# Spacetime symmetries vs. diffeomorphism invariance

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1. Feb 18, 2014

### physicus

This is a very basic question, but I cannot get my head around the following: Any physical system should be invariant under changes of coordinates, because these are just a way of parametrizing the manifold/space in which my physical system is embedded.

Now, let us consider a system that obeys Lorentz invariance (as an example of a space-time symmetry) in addition. This should be an additional constraint on the system and basically says that the system is invariant under any transformation $x^mu \to x'^\mu = \Lambda^\mu{}_\nu x^\nu$ such that $\Lambda^\mu{}_\nu\Lambda^\rho{}_\sigma\eta_{\mu\rho}=\eta_{\nu\sigma}$.
It seems that this is just a special case of a coordinate transformation and therefore Lorentz invariance was just a special case of invariance under changes of coordinates. This can of course not be true, so is a Lorentz transformation not just a coordinate transformation or does the term "invariant" mean different things in the two cases?

Thanks for any clarification!

2. Feb 18, 2014

### WannabeNewton

3. Feb 18, 2014

### bcrowell

Staff Emeritus
This isn't a good description of diffeomorphism invariance. Diff invariance isn't about invariance of the properties of systems, it's about invariance of the form of the laws of physics. A hammer doesn't have the same properties under a rotation, for example; its orientation is one of its properties, and will have been changed by the rotation.

Here's a counterexample to the idea that Lorentz invariance is a special case of diff invariance. You can write all of Euclidean geometry using index-gymnastics notation, so that all the postulates and theorems are unchanged in form when you do an arbitrary diffeomorphism. But Euclidean geometry clearly isn't Lorentz invariant. We can't even define what a Lorentz transformation would be in the context of Euclidean geometry.

Lorentz invariance is equivalent to saying that (1) our theory is a metric theory, (2) [deleted, dumb mistake], and (3) the metric's eigenvalues have the signs +---. #3 implies that we can always locally find coordinates such that the metric has the standard form (+1,-1,-1,-1). We can then define a Lorentz transformation as one that keeps the metric in this preferred form.

Examples: Galilean relativity violates #1, because it doesn't have a metric. Euclidean geometry violates #3.

Last edited: Feb 18, 2014
4. Feb 18, 2014

### dEdt

Here's my two cents:

The laws of physics are invariant under any (differential) coordinate transformation. A Lorentz transformation is of course just a special class of coordinate transformations, so the laws of physics must also be invariant under Lorentz transformations. What's special is about Lorentz transformations is that if you have a local inertial coordinate system ie $g_{\alpha\beta}=\eta_{\alpha\beta}$ in a small region of spacetime, then you stay in a local inertial coordinate system after applying a Lorentz transformation ie the metric coefficients stay the same.

The reason why this is significant is that for practical purposes we're often restricting our attention to small regions of spacetime, such as when we analyze the results of some collision taking place in a particle accelerator. In such cases we can effectively ignore the curvature of spacetime and treat the metric as being constant. Then there exists a special class of coordinate systems, called inertial coordinate systems, where the local laws of physics are simplest, and the only coordinate transformations that preserve the laws of physics are Lorentz transformations.

5. Feb 18, 2014

### bcrowell

Staff Emeritus
This is the only part of your post that I disagree with. This is sort of the way Einstein and many others would have described the situation ~100 years ago. But in fact the laws of physics are equally simple in all coordinate systems, if you write them in tensor notation.

6. Feb 18, 2014

### TrickyDicky

For real?

7. Feb 18, 2014

### physicus

Does statement (2) mean that the metric is generally an (0,2) tensor under diffeomorphisms, but a scalar in a Lorentz invariant theory? Also, the notion of Lorentz invariance is "independent" of Lorentz transformations, i.e. the notion of Lorentz transformations is secondary? They are just the means of switching between inertial frames. How does it follow then that physical objects must live in representations of that group of transformations that keep the metric in the preferred form?

How would one phrase the same conditions for conformal invariance of a theory in Minkowskian spacetime? Would you just weaken condition (2) and allow for a rescaling of the metric. And conformal transformations are then just transformations that only rescale the metric?

This sounds like it is only "practical" to consider certain diffeomorphisms that preserve the exact form of the metric. But this cannot be the whole story, because imposing Lorentz symmetry on a system really restricts it, or does it not?

8. Feb 18, 2014

### dEdt

If you impose Lorentz symmetry on the laws of physics without having first imposed diffeomorphism invariance, then you've severely restricted the possible forms of the laws of physics. This is what Einstein did in 1905. But if you later realize that the laws of physics are diffeomorphism invariant, as Einstein realized between 1905 and 1915, then 1) the laws of physics are constrained even more than you previously thought, and 2) Lorentz invariance is simply a consequence of diffeomorphism invariance plus the existence of a metric, so it doesn't constrain the laws of physics.

