- #1
Apashanka
- 429
- 15
For a curve parametrised by ##\lambda## where ##\lambda## is along length of the curve and is 0 at one end point.
At each ##\lambda## say tangent vector V and A be the two possible vectors of the tangent space.
where ##V=V^\mu e_\mu## and ##A=A^\nu e_\nu##, {e} are the basis vectors.
Now ## \nabla_A V=A^\mu \nabla_\mu(V^\nu e_\nu)=A^\mu(\nabla_\mu V^\nu)e_\nu+A^\mu V^\nu(\nabla_\mu e_\nu)##Now if ##\nabla_\mu V^\nu=0## then the covariant derivative is still not zero .
Similarly from the energy-momentum conservation ##\nabla_\mu T^{\mu \nu}=0 ## from this can we say that in general the covariant derivative of this energy --momentum is non-zero by just comparing to rank 1 tensor??
May be there is something wrong above (I apologise) as I am trying to learn these things...
Thank you
At each ##\lambda## say tangent vector V and A be the two possible vectors of the tangent space.
where ##V=V^\mu e_\mu## and ##A=A^\nu e_\nu##, {e} are the basis vectors.
Now ## \nabla_A V=A^\mu \nabla_\mu(V^\nu e_\nu)=A^\mu(\nabla_\mu V^\nu)e_\nu+A^\mu V^\nu(\nabla_\mu e_\nu)##Now if ##\nabla_\mu V^\nu=0## then the covariant derivative is still not zero .
Similarly from the energy-momentum conservation ##\nabla_\mu T^{\mu \nu}=0 ## from this can we say that in general the covariant derivative of this energy --momentum is non-zero by just comparing to rank 1 tensor??
May be there is something wrong above (I apologise) as I am trying to learn these things...
Thank you