- #1

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My only question is, is there a name for this result?

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- Thread starter snoopies622
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- #1

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My only question is, is there a name for this result?

- #2

CompuChip

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Metric?

Adjointness (or whatever you call "raising/lowering indices", i.e. going from co- to contravariant and vv)

- #3

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(The quantity [tex] A_a= g_{ap} A^p [/tex] is called the metric-dual of [tex] A^a [/tex].)

Given [tex]A^a{}_b{}^{cd}[/tex], you are forming the scalar using the metric [and its inverse]:

[tex]g_{ap} g^{bq} g_{cr} g_{ds} A^a{}_b{}^{cd} A^p{}_q{}^{rs}[/tex]

- #4

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I was wondering because in another thread someone mentioned using [tex]R^{abcd}R_{abcd}[/tex] (the "Kretschmann scalar"?) in order to show that the event horizon of a Schwarzschild black hole is not a real singularity, and it occurred to me that such a thing might be useful in other circumstances as well, so surely there must be a name for it...

By the way, am I correct in my belief that the "square-norm" of any metric tensor is the number of dimensions of its manifold?

- #5

CompuChip

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- #6

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By definition, [itex]g^{ab} g_{bc} = \delta^a_c[/itex] so if you contractawithcyou get [itex]g^{ab} g_{ba} = \sum_{i = 1}^d 1 = d [/itex].

Ah, yes, of course. Thanks, CompuChip.

- #7

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By the way, am I correct in my belief that the "square-norm" of any metric tensor is the number of dimensions of its manifold?

As seen from CompuChip's response, the metric tensor must have an inverse for that calculation. (Note: The metric of a Galilean spacetime is degenerate.)

- #8

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..The metric of a Galilean spacetime is degenerate.

I don't know what this means. Are you referring to a Euclidean metric? Minkowskian? And whether or not their matrix representations have inverses? Don't they?

- #9

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I don't know what this means. Are you referring to a Euclidean metric? Minkowskian? And whether or not their matrix representations have inverses? Don't they?

The nondegenerate (i.e. invertible) metric of an n-dimensional Euclidean space has

the diagonal form (+1,+1,...,+1,+1) in rectangular coordinates.

The nondegenerate (i.e. invertible) metric of an n-dimensional Minkowskian spacetime has

the diagonal form (-1,+1,...,+1,+1) in rectangular coordinates.

The degenerate (i.e. non-invertible) metrics of an n-dimensional Galilean spacetime has

the diagonal forms (0,+1,...,+1,+1) [for the spatial metric] and (+1,0,...,0,0) [for the temporal metric] in rectangular coordinates.

- #10

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Oh, OK. Thanks.

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