Diffeomorphism Invariance in GR

[rogerl]

Does anyone know of any website that has animations of what this Diffeomorphism Invariance in General Relativity can do? I read a lot of articles about it but can't seem to get the essence or visualize how it actually occurs exactly. Thanks.

 

[rogerl]

Or I'll try using words to understand it. It is said that:

"Diffeomorphism Invariance, is closely related to background independence. This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations. A point in spacetime is defined only by what physically happens at it, not by its location according to some special set of coordinates (no coordinates are special). Diffeomorphism Invariance is very powerful and is of fundamental importance in General Relativity"

Questions: What does it mean that "A point in spacetime is defined only by what physically happens at it, not by its location according to some special set of coordinates?"? It's like saying that a satellite can be anywhere on Earth and location is not important. But it is. Please give a clear example of what Diffeomorphism Invariance mean. Thanks.

 

[haushofer]

The main point is that with a fixed background, you can shift fields with respect to that background. That fixed background defines points which have a physical meaning, and this can be covered by different coordinate systems. So coordinates do not have physical meaning.

Without fixed background, as in GR, you don't have this. If you transform all the physical fields, you also tranform the metric. Points loose their meaning and only distances are physically meaningful.

 

[rogerl]

The main point is that with a fixed background, you can shift fields with respect to that background. That fixed background defines points which have a physical meaning, and this can be covered by different coordinate systems. So coordinates do not have physical meaning.

Without fixed background, as in GR, you don't have this. If you transform all the physical fields, you also tranform the metric. Points loose their meaning and only distances are physically meaningful.

Can you please give an example of a field that is shifted with respect to the background in each case?

 

[atyy]

"Diffeomorphism Invariance, is closely related to background independence. This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations. A point in spacetime is defined only by what physically happens at it, not by its location according to some special set of coordinates (no coordinates are special). Diffeomorphism Invariance is very powerful and is of fundamental importance in General Relativity"

There are several meanings of diffeomorphism invariance. Give us the reference so we know what's being discussed.

 

[rogerl]

There are several meanings of diffeomorphism invariance. Give us the reference so we know what's being discussed.

Its from the sci-am article "Atom of Space and Time" by Smolin. But I'm asking about plain generic GR meaning.

 

[atyy]

Its from the sci-am article "Atom of Space and Time" by Smolin. But I'm asking about plain generic GR meaning.

There are two different things in GR both called diffeomorphism invariance.

One is the ability to use arbitrary coordinates, also called "general covariance". This is not specific to GR, and is true of all theories, even special relativity and Newtonian physics. When Smolin says "This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations.", he seems to be referring to general covariance. However, it is not true that general covariance applies only to general relativity.

The special thing about GR is that the 4D spacetime metric is modified by matter such that specifying the spacetime metric completely specifies the distribution of energy in spacetime. This is also called background independence, because there is no fixed background that is unmodified by matter.

 

[rogerl]

There are two different things in GR both called diffeomorphism invariance.

One is the ability to use arbitrary coordinates, also called "general covariance". This is not specific to GR, and is true of all theories, even special relativity and Newtonian physics. When Smolin says "This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations.", he seems to be referring to general covariance. However, it is not true that general covariance applies only to general relativity.

The special thing about GR is that the 4D spacetime metric is modified by matter such that specifying the spacetime metric completely specifies the distribution of energy in spacetime. This is also called background independence, because there is no fixed background that is unmodified by matter.

What. You mean diffeomorphism invariance is synonymous to background independence in GR? I know the latter involves the matter coupling to spacetime affecting space and time. In the former diffeomorphism invariance it has to do with the coordinates such that as one described "If you transform all the physical fields, you also tranform the metric. Points loose their meaning and only distances are physically meaningful.". So even in GR, there is distinction between the two. Do you believe they are just 100% synonyms?

 

[atyy]

What. You mean diffeomorphism invariance is synonymous to background independence in GR? I know the latter involves the matter coupling to spacetime affecting space and time. In the former diffeomorphism invariance it has to do with the coordinates such that as one described "If you transform all the physical fields, you also tranform the metric. Points loose their meaning and only distances are physically meaningful.". So even in GR, there is distinction between the two. Do you believe they are just 100% synonyms?

That's not what I said. I said there are 2 *different* things both called "diffeomorphism invariance", depending on the writer.

One is "general covariance", which means we can use any coordinate system we please, and which is a property of all theories, not just GR.

