Spacetime symmetries vs. diffeomorphism invariance

[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!

 

[WannabeNewton]

The difference lies in the distinction between the concepts of covariance and invariance; they are indeed two different things. See the following excerpt: http://books.google.com/books?id=eY...iance vs general invariance straumann&f=false

 

[bcrowell]

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.

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.

 

[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.

 

[bcrowell]

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.

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.

 

[TrickyDicky]

But in fact the laws of physics are equally simple in all coordinate systems, if you write them in tensor notation.

For real?

 

[physicus]

Thank you for your answers!

Lorentz invariance is equivalent to saying that (1) our theory is a metric theory, (2) the metric is a scalar under diffeomorphisms, 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.

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?

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.

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?

 

[dEdt]

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?

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.

 

[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

 

[PAllen]

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.

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.

 

[dEdt]

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.

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.

 

[bcrowell]

Does statement (2) mean that the metric is generally an (0,2) tensor under diffeomorphisms, but a scalar in a Lorentz invariant theory?

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

 

[PAllen]

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.

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

 

[dEdt]

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

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.

 

[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.

 

[PAllen]

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.

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.

 

[dEdt]

But these examples of apply to models, where we can rigorously apply measures of complexity (counting degrees of freedom, information criteria, e.g.).

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.

 

[dEdt]

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.

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.

 

[PAllen]

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.

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.

 

[dEdt]

No, Newtonian mechanics (not gravity) requires only be be written with vectors and covariant derivatives to be diff invariant.

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

 

[PAllen]

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

What do you mean by the difference? Newtonian mechanics expressed in vector/tensor form:

- takes the same simple (IMO) form in any coordinates
- any scalar computed is invariant (same value computed any an coordinates)
- vectors and tensors are covariant

I don't see a difference from GR without something extra (e.g. metric signature with spacetime rather than space + time (= fibre bundle); or there can't exist coordinates that globally simplify the machinery (make the connection vanish, for example).

 

[WannabeNewton]

The corollary is that the distinction between "invariant" and "covariant" is also one of aesthetics and lacks a rigorous definition.

There certainly is a rigorous distinction between the two. All of this is discussed rigorously and precisely in the link I provided in post #2.

There's a lot of pointless semantics being thrown around that might obfuscate the original question. As PAllen already noted, Newtonian gravity can be made covariant by reformulating the theory in terms of dynamical curved space-time-simple as that. Covariance is a featureless property of physical theories. Invariance is where the bread and butter of the theory lies as far as symmetries go.

 

[TrickyDicky]

This excerpt from Wikipedia should clear up any further confusion:

"Invariance vs. covariance

Invariance, or symmetry, applies to objects, i.e. the symmetry group of a space-time theory designates what features of objects are invariant, or absolute, and which are dynamical, or variable.

Covariance applies to formulations of theories, i.e. the covariance group designates in which range of coordinate systems the laws of physics hold.

This distinction can be illustrated by revisiting Leibniz's thought experiment, in which the universe is shifted over five feet. In this example the position of an object is seen not to be a property of that object, i.e. location is not invariant. Similarly, the covariance group for classical mechanics will be any coordinate systems that are obtained from one another by shifts in position as well as other translations allowed by a Galilean transformation.

In the classical case, the invariance, or symmetry, group and the covariance group coincide, but, interestingly enough, they part ways in relativistic physics. The symmetry group of the general theory of relativity includes all differentiable transformations, i.e., all properties of an object are dynamical, in other words there are no absolute objects. The formulations of the general theory of relativity, unlike those of classical mechanics, do not share a standard, i.e., there is no single formulation paired with transformations. As such the covariance group of the general theory of relativity is just the covariance group of every theory."

 

[bapowell]

There certainly is a rigorous distinction between the two. All of this is discussed rigorously and precisely in the link I provided in post #2.

No, I got that. I was just pointing out that this is the conclusion that follows if we adopt "complexity" of the equations of motion as our criterion.

 

[WannabeNewton]

No, I got that. I was just pointing out that this is the conclusion that follows if we adopt "complexity" of the equations of motion as our criterion.

Oh I see, I definitely misread your statements then. My apologies!

 

[dEdt]

What do you mean by the difference? Newtonian mechanics expressed in vector/tensor form:

- takes the same simple (IMO) form in any coordinates
- any scalar computed is invariant (same value computed any an coordinates)
- vectors and tensors are covariant

I don't see a difference from GR without something extra (e.g. metric signature with spacetime rather than space + time (= fibre bundle); or there can't exist coordinates that globally simplify the machinery (make the connection vanish, for example).

I'll give an example from special relativity to illustrate what I mean:

The law of inertia can be expressed in coordinate independent form as $$v \cdot\nabla v=0,$$ where $v$ is the particle's 4-velocity. This equation involves combining a vector and a tensor to produce a scalar. However, it is not coordinate invariant. For, in an inertial coordinate system the equation reduces to $$v^\beta \partial_\beta v^\alpha=0,$$ while in a generic coordinate system the equation is more complicated: $$v^\beta \partial_\beta v^\alpha +v^\beta \Gamma^{\alpha}_{\beta\gamma}v^\gamma=0.$$

So despite the coordinate independence of the original equation, it still takes a simpler form in certain privileged coordinate systems, and hence the law isn't coordinate invariant.

 

[samalkhaiat]

I'm not arguing that Occam's razor is a law of nature,

Why "NOT"? From particle interaction to galaxy formation, we see "nature uses as little as possible".
I would say that nature uses Occam's in the same way it uses the principle of least action. Is the action principle a law of nature?

Because of this, the principle ... is not a "rough guide", but a robust constraint on the laws of physics.