9. Feb 18, 2014

### WannabeNewton

physicus, did you read the excerpt from Straumann's text? It's all explained there. What you're hinting at is a very subtle but important distinction that is often completely butchered in texts so it's good that you're asking it now

If you want something more lengthy but less mathematically sophisticated, see here: http://www.pitt.edu/~jdnorton/papers/decades.pdf

10. Feb 18, 2014

### PAllen

Actually, diffeomorphism invariance does not constrain physics at all ( for classical theories). It only constrains the formalism you use to express them. For example, Newtonian gravity can be expressed as a diffeomorphism invariant theory. However, it is not Lorentz invariant. Bcrowell, WannabeNewton, have already clarified this for you. To get Lorentz invariance, you not only need to a metric, you need one of the right signature.

11. Feb 18, 2014

### dEdt

The claim that all laws of physics can be made coordinate invariant, which has been made since Kretschmann's 1917 paper, is both true and in my view irrelevant. While it may be true that every law of physics can be reconstructed to be coordinate invariant, such a procedure will in general make the law more complex. If you don't believe me, just compare Newton's Law of Gravitation in its original form with its general covariant form.

The content of diffeomorphism invariance is that the laws of physics in their simplest form are coordinate invariant. This is a highly nontrivial constraint on the laws of physics, and before General Relativity no law of physics could meet it.

12. Feb 18, 2014

### bcrowell

Staff Emeritus
Oops, sorry. Obviously the metric is a tensor, not a scalar. I should have said that the line element ds2 was a scalar.

13. Feb 19, 2014

### PAllen

But that makes it a philosophic principle. Try giving a formal definition of simplest form.

14. Feb 19, 2014

### dEdt

If you're arguing that the concept of simplicity cannot be used in science because it lacks a formal definition, then you're dead wrong. Science is predicated on Occam's razor, and consequently scientists are constantly forced to assess the simplicity or complexity of proposed laws of nature. Just take the geocentrism/heliocentrism debate as an example. There's absolutely nothing wrong with constructing a model of the Universe where the Earth is at the centre, except that such is model is woefully complex. In contrast, a model of the Universe where the Sun is at the centre is much simpler, and thus preferably to a geocentric Universe. One might even take the simplicity of the heliocentric model as a sign that it is fundamentally correct.

So while "simplicity" might not have a definition sufficiently rigorous definition to satisfy a mathematician, it is still a largely unambiguous term and thus has a rightful place in science. Indeed, science is filled with terms that lack "formal definitions" but that are nonetheless extremely useful. For example, I challenge you to give a formal definition for an inertial reference frame (pre-GR) or life.

15. Feb 19, 2014

### bapowell

But these examples apply to models, where we can rigorously apply measures of complexity (counting degrees of freedom, information criteria, e.g.). You are instead talking about the "simplicity" of an equation, which can probably at best be judged on an aesthetic level only. The corollary is that the distinction between "invariant" and "covariant" is also one of aesthetics and lacks a rigorous definition.

Last edited: Feb 19, 2014
16. Feb 19, 2014

### PAllen

Occam's razor is not a scientific law, and is hardly inviolate, and many times scientists disagree on which of two competing theories is simpler (before one is established by experiment). You are arguing that 'rough guides' for theory development are principles. This is not scientific.

17. Feb 19, 2014

### dEdt

I would use (basically) those same criteria when judging the simplicity of an equation, as you put it. Take Newtonian gravitation as an example. One way of formulating Newton's law of Gravitation is by Poisson's equation:
$$\nabla^2 \phi=4\pi G \rho,$$ with $$\ddot{\vec x}=-\nabla \phi.$$

This formulation is not coordinate invariant. Alternatively, you can use the Newton-Cartan theory, where
$$R_{\alpha\beta} = 4 \pi G \rho \Psi_\alpha \Psi_\beta.$$

It's commonly argued that this theory is coordinate invariant, which I disagree with, but regardless it's clear that these equations are more complex than Poisson's equation.

18. Feb 19, 2014

### dEdt

I'm not arguing that Occam's razor is a law of nature, just that its use illustrates that there is a common and robust understanding of what constitutes simplicity.

If, in order to make an equation coordinate invariant, you have to start introducing new terms in the equations, or new dynamical fields, or start using more elaborate mathematical machinery, etc., then everyone will agree that you've made the equations more complex. Indeed, every example outside of GR to make a law of physics coordinate invariant leads to new equations that are, according to any reasonable person, more complicated than the original equations. If you have a counter-example ie an example of someone reasonably arguing that the new equations aren't more complicated than originals, I'd like to see it, but as far as I know there is none. Because of this, the principle of coordinate invariance is not a "rough guide", but a robust constraint on the laws of physics.

19. Feb 19, 2014

### PAllen

No, Newtonian mechanics (not gravity) requires only be be written with vectors and covariant derivatives to be diff invariant. So now you are going to argue that in some objective sense covariant derivative makes it more complex because it could be written with partial derivative using the right coordinates? So your version of general covariance becomes: it must not be possible to globally make the connection vanish? That's awfully peculiar as a fundamental principle.

20. Feb 19, 2014

### dEdt

Not true. Coordinate independent is not the same as coordinate invariant.