# Questions tagged [tensor-calculus]

Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

2,198 questions
Filter by
Sorted by
Tagged with
12 views

### How to obtain the Newtonian limit of specific energy $\mathcal{E}=-u_t$ of general relativity?

In general relativity, the conserved specific energy is expressed by the equation $$\mathcal{E}=-u_t$$, where $u_t$ is the time-component of the four-velocity. This is the total energy of the system ...
• 1,612
1 vote
73 views

### Variation of the Gauss-Bonnet action and Palatini identity for the purely covariant Riemann tensor

I'm taking the variation of the Gauss-Bonnet action $$\mathcal{L}_{GB} = \frac{1}{2}\left(R^{2} - 4R^{\mu\nu}R_{\mu\nu} + R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}\right)$$ to obtain the equations of ...
69 views

• 313
81 views

### Christoffel symbols for certain metric using different formulas

Let M be differentiable manifold that represents spacetime, s.t. $\Gamma^{\lambda}_{\mu\nu}$ is the Christoffel symbol/connection coefficient. The general formula for the christoffel symbol is defined ...
• 313
17 views

### Variation of tensor $\sqrt{1-A_{MP} B^{P}{}_{N}}$ in terms of $A$ and $B$

I want to vary the tensor $X_{MN}=\sqrt{\delta_{MN}-A_{MP} B^{P}{}_{N}}$ in terms of $A$ and $B$. Perturbatively I can do this since X_{MN} = \delta_{MN} - \frac{1}{2} (AB)_{MN} - \...
• 3,076
36 views

### A little clarification on Cartesian tensor notation

Goldstein pg 192, 2 ed In a Cartesian three-dimensional space, a tensor $\mathrm{T}$ of the $N$th rank may be defined for our purposes as a quantity having $3^{N}$ components $T_{i j k}$.. (with $N$ ...
• 688
65 views

### Confusion on two tensors constructed from Riemann curvature tensor and its dual

Assuming the metric signature is $(-+++)$ and solves vacuum Einstein equation, we start from Riemann curvature tensor $R_{\mu \nu \rho \sigma}$ and its dual ${}^*\!R_{\mu \nu \rho \sigma}$ and ...
• 855
1 vote
162 views

• 101
1 vote
120 views

• 313
1 vote
63 views

### "Proof" that zero curvature implies $\partial_a \Gamma^b_{cd}$ is symmetric in $a$ and $c$

I know the claim is wrong. I just want to know where this "proof" goes haywire: Assume curvature is 0 implies Parallel transport is path independent implies Path integration of Christoffel ...
• 917
1 vote
112 views

### About covariant derivative on scalar functions

My question is . When we discuss covariant derivative on any tensorial object we say we apply an operator which reduces to ordinary partial derivative while acting on scalar function. Why it reduces ...
36 views

• 1,028
75 views

• 13
976 views

### Why is it natural to make the electromagnetic field an antisymmetric, rank 2 tensor?

As far as I know, the electromagnetic field strength tensor is defined to be the simplest object involving the electric and magnetic fields that transforms properly under Lorentz transformations. ...
• 233
70 views

### How we find the contorsion tensor?

I know that the formula for contorsion tensor is $$K^{\mu\nu}_a=\frac12({T_a}^{\mu\nu}+T^{\nu\mu}_a-T^{\mu\nu}_a)$$ I want to know how I can find ${T_a}^{\mu\nu}$. What kind of contraction do I follow ...
225 views

55 views

### Symmetries of Dual Riemann tensor

In four dimensions, the dual of the Riemann tensor is defined as \begin{align} ^*R^{\mu}{}_{\nu}{}^{\gamma\delta}=\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}R^{\mu}{}_{\nu\alpha\beta}\,. \end{align}...
73 views

### Measuring the spacetime curvature of an object

Suppose that I want to measure the amount of curvature of spacetime that an object in space creates, like the star Rigel? How would I proceed to do so? Would I use the Ricci tensor $R_{\mu\nu}$ or the ...
• 313
48 views

### Proof that the maximal slicing condition maximizes the volume

I am struggling with proving that the maximum slicing condition $$K = 0$$ not only extremizes, but maximizes the volume $\mathcal{V}$ enclosed within a closed two-dimensional surface $S$. The ...
• 681
I have seen many times on different sources that the equation: $$\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}F_{\rho\sigma}=0$$ Is true identically following from it being a product of an anti-symmetric ...