Indeed, without the principle of general covariance, there is NO logically consistent path to the E-H action. See this
View attachment E-H action.pdf

Sam

 

[samalkhaiat]

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!

Read though this
View attachment Symmetry Principles and the Axiomatic Structure of Physical Theories.pdf

Sam

 

[PAllen]

I'll give an example from special relativity to illustrate what I mean:

The law of inertia can be expressed in coordinate independent form as $$v \cdot\nabla v=0,$$ where $v$ is the particle's 4-velocity. This equation involves combining a vector and a tensor to produce a scalar. However, it is not coordinate invariant. For, in an inertial coordinate system the equation reduces to $$v^\beta \partial_\beta v^\alpha=0,$$ while in a generic coordinate system the equation is more complicated: $$v^\beta \partial_\beta v^\alpha +v^\beta \Gamma^{\alpha}_{\beta\gamma}v^\gamma=0.$$

So despite the coordinate independence of the original equation, it still takes a simpler form in certain privileged coordinate systems, and hence the law isn't coordinate invariant.

Well, this is exactly what I was referring to as needing to add to the requirement of coordinate independence. The law is definitely coordinate invariant if you express it appropriately (and quite naturally). What you want to add is an additional requirement: that there is no simpler expression of the law that is true in some coordinate systems. To me, that is not coordinate invariance or general covariance, but something else. Anderson and others have worked on formulating this something else, but they admit that it is something beyond coordinate invariance.

 

[stevendaryl]

Here is a way of thinking about theories that perhaps helps to clarify the sense in which GR is simpler than Newtonian gravity, in terms of covariance.

As people have pointed out, any theory of physics can be written in a covariant form. However, when one writes Newtonian physics in a covariant form, one finds that there is a "nondynamic" scalar field--Newton's universal time. It's nondynamic, in the sense that matter and energy and so forth don't affect it.

If you write Special Relativity in a covariant form, you will find that it has no nondynamic scalar fields. But it has a nondynamic tensor field, namely the metric tensor.

In General Relativity, there are no nondynamic scalar, vector or tensor fields. All the geometric fields are dynamic, they are affected by matter and energy.

 

[PAllen]

Here is a way of thinking about theories that perhaps helps to clarify the sense in which GR is simpler than Newtonian gravity, in terms of covariance.

As people have pointed out, any theory of physics can be written in a covariant form. However, when one writes Newtonian physics in a covariant form, one finds that there is a "nondynamic" scalar field--Newton's universal time. It's nondynamic, in the sense that matter and energy and so forth don't affect it.

If you write Special Relativity in a covariant form, you will find that it has no nondynamic scalar fields. But it has a nondynamic tensor field, namely the metric tensor.

In General Relativity, there are no nondynamic scalar, vector or tensor fields. All the geometric fields are dynamic, they are affected by matter and energy.

I agree with this, and it is what the reference WNB gave in post#2 is formalizing (based on the approach given by Anderson in 1967 - where I first encountered it).

I realized there is a further terminological confusion going on, which I have contributed to (). Within covariantly formulated theories, one may produce scalar invariants. However this sense of an invariant is different from formulating a principle of general invariance that is distinct from covariance.

I have always preferred direct terminology like "no absolute objects" as the way way to formulate a theory filtering principle as opposed to a language for formulating any theory (general covariance) - rather than using the historically confused 'general invariance' for this. Some have used that the symmetry group of a theory ( as defined by Anderson - equivalently Straumann ) is the diffeomorphism group as the substantive (theory filtering) principle. To me, though, it all gets back to absolute objects, because absolute objects underpin this definition of the symmetry group of a theory.

 

[PAllen]

Why "NOT"? From particle interaction to galaxy formation, we see "nature uses as little as possible".
I would say that nature uses Occam's in the same way it uses the principle of least action. Is the action principle a law of nature?

Why are there three particle generations not one? Why are there 4 forces, not fewer? Occam's razor is, at best, a guide, just another way of saying simpler, more elegant theories are usually right.

Least action is a very different case. From classical, pre-relativity physics, to GR, to QFT, all experience, without exception, is consistent with "theories of nature are described by action principles".

 

[dEdt]

Well, this is exactly what I was referring to as needing to add to the requirement of coordinate independence. The law is definitely coordinate invariant if you express it appropriately (and quite naturally). What you want to add is an additional requirement: that there is no simpler expression of the law that is true in some coordinate systems. To me, that is not coordinate invariance or general covariance, but something else. Anderson and others have worked on formulating this something else, but they admit that it is something beyond coordinate invariance.

It seems like we're not actually disagreeing with each other, just that we're using different definitions for "coordinate invariant".

 

[samalkhaiat]

Why are there three particle generations not one? Why are there 4 forces, not fewer?

Maybe you should ask Occam. Still 3 generations is more natural than 3 millions resonances. Laws of nature show decoupling at different length scales, and this exactly why we are still working towards unification.

Least action is a very different case. From classical, pre-relativity physics, to GR, to QFT, all experience, without exception, is consistent with "theories of nature are described by action principles".

Isn’t this what I was trying to tell you? So, Is the action principle a law of nature?

Sam

 

[PAllen]

Maybe you should ask Occam. Still 3 generations is more natural than 3 millions resonances. Laws of nature show decoupling at different length scales, and this exactly why we are still working towards unification.




Isn’t this what I was trying to tell you? So, Is the action principle a law of nature?

Sam

Yes, I'm agreeing with that (action principle), emphatically. While I think Occam's razor is not a law of nature but only a guide, effectively the same as 'seek the simplest possible laws', with a lot of judgment about what that means.