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Diffeomorphism Invariance in GR 
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#1
Mar2811, 04:06 AM

P: 239

Does anyone know of any web site 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.



#2
Mar2811, 04:23 AM

P: 239

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. 


#3
Mar2811, 06:33 AM

Sci Advisor
P: 888

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. 


#4
Mar2811, 07:37 AM

P: 239

Diffeomorphism Invariance in GR



#5
Mar2811, 09:47 AM

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P: 8,395




#6
Mar2811, 05:43 PM

P: 239




#7
Mar2811, 10:21 PM

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P: 8,395

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. 


#8
Mar2811, 11:18 PM

P: 239




#9
Mar2811, 11:26 PM

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P: 8,395

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). 


#10
Mar2911, 04:12 AM

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P: 888

[tex] \delta \phi = \xi^{\mu}\partial_{\mu} \phi(x) [/tex] Now, if you put a scalar field phi on an arbitrary background g and shift all the dynamical fields, you get [tex] \delta \phi = \xi^{\mu}\partial_{\mu} \phi(x), \ \ \ \delta g_{\mu\nu}(x) = 2 \partial_{(\mu}\xi_{\nu)}(x) [/tex] 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". 


#11
Mar2911, 06:52 AM

P: 239

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. 


#12
Mar3011, 02:15 AM

P: 239

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. 


#13
Mar3011, 05:59 PM

PF Gold
P: 691

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. 


#14
Mar3011, 07:56 PM

P: 239

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. 


#15
Mar3011, 08:00 PM

Sci Advisor
P: 8,395

Newtonian spacetime has a metric.



#16
Mar3011, 08:21 PM

P: 239




#17
Mar3111, 12:17 AM

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P: 8,395

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/grqc/0506065 .



#18
Mar3111, 05:04 AM

Sci Advisor
P: 888

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


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