Maple: Tensors and arbitrary dimensions

[wildemar]

Hello there,

I'm currently trying to get my head around General Relativity for a term paper; the twist is that I'm dealing with an arbitrary amount of dimensions, that is 4+d, where d is unspecified.

Now the maple tensor package does calculation with some fixed amount of dimensions just fine, obviously. So currently I'm simply using 4+2 dimensions, the two extra ones representing the "arbitrary amount". As was to be expected, this has come to bite me, since the actual amount of extra dimensions comes up in the equations down the line, i.e. I have terms like "2a + 2b", where the 2 in front of the b is actually the number of extra dimensions which would change if I used some other amount and the 2
in front of the b is actually a regular coefficient. I guess you see the problem now: Without actually doing all the calculations by hand, there is no way to figure what numbers are dependent on the dimensions an what
numbers are not.

So I'm asking: Do you know any tricks that would enable me to specify spacetimes (that is, manifolds) with arbitrary dimension?

regards,
/W

 

[shoehorn]

Hello there,

I'm currently trying to get my head around General Relativity for a term paper; the twist is that I'm dealing with an arbitrary amount of dimensions, that is 4+d, where d is unspecified.

You're not dealing with general relativity unless d = 0. The dimensionality of the spacetime manifold is, unsurprisingly, hugely important

Now the maple tensor package does calculation with some fixed amount of dimensions just fine, obviously. So currently I'm simply using 4+2 dimensions, the two extra ones representing the "arbitrary amount". As was to be expected, this has come to bite me, since the actual amount of extra dimensions comes up in the equations down the line, i.e. I have terms like "2a + 2b", where the 2 in front of the b is actually the number of extra dimensions which would change if I used some other amount and the 2
in front of the b is actually a regular coefficient. I guess you see the problem now: Without actually doing all the calculations by hand, there is no way to figure what numbers are dependent on the dimensions an what
numbers are not.

So I'm asking: Do you know any tricks that would enable me to specify spacetimes (that is, manifolds) with arbitrary dimension?

regards,
/W

You're missing the point somewhat. Maple (and, by extension, even more powerful packages like grtensorII) are essentially matrix-based; that is, they perform tensor calculations by manipulating multi-dimensional arrays in memory. Unfortunately, in order for this approach to be feasible and useful, you need to specify in advance the number of dimensions you're working with. As a result, I'm not sure how you'd go about the "top-down" approach to the problem that you seem to want.

There may be some specific approaches or techniques which you may find useful, but without specific knowledge of the models you're trying to implement I can't say much more at this point.

 

[wildemar]

You're not dealing with general relativity unless d = 0. The dimensionality of the spacetime manifold is, unsurprisingly, hugely important.

Huh? I didn't know the definition was that narrow. Because everything else I do certainly falls into the category of GR, Einstein Equation and all. And since the extra dimensions compactify quite quickly, after some time one gets an essentially 4-dimensional spacetime.

I'm curious now: How would I describe what I do, then? It's still higher dimensional relativistic cosmology though, right? (OK, I realize now that I didn't say I was doing cosmology. But I am; I'm basically adding extra dimensions to the FRW-Metric.)

You're missing the point somewhat. Maple (and, by extension, even more powerful packages like grtensorII) are essentially matrix-based; that is, they perform tensor calculations by manipulating multi-dimensional arrays in memory. Unfortunately, in order for this approach to be feasible and useful, you need to specify in advance the number of dimensions you're working with. As a result, I'm not sure how you'd go about the "top-down" approach to the problem that you seem to want.

I thought as much. And my professor said basically the same thing. I just thought I'd give it a shot and try to ask a few more people, just in case there is a purely symbolic way of doing this (the paper I'm reading for this does it symbolically).

I guess I could still run the same calculations with several fixed dimensions and see in what way the results differ. I was hoping to avoid this, but I guess it's not a big pain.

There may be some specific approaches or techniques which you may find useful, but without specific knowledge of the models you're trying to implement I can't say much more at this point.

I'm not quite sure what you need to know about this, but let me try.

There are two extensions to the standard model that I employ: Adding a Gauß-Bonnet term to the Einstein Equation:
$G_{a b} + \lambda g_{a b} + \mathcal{G}_{a b} = \frac{1}{\kappa} T_{a b}$
and adding several dimensions to the FRW-Metric:
$ds^2 = -dt + a(t)^2 \left( \frac{dr^2}{1 - K r^2} + r^2 \left( d\theta^2 + \sin^2\theta \ d\phi^2 \right) \right) + b(t)^2 \gamma_{m n}(y) dy^m dy^n$

Both the regular and the additional dimensions are assumed to be flat, that is $K=0$ and $\gamma_{m n}$ is the unit matrix.

I case you want to bother checking the paper that I'm referring to, it's "Solutions of higher dimensional Gauß-Bonnet FRW cosmology" by K.Andrew, B.Bolen and Ch.A.Middleton, Gen Relativ Gravit (2007) 39:2061-2071, DOI 10.1007/s10714-007-0502-7, http://arxiv.org/abs/0708.0373" .

thanks for your time, btw. :)
/W