Formalism for an extended object

[alejandrito29]

i am read a paper of name: "formalism for an extended object"(in spanish)..
a sub manifold of coordinates $$y^i$$ i=0...p embebed in a manifold with coordinates $$x^u$$ u=0...D with metric $$g_{uv}$$
the induced metric is:

$$h_{ij}=d_ix^ud_jx^vg_{uv}$$

The paper says that the energy momentum tensor is:
$$T^{uv}(Z^a)= \int\! dy^{p+1} \, \frac{\sqrt{h}}{\sqrt{g}}h^{ij}d_ix^ud_jx^v \delta (x^a-Z^a)$$

but the paper does not say : ¿what is $$Z^a$$ and $$x^a$$?

 

[pervect]

They are just dummy variables. Look at the Fourier transform, for instance

$$F(s) = \int \exp{(2 \pi i t s f(t))} dt$$

s and t are just dummy variables, similar to Z and x. While the Fourier transform transforms between the frequency and time domain, the transform you quote from the paper transforms the stress energy tensor between the representation in the original manifold coordinates and the submanifold coordinates. Though I'm not quite positive that I have the direction correct, it might do the reverse...

 

[alejandrito29]

the transform you quote from the paper transforms the stress energy tensor between the representation in the original manifold coordinates and the submanifold coordinates. .

mmm thanks, but, i don't understand

x: manifold coordinate
y: submanifold coordinate

¿and z?
¿yo says :the transform is between the representation inthe original manifold (x?) and the submanifor coordinates(y?) ?
¿what is z?