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0answers
57 views

Why doesn't this proof change indices?

In this pdf, in the second line of the proof, $\sigma$ was plugged in where it appears as $$\frac{\partial x^\sigma}{\partial y^{\rho'}}$$ Meanwhile in converting the coordinates of $g^{\mu'\rho'}$, ...
1
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1answer
81 views

Proof that terms in decomposition of a tensor are symmetric and antisymmetric

Any tensor of rank 2 can be rewritten as: $$A_{bc} = \frac{1}{2}(A_{bc} + A_{cb}) + \frac{1}{2}(A_{bc}-A_{cb})$$ I can understand how that works. My question is: Prove that (independently): ...
4
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2answers
117 views

Geometric meaning of parallel transport

The definition of parallel transport of a vector $v^b$ along a curve $C$ with tangent field $\it{t}^a$ is given by Wald's GR as $$t^a \nabla_a v^b = 0$$ Is it correct to think of $\nabla_a v^b$ as ...
1
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1answer
30 views

Tensors applied to vector and dual vector fields in GR

This is a specific question about tensor manipulation in Wald's GR. I'm asking for clarification of a trivial step, because I'm working through the text outside the context of a class, without prior ...
2
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2answers
69 views

How to define pseudovector mathematically?

The textbook describes pseudovector like this: Let $a,b$ be vectors and $c=a\times b$, $P$ be the parity operator. Then $P(a)=-a,P(b)=-b$ by definition. But $P(c)=c$ since both $a$ and $b$ reverse ...
1
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2answers
221 views

Differentiation in general relativity

If we have: $$ \frac{d\phi^a}{d\tau}= \frac{\partial \phi^a}{\partial x^\mu} \frac{dx^\mu}{d\tau} \tag{1}$$ Differentiating it, we get: $$ \frac{\partial \phi^a}{\partial ...
3
votes
0answers
93 views

Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two ...
1
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1answer
60 views

Demostrating possible equivalence of two tensors

Is there anyway to see by inspection that a form like $$a(x^2 )^{-3} (g _{μσ} x_{\rho} x_{ ν} + g_{μρ} x_{σ} x_{ ν} +g_{νσ} x_{ρ} x_{ μ} + g_{ νρ} x_{ σ} x_{ μ} ) $$ may be equivalent to (i.e ...
3
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3answers
177 views

How to prove the Levi-Civita contraction?

I want to prove the following relation \begin{align} \epsilon_{ijk}\epsilon^{pqk} = \delta_{i}^{p}\delta_{j}^{q}-\delta_{i}^{q}\delta_{j}^{p} \end{align} I tried expanding the sum \begin{align} ...
1
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0answers
30 views

Derivative of a constant tensor field along a path

I understand that the time derivative of a tensor filed $\mathbf T$ along a curve $\gamma\left(t\right)$ can be shown to be, $$ \frac { d\mathbf{T}}{ dt } = \left(\frac { dT^{ i } }{ dt } + V^{ k ...
3
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1answer
76 views

Christoffel symbol

For two nearby points in General Theory of Relativity. The change in the vector components when parallel transported is given by Now, since the parallel transport change must depend on the path ...
2
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0answers
81 views

Calculation of Einstein Equation

I have a 3d system with Lagrangian $$e_3^{-1} L_3 = -\frac{1}{2} R_3 + \delta_{ab} \partial_\rho q^a \partial^\rho q^b + \frac{1}{2H} V(q)$$ From this I want to calculate the Einstein equation by ...
0
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0answers
56 views

Jacobian for Kronecker delta

I was revising on a bit of tensor calculus, when I stumbled upon this: $$\delta^i_j = \frac{\partial y^i}{\partial x^\alpha} \frac{\partial x^\alpha}{\partial y^j}$$ And the next statement reads, ...
2
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2answers
71 views

Definition of derivative operator on a manifold

I'm hoping to understand the motivation for certain parts of the definition of a derivative operator $\nabla$ on a manifold $M$. In Wald's General Relativity, two clauses of the definition are: ...
3
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1answer
33 views

Are the cylindrical and spherical form of Jeans' equations equivalent?

The question kind of says it all, what I really want to know is are the differences in their forms only due to the co-ordinate transform? And as such should a suitable spherical system satisfy ...
-1
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1answer
102 views

Why can't we do some basic algebra in tensor calculus?

I have a very, very stupid question on the basics of tensor calculus. Consider $R_{ij} = 0$. 1)If I expand the ricci tensor $R_{ij}= g^{lm}R_{iljm}=0$. Now, my question is that, why can't we divide ...
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1answer
77 views

Interpretations of (r,s) tensors [duplicate]

A tensor of type (r,s) on a vector space V is a C-valued function T on V×V×...×V×W×W×...×W (there are r V's and s W's in which W is dual space of V) which is linear in each argument. We take (0, 0) ...
0
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1answer
62 views

Showing a fourth rank tensor in $\epsilon$'s reduces to one in the metric $g$

Consider the fourth rank tensor $$S_{\mu \nu \rho \sigma} = a(\epsilon_{\mu \sigma}\epsilon_{\nu \rho} + \epsilon_{\mu \rho}\epsilon_{\nu \sigma})f(x^2),$$ in 2D where $a$ is a constant and $f(x^2)$ ...
2
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1answer
55 views

Derivation of f(R) field equations, problem with integration by parts

I am following the derivation of the field equations on the the Wikipedia page for $f(R)$ gravity. But I do not understand the following step: $$ \delta S = \int \frac{1}{2\kappa} \sqrt{-g} ...
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0answers
37 views

Just a contraction of indices

I came across a contraction which is not giving the desired result. This is a toy problem in how to get a supergravity theory in low energy limit of a superstring theory using the vanishing of beta ...
3
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1answer
58 views

Show that getting parallel transported does not change angle between them- Tensors [closed]

I must tell you that I have never seen this kind of question in Tensor Analysis. Our professor had set up this question in our exam, but I don't know whether it belongs to tensors or not. The question ...
3
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3answers
140 views

Relation between component and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
1
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2answers
127 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 ...
2
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4answers
203 views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
2
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1answer
49 views

Formular for interior product, example

In Nakahara's Geometry,Topology and Physics, the interior product is defined like this : $$i_X: \Omega^{r}(M) \rightarrow \Omega^{r-1}(M).$$ Where $ X \in X(M)$ and $\omega \in \Omega^{r}(M)$ ...
5
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1answer
94 views

Proca Lagrangian manipulation

How can I show that the Lagrangian density $$\mathcal{L} = -\frac{1}{2}\partial_\alpha \varphi_\beta \partial^\alpha \varphi^\beta + \frac{1}{2} \partial_\alpha \varphi^\alpha \partial_\beta ...
1
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0answers
62 views

Variation of the purely covariant Riemann tensor

I need to find the variation of the purely covariant Riemann tensor with respect to the metric $g^{\mu \nu}$, i.e. $\delta R_{\rho \sigma \mu \nu}$. I know that, $R_{\rho \sigma \mu \nu} = g_{\rho ...
5
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1answer
124 views

Levi Civita covariance and contravariance

I read some older posts about this question, but I don't know if I'm getting it. I'm working with a Lagrangian involving some Levi Civita symbols, and when I calculate a term containing ...
5
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1answer
118 views

E&M and geometry - a historical perspective

Recently, I was contemplating the beautiful formulation of electromagnetism (specifically Maxwell's equations) in terms of differential forms: $$F=\mathrm{d} A\implies \mathrm{d}F=0 ...
1
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1answer
52 views

Duality in arbitrary finite dimension using the Levi-Civita tensor

In 4-D flat metric E&M context, given a rank $p$ tensor, one can construct dual of $4-p$ rank tensor by Levi-Civita tensor. Here dual is not in the same sense of mathematical dual. I do not know ...
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2answers
481 views

Why isn't invariant notation common?

In principle, one can write quantities in a manifestly invariant - rather than covariant - fashion in e.g. special relativity. For example, rather than writing just $x^\mu$, we could write the basis ...
2
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2answers
111 views

Why is the anti-symmetric tensor more important than symmetric tensors?

In differential geometry, the differential forms are anti-symmetric tensors. So, why is the anti-symmetric tensor like $ d x_1 \otimes dx_2 - d x_2 \otimes d x_1 $, more important than the ...
10
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4answers
150 views

Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
10
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2answers
164 views

Inverting the equation for $T_{\mu\nu}$ in terms of $F_{\mu\nu}$

The Stress-Energy Tensor for electromagnetism is given by: $$ T_{\mu \nu} = F_{\mu}\,^{\alpha}F_{\nu\alpha}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} $$ How can I find $F_{\mu\nu}$ in ...
1
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2answers
101 views

Planetary motion: integration of equation of motion

I was reading Planetary Motion (page 117) in Barry Spain's Tensor calculus, and stupidly enough, I didn't understand this. The equations are : $$\frac{d^2\psi}{d\sigma^2} + ...
2
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2answers
100 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 ...
3
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2answers
131 views

Why are totally antisymmetric tensors more useful than totally symmetric tensors?

In an arbitrary number of dimensions, one can naturally define