The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
0answers
123 views

Lecture Notes confusion: Constructing the Einstein Equation

This question is on the construction of the Einstein Field Equation. In my notes, it is said that The most general form of the Ricci tensor $R_{ab}$ is $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ ...
1
vote
0answers
119 views

How to integrate twice of this viscous term?

I am reading a paper, and I do not understand why the author said the following term when integrated twice will become, $\int\limits_\Omega {{\rm{d}}\Omega {{\bf{\psi }}^{\bf{u}}}\cdot\nabla ...
0
votes
2answers
131 views

Weinberg's proof that Levi-Civita Symbol is a tensor

In Gravitation and Cosmology, S.Weinberg states the following: $$\Lambda_{\epsilon}^{\alpha}\Lambda_{\zeta}^{\beta}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa ...
0
votes
2answers
143 views

Raising indices in Killing equation or not?

I'm having issues with computation of Killing equation. I'm using Mathematica to check if the given vectors are Killing vectors or not, and by hand for simple vector like $\xi=\partial_t$ I get the ...
0
votes
3answers
485 views

Need some basic help with notation and the Christoffel symbols

Apologies in advance if some of the questions below seem overly simple. In an introductory GR book, I find the following expression for the autoparallel of the affine connection (the upper bound of ...
0
votes
1answer
66 views

Kronecker delta in inertial tensor

I feel confused in (11.9) how does the book prove the following identity: $$\sum\limits_{i} w_{i}x_{\alpha,i} \sum\limits_{j} w_{j}x_{\alpha,j} = ...
0
votes
1answer
133 views

Stress-energy tensor explicitly in terms of the metric tensor

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $R_{\mu\nu}$ and scalar curvature ...
0
votes
1answer
333 views

Contraction of the metric tensor

This is perhaps a simple tensor calculus problem -- but I just can't see why... I have notes (in GR) that contains a proof of the statement In space of constant sectional curvature, $K$ is ...
0
votes
1answer
79 views

Writing a tensor with respect to a particular basis

When defining tensors as multilinear maps, I am having trouble understanding why a tensor, let's say of type (2,1), can be written in the following way: $$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu ...
0
votes
1answer
2k views

Conversion of motion equation from Cartesian to Polar coordinates: Is covariant differentiation necessary?

I have earlier posted the same question here on math stackexchange but without any answer. As the question concerns tensors, I guess that I have come to the right ...
0
votes
1answer
87 views

Square of a tensor

I think, $$\sigma_{ij}\sigma^{ij} = \sigma^2.$$ However, on the Wikipedia page on Raychaudhuri equation, It was mentioned: $$\sigma^2=\frac{1}{2}\sigma^{ij}\sigma_{ij}$$ I am confused, but I think ...
0
votes
3answers
53 views

Lowering and Raising Kronecker Delta

When an index of the Kronecker-delta tensor $\delta_a^b$ is lowered or raised with the metric tensor $g_{ab}$, i.e. $g_{ab}\delta^b_c$ or $g^{ab}\delta_b^c$, is the result another Kronecker-delta ...
0
votes
1answer
66 views

Why is it suffice to show Tensorial identity on a tensor composed of two vectors?

I've encounter many proves of Tensorail identity that begin with assuming our tensor can be written in form of: $T^{\alpha\beta}=u^{\alpha}v^{\beta}$ . As helpful is it might be, I'm not sure if its ...
0
votes
1answer
47 views

Zero-zero (lower indicies) term for affine connection ($\Gamma_{00}^\lambda$), why do some terms dissapear?

More simply a tensor algebra question, but in General relativity I have the following when I calculate $\Gamma_{00}^\lambda$:- $$ \Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left( \frac{\partial ...
0
votes
3answers
376 views

Relativistic basic question - four vector, Lorentz matrix

I have heard relativistics only very compressed during my student time. Now I looked up the definitions again and a question comes into my mind: A contravariant vector is transformed like this: ...
0
votes
0answers
60 views

Mathematica package for supergravity and string theory

I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...
0
votes
0answers
54 views

Proving the contracted Bianchi identity [closed]

I know how to derive the contracted Bianchi identity $$\nabla_\mu(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R) = 0.$$ But I am having trouble proving that the LHS of this is 0. Can someone help me please? ...
0
votes
0answers
58 views

Ricci/Tensor Calculus and Matrix Calculus

What are the differences in the types of problems Ricci/Tensor Calculus and Matrix Calculus can solve? One formulation is generally used by the physics community and the other by statisticians and ...
0
votes
0answers
68 views

Einstein frame vs. Matter frame

What is the difference between Einstein frame and Matter frame in General Relativity? -A brief comment on each could be useful too. These two frames were used in this manuscript ...
0
votes
0answers
69 views

How should the implicit sum $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted?

$C$ is a 3x3x3x3 tensor. How should the expression $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted? This is my guess: $$ \sum_{i=1}^3\sum_{j=1}^3 \sum_{k=1}^3\sum_{l=1}^3 C_{ijkl}u_{i,j}u_{k,l} $$
0
votes
0answers
68 views

Angular Momentum with Upper Index

I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i ...
0
votes
0answers
27 views

How to calculate independent componets of tensor for crystal symmetry?

I have a question as title says. For a formula $j_i = \beta_{ijl} e_j e_l^*$, where $i$, $j$, $k$ are coordinate and $\beta_{ijk}$ is the tensor, if I know the tensor should be satisfied for crystal ...
0
votes
0answers
253 views

covarient derivative of electromagnetic field tensor

I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is Divergence-less directly(Without using general lie derivative method which can prove any energy momentum ...
0
votes
1answer
115 views

Positive Permutation Tensor

I have seen that one can make an operator with $$ L^i=\epsilon^{ijk}x_{j}\partial_{k} $$ But what if I want to make instead items that are sums, instead of differences. For instance ...
0
votes
0answers
79 views

How do I write the energy of a constant, uniform 2D charge distribution?

Let's consider a 2D electromagnetic field defined in a square domain $[0,\Lambda]^2$, with periodic boundary conditions, with a constant charge distribution, uniform all over the aforementioned ...
0
votes
1answer
170 views

Determine the tensor of contraint and deformation of a cube under compression

We have a cube under compression with dimension l1*l2*l3, is put between 2 rigid plates in the axis 1 (two plates block the deformation of the cube in thí axis), the cube is also put on a rigid plate, ...