The other is "background independence" or "no prior geometry", which essentially just means that matter curves spacetime, which is special to GR (at least in comparison to earlier theories).

 

[haushofer]

Can you please give an example of a field that is shifted with respect to the background in each case?

Put a scalar field on a Minkowski background. Then the only dynamical field is the scalar field, and via a coordinate transformation you can shift the field phi wrt the background eta:

<br /> \delta \phi = -\xi^{\mu}\partial_{\mu} \phi(x)<br />

Now, if you put a scalar field phi on an arbitrary background g and shift all the dynamical fields, you get

<br /> \delta \phi = -\xi^{\mu}\partial_{\mu} \phi(x), \ \ \ \delta g_{\mu\nu}(x) = 2 \partial_{(\mu}\xi_{\nu)}(x)<br />

The physics is still the same. You shifted the scalar field, but you shifted the geometry along with it because the metric is also dynamical, and thus shifted. Nothing happened, because points don't have a physical meaning without fixed background; it's a redundancy in your description. And thus a gauge "symmetry".

 

[rogerl]

Put a scalar field on a Minkowski background. Then the only dynamical field is the scalar field, and via a coordinate transformation you can shift the field phi wrt the background eta:

<br /> \delta \phi = -\xi^{\mu}\partial_{\mu} \phi(x)<br />

Now, if you put a scalar field phi on an arbitrary background g and shift all the dynamical fields, you get

<br /> \delta \phi = -\xi^{\mu}\partial_{\mu} \phi(x), \ \ \ \delta g_{\mu\nu}(x) = 2 \partial_{(\mu}\xi_{\nu)}(x)<br />

The physics is still the same. You shifted the scalar field, but you shifted the geometry along with it because the metric is also dynamical, and thus shifted. Nothing happened, because points don't have a physical meaning without fixed background; it's a redundancy in your description. And thus a gauge "symmetry".

Why. In Newtonian world, space is empty and can't form a background so there is no fixed background like the aether.. so naturally if you move say 2 magnets anywhere on earth. It has the same attraction.

 

[rogerl]

Put a scalar field on a Minkowski background. Then the only dynamical field is the scalar field, and via a coordinate transformation you can shift the field phi wrt the background eta:

<br /> \delta \phi = -\xi^{\mu}\partial_{\mu} \phi(x)<br />

Now, if you put a scalar field phi on an arbitrary background g and shift all the dynamical fields, you get

<br /> \delta \phi = -\xi^{\mu}\partial_{\mu} \phi(x), \ \ \ \delta g_{\mu\nu}(x) = 2 \partial_{(\mu}\xi_{\nu)}(x)<br />

The physics is still the same. You shifted the scalar field, but you shifted the geometry along with it because the metric is also dynamical, and thus shifted. Nothing happened, because points don't have a physical meaning without fixed background; it's a redundancy in your description. And thus a gauge "symmetry".

I'm trying to analyze your example. Is your "an arbitary background g" a Newtonian background or a general relativistic background??

When you said "The physics is still the same", are you referring to the first or the second formula?

Whatever, the GR background is dynamical versus the SR fixed background. But how about a Newtonian background. It is not fixed like Minkowski background. Objects are not coupled to Newtonian background because it is empty. But objects are coupled to the minkowski background because this manifold (being substance like) can stick to the object.

So it seems the best space time for Diffeomorphism Invariance is in the following order:

General Relativistic curved space -> Newtonian Spacetime -> Minkowski SR Spacetime

GR curved space is dynamical. Newtonian spacetime is empty. Minkowski space is fixed.

 

[Cleonis]

[...]
GR curved space is dynamical. Newtonian spacetime is empty. Minkowski space is fixed.

I understand why you say it that way, but I think it is an unhelpful direction to take.


For a meaningful comparison of the three: GR spacetime, Minkowski spacetime, and Newtonian spacetime, one must cast each of those three in its most developed form.

We still value Newtonian mechanics because it is a limiting case of relativistic physics.
For a modern view of Newtonian spacetime one must start with relativistic spacetime, and then work down to Newtonian spacetime.

In the case of relativistic spacetime John Wheeler coined the following summary: "Spacetime is telling matter/energy how to move, matter/energy is telling spacetime how to curve."

To obtain Newtonian spacetime we remove only the spacetime curvature aspect. That leaves us with the following property of Newtonian spacetime: "Spacetime is telling matter how to move."

Spacetime telling matter how to move is inertia;
Newton's laws of motion describe the properties of inertia. The laws of motion and the laws of inertia are one and the same thing.

Let me elaborate on that.
The general theory of relativity unifies the description of inertia and the description of gravitation into a single theory. That is: in relativistic physics the phenomenon of inertia is described as a property of spacetime
Hence, if you take Newtonian spacetime as a limiting case of relativistic spacetime, then the phenomenon of inertia is a property of spacetime.

Comparison
GR spacetime, Minkowski spacetime and Newtonian spacetime have the following in common: inertia is a property of the spacetime.

The differences are in the metric.
- Newtonian space is a euclidean space. Newtonian space is immovable. Newtonian time flows uniformly and universally.
- Minkowski spacetime is described by the Minkowski metric. Minkowski spacetime is immovable
- GR spacetime is dynamic, it curves in the presence of inertial mass. At every point of GR spacetime the tangent space has the Minkowski metric.

history
It's possible that in the history of physics there have been people who thought of Newtonian spacetime as just empty nothingness. But thinking that way is a dead end.

 

[rogerl]

I understand why you say it that way, but I think it is an unhelpful direction to take.


For a meaningful comparison of the three: GR spacetime, Minkowski spacetime, and Newtonian spacetime, one must cast each of those three in its most developed form.

We still value Newtonian mechanics because it is a limiting case of relativistic physics.
For a modern view of Newtonian spacetime one must start with relativistic spacetime, and then work down to Newtonian spacetime.

In the case of relativistic spacetime John Wheeler coined the following summary: "Spacetime is telling matter/energy how to move, matter/energy is telling spacetime how to curve."

To obtain Newtonian spacetime we remove only the spacetime curvature aspect. That leaves us with the following property of Newtonian spacetime: "Spacetime is telling matter how to move."

Spacetime telling matter how to move is inertia;
Newton's laws of motion describe the properties of inertia. The laws of motion and the laws of inertia are one and the same thing.

Let me elaborate on that.
The general theory of relativity unifies the description of inertia and the description of gravitation into a single theory. That is: in relativistic physics the phenomenon of inertia is described as a property of spacetime
Hence, if you take Newtonian spacetime as a limiting case of relativistic spacetime, then the phenomenon of inertia is a property of spacetime.

Comparison
GR spacetime, Minkowski spacetime and Newtonian spacetime have the following in common: inertia is a property of the spacetime.

The differences are in the metric.
- Newtonian space is a euclidean space. Newtonian space is immovable. Newtonian time flows uniformly and universally.
- Minkowski spacetime is described by the Minkowski metric. Minkowski spacetime is immovable
- GR spacetime is dynamic, it curves in the presence of inertial mass. At every point of GR spacetime the tangent space has the Minkowski metric.

history
It's possible that in the history of physics there have been people who thought of Newtonian spacetime as just empty nothingness. But thinking that way is a dead end.

I think a more logical approach is say that nature is relativistic thanks to the Dirac Equation. So in choosing between Minkowski and GR metric, the latter is surely background independent. But with Newtonian space time. There is no metric, space and time is not connected.. so there is nothing in the graph that shift matter around it, therefore it is also background independent. However since matter follows the Dirac Equation, then Newtonian space time is automaticaly rejected as not part of nature.

 

[atyy]

Newtonian spacetime has a metric.

 

[rogerl]

Newtonian spacetime has a metric.

In minkowski spacetime. Space and time is united as one and become spacetime. In Newtonian, space and time is separate, there is no spacetime manifold.. so what kind of metric it has?

 

[atyy]

Special relativity and Newtonian mechanics both have global inertial frames, which correspond to fixed background tensor fields. The tensor field in special relativity is the Minkowski metric diag(-1,1,1,1). In Newtonian mechanics, one can set up two constant tensor fields diag(1,0,0,0) and diag(0,1,1,1) that represent "time" and "space". For more detail, see Proposition 3.2.1 on p49 of http://arxiv.org/abs/gr-qc/0506065 .

 

[haushofer]

In minkowski spacetime. Space and time is united as one and become spacetime. In Newtonian, space and time is separate, there is no spacetime manifold.. so what kind of metric it has?

It has two separate metrics, a temporal and a spatial one. This can be argued by the fact that the Galilei group only leaves invariant two separate, degenerate metrics. The reason, ofcourse, is the appearance of absolute time.

The theory based on this construction is called Newton-Cartan. It is a geometric reformulation of Newton.