## Tensors

#### Oz and John Baez

Poor old Oz is depressed. He fears that the Wiz is going to be terribly, terribly cross with him. He needs that most exquisitely boring of subjects, an elementary Tensor revision course. Abject embarrassment!

The Wiz, rightly, tends to gloss over details. Oz also has a poor memory at times and it takes a little while for things to sink in.

Basically Oz has been looking at ways to find which elements of Rabcd correspond to his 16 elements of Rab. He realises that he is not as sure as he should be about some basic tensor stuff. He might have come a fair way, but his knowledge is still appallingly elementary. Basically he does it mechanically, and not very well at that.

He hopes that Wiz will find the time to answer a few basic questions, or at least some other wandering Wiz can deign to... he walks up to the Wiz's door and knocks.

"Come in," says the Wiz.

"Could review a bit of, umm, tensor stuff with me?" asks Oz, looking down at the floor and blushing.

"Boring, boring, boring. Next we'll be reviewing partial differential equations and Fourier transforms. But okay...."

"Okay." Oz scratches on the floor:

Tcc = g^(ca)Tac = scalar.

"I feel that there should also be a product that is another tensor. I suspect that gabTcd would be a tensor of form Uabcd...."

The wizard nods. "A product of g and T that's a tensor? Sure, like gabTcd. That's rank (0,4): eats four vectors for breakfast and spits out a scalar at noon."

"By the way, why are you using a mix of {}'s and ()'s? There's no point in doing that here, and it could even be confusing if I didn't know that you couldn't possibly know what () meant in this context. We just use 's and 's as a substitute for writing batches of superscripts or subscripts, respectively."

"I also wonder if the reversal of the sub and superscript order is important. I also wonders if gcaT^(ac) is different, and why?"

"Well, the order is very important in general, but the metric and the stress energy just HAPPEN to be symmetric: gab = gba, and Tab = Tba. Figure out why."

"Also: show that gac Tac = gac Tac, using the fact that gab is the inverse matrix of gab, and maybe some other facts."

Oz scratches another tensor in the dust on the floor:

Rabcd

"This takes in three vectors and outputs one. I think this could be evaluated as something that takes in three vectors ub,vc,wd to output a vector qa like this:

qn = sum(i=0 to 3){sum(j=0 to 3)[sum(k=0 to 3) R(n,i,j,k) ui vj wk ]}

Is this right (if you can work out what I actually said!)?"

"Almost; you mean Rnijk where you wrote R(n,i,j,k). Also, the Einstein summation convention is designed to prevent such an obscene proliferation of summation signs. The quick way to write what you wrote is:

qn = Rnijk ui vj wk"

Oz nods. "In other words Rabcd is a 4x4x4x4 matrix of 256 elements."

"Yes, but thankfully, it possesses enough symmetries to reduce down to only 20 independent elements. Exercise: show from the the definition that Rabcd = -Racbd. You know, that `skew-symmetric in the first two slots' business."

Oz asks, "Now when we raise an index how does that alter the terms in the array? I suspect it's not trivial."

"Well, index raising is done using the metric and the Einstein summation convention, so e.g. if we want to raise the first index on Xabcd we do this:

Xabcd = gan Xnbcd

where we sum over n. So yes, this can alter the terms in a severe way."

"Now I suspect that to get from Rab to a form like Ra_{bcd) I would need to blend a gcd with Rab to get a Rabcd then raise an index to get Rabcd."

"That's right."

"I think this will be very, very messy."

"Well, to get from Rab to something of the same general form as the Riemann tensor we can just form something like Rbd deltaac. Note this has the same number of indices up and down as the Riemann tensor, which is Rabcd. In fact, you are right that there is some Rbd deltaac lurking in the Riemann tensor, waiting to burst free, and also some Weyl tensor stuff... but you are also right that the exact formula is a mess."

"What's that deltaac thing? " asks Oz.

"Well, it's the Kronecker delta, which is 1 if a = c and 0 otherwise, but it's closely related to the metric, since we get it by raising one index on the metric: gac = deltaac."

"Also... gab is which metric?" asks Oz.

The wizard becomes a bit angry. "It's whatever the hell metric is the actual metric on spacetime that you happen to be studying!!!!!!!!"

"Crucial, crucial point. Remember, the metric describes the geometry of spacetime. All sorts of things depend on the metric. You gotta use the metric you actually are studying, you can't just pull one out of the hat."

"Can it be simply the minkowskian one, despite the evidence that the metric is likely to be non-minkowskian?"

"No way! That'd be like randomly picking an electric field and stuffing it into the problem you're solving, instead of using the physically correct one."

"I also suspect that this brute force way of doing it is silly. All we need to know is in Rab, and a diagonalised form at that. How should I proceed trying to see what this means?"

"It depends on what you're doing." The wizard pauses, subliminally aware of some disturbance outside in the hall. Continued...