3
votes
1answer
80 views

Why do we do partial and not covariant differentiation with $x^{\nu}$?

Why when taking the velocity vector we make $$u^{\nu}=\frac{d}{d\tau}x^{\nu}$$ and not $$u^{\nu}=\frac{\nabla}{d\tau}x^{\nu}$$ where in the last equation I meant the covariant derivative. Why?
2
votes
2answers
81 views

Covariant derivative of a covariant tensor wrt superscript

Is it true that when you take the covariant derivative of a covariant tensor, do you always have to do with a subscript? What if you do it wrt a superscript?Does the first term (with the partial ...
2
votes
1answer
80 views

Are covariant derivatives of Killing vector fields symmetric?

I'm reading the Lecture Notes on General Relativity by Matthias Blau, and in section 9.1 (point 1) he writes: Let $K^\mu$ be a Killing vector field, and ${x^\mu(\tau)}$ be a geodesic. Then the ...
0
votes
1answer
69 views

How to derive the schwarzchild metric?

I'm having trouble differentiating the following when making a change of co-ordinates to determine the Schwarzchild metric. $$r'^{2}=r^{2}C(r)$$ Then taking the total derivative of both sides, the ...
0
votes
1answer
64 views

Covariant derivative of a vanishing tensor component [closed]

Is the covariant derivative of a vanishing tensor component necessarily zero?
1
vote
1answer
77 views

Covariant derivative as a tensor

$$\nabla_{j} v^{i}~=~g^{ik}\nabla_{j}v_{k}.$$ Does this equality involve an intermediate step, where I take the metric inside the derivative, and then use the fact that covariant derivative of the ...
1
vote
2answers
153 views

What is path of light in the accelerating elevator?

Mathematically, (by mathematically I means by equations) what is path of light in the accelerating elevator? What is the difference between an ordinary derivative and covariant derivative (which is ...
0
votes
1answer
274 views

How to find the intrinsic covariant derivative component?

How to find the intrinsic covariant derivative component? In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant ...