom you need for GR is that space-time is represented by a 3+1-dimensional smooth manifold with a metric that's a solution of Einstein's equation.

For SR, I believe that what we need (other than Minkowski space) is the Euler-Lagrange equations obtained from a Lorentz-invariant action.

I haven't thought this through 100%, so don't just take my word for it. If someone has a better answer, I hope they will post it here.

 

[HallsofIvy]

It's not at all clear to me why one would want to. Perhaps you really mean "axiomatizing the mathematical formalism for relativity".

 

[Fredrik]

The way I see it, that's exactly the same thing. Relativity wouldn't be a theory without the mathematical formalism. The mathematical formalism is the theory. Without an axiomatic formulation we can't even define what relativity is.

 

[jcsd]

Not sure exactly what you getting at, there's no hard and fast difference between an axiom and a postulate in physics. Special relativity (in it's original formulation) is derived from what are usually called it's two postulates, but if you called them axioms you wouldn't necessarily be wrong.

 

[Fredrik]

Those postulates (or axioms if you prefer to call them that), say a lot about space and time but nothing about matter. If we consider special relativity to be nothing more than a statement about the properties of space and time, then we don't need any other postulates. I'm aware of that, and I think the OP is too, but is this how we want to define "special relativity"?. As a theory of a space-time that's completely empty?

I suppose we could do that, but then we would need another name for the theory that describes matter in that space-time, something like "special relativistic mechanics". This is a theory that requires additional postulates/axioms. (Something like Newton's laws).

 

[Hurkyl]

If one were to formalize SR as an axiomatic theory, what axioms would be required other than Minkowski space?

It seems like conservation of 4-momentum is the only other independent axiom.

Keep in mind that Minkowski geometry doesn't even provide words like "particle", "force", "mass", "momentum", "electric field", et cetera, let alone any information about what properties they might satisfy.

If you're going to formalize SR and want to use Minkowski geometry as a starting point, at the very least you're going to need to expand the language to include these additional terms, and write down some axioms relating them to geometry and to each other.

 

[jcsd]

Those postulates (or axioms if you prefer to call them that), say a lot about space and time but nothing about matter. If we consider special relativity to be nothing more than a statement about the properties of space and time, then we don't need any other postulates. I'm aware of that, and I think the OP is too, but is this how we want to define "special relativity"?. As a theory of a space-time that's completely empty?

I suppose we could do that, but then we would need another name for the theory that describes matter in that space-time, something like "special relativistic mechanics". This is a theory that requires additional postulates/axioms. (Something like Newton's laws).

How can pure SR talk about empty space only? The second postulate that is light has a constant velocity in all inertial frames, so there's soemthign right there populating spacetime. I don't see that you need to add much more than definitions in order to get simple special relativistic kinematics.

 

[jcsd]

Special relativity provides a background, the first psotulate stats that the laws o fphysics are invaraint in all inertial frames o whcih alws you choose to start from is up to you. You can't get relativistic quantum mechanics without the axioms of quantum mechanics.

 

[jdstokes]

Fredrik,

I agree with you that the action should be our starting point to axiomatize the theory.

Let M be a 4-dimensional vector space equipped with a symmetric non-degenerate bilinear form with signature (+,-,-,-). This gives rise to differential invariant

$ds^2 = dt^2-dx^2-dy^2-dz^2$

which has corresponding Lagrangian

$L = \sqrt{t'^2-x'^2-y'^2-z'^2}$

A particle is just a point along a world-line.

Axiom: Particles move along trajectories which minimize the action $S = \int L d\lambda$.

Choose the parameter $\lambda$ to be the proper time so we get the equivalent lagrangian

$L = \frac{1}{2}(t'^2-x'^2-y'^2-z'^2)}$

We automatically get the conserved quantity $H = \frac{1}{2}(p_t^2 - \vec{p}^2)$

Since time is conjugate to the energy we can re-write this as

$m^2 = E^2 -\vec{p}^2$

and use this as a definition of mass.

Conservation of 4-momentum is not necessary in this formalism because it follows directly from fact that L is autonomous.

 

[Fredrik]

Keep in mind that Minkowski geometry doesn't even provide words like "particle", "force", "mass", "momentum", "electric field", et cetera, let alone any information about what properties they might satisfy.

If you're going to formalize SR and want to use Minkowski geometry as a starting point, at the very least you're going to need to expand the language to include these additional terms, and write down some axioms relating them to geometry and to each other.

The best way to do this seems to write down the Lagrangians of the point particles and fields that we want to describe, and use them to define the words. For example, write down the Lagrangian that leads to Maxwell's equations, and define the electric and magnetic fields as the appropriate components of the electromagnetic field tensor.

How can pure SR talk about empty space only? The second postulate that is light has a constant velocity in all inertial frames, so there's soemthign right there populating spacetime.

That's a good point. I actually hadn't even thought of that. This means that the theory that's implied by Einstein's postulates (and the hidden assumptions needed to define an inertial frame) isn't really equivalent to the claim

"Space-time can be represented mathematically by Minkowski space",​

as I used to think. It's actually equivalent to the claim

"Space-time can be represented mathematically by Minkowski space and there's something called light in that space-time that in every inertial frame moves at the speed that's the same in all in inertial frames".​

This is of course very awkward and makes me dislike Einstein's postulates even more than I did before. (The reason I never liked them is that it's not 100% clear what they say. They are a good starting point, but they don't deserve to be called axioms or postulates).

Special relativity provides a background, the first psotulate stats that the laws o fphysics are invaraint in all inertial frames o whcih alws you choose to start from is up to you. You can't get relativistic quantum mechanics without the axioms of quantum mechanics.

No one has even mentioned quantum stuff here. We're talking about classical physics. But if we replace the word "quantum" with "classical" everywhere in your post, it would make an excellent point. We can't get (special) relativistic classical mechanics without the axioms of classical mechanics. The additional axiom we need is the principle of least action. (Do we need anything else? How about Newton's third law?)

However, what we get is still just a framework, but it's a framework that allows us to start writing down Lagrangians. Each Lagrangian defines exactly one "special relativistic classical mechanical" theory.