So uh tensorial notation from my string theory course (technically physics, but oh well), something bothers me with the Polyakov action. At some point we want to know the variation of sqrt(-det(g) ) when g varies, g being a 2d metric.
I am told that d(sqrt( -det(g) )) = +1/2 * sqrt( -det(g) ) * gab * dgab = -1/2 * sqrt( -det(g) ) * gab * dgab
So gab * dgab = - gab * dgab, and that sign change is bugging the hell out of me. So, can someone explain that to me, or is it an error?
I'm a bit rusty on this stuff. If I recall correctly, gab * gab = gbb = tr g. Is there any reason in your context to assume that the trace of your metric is constant? Because then your identity just follows from the product rule. That is, differentiating the trace would give gab * dgab + gab * dgab = 0.
edit: oh, gbb isn't the trace of g, is it? It's the trace of the Kronecker delta, or whatever is appropriate for your signature. So yeah, it should be a constant, so what you're looking for comes right out of the product rule.
edit 2: The signature doesn't come into play for the scalar gbb. That is just the trace of the Kronecker symbol, which is the number of dimensions. In this case, 2.
Ugh, you are obviously right.
[ By definition in the tensor notation, gab
, where the -1 means the inverse of the metric, not the inverse of gab
, tensor notation can be very unclear. So yeah, gab
= n (where n dimension), and derivation indeed gives the result. ]
What bothered me in the equation is that, "usually", Aa
because of the symmetry of the metric (sometimes you don't use a metric to lower and raise indices though). Somehow I can't make sense of why this is different when using dg
. Note that in this case, g
is a variable that happens to be a metric, so there's no reason why you should use g
to raise and lower indices, in fact we had defined another metric beforehand. If we call hab
the object that we use to lower indices (probably the predefined metric, but when reading the notes again, I noticed it wasn't specified), we get:hac
= - hca
And somehow that doesn't feel right, 95% of the time h
is symmetric, and in the cases where it isn't, I've only encountered antisymmetric h
. I guess there must be a reason that prevents me from raising and lowering indices for dg
, but I don't see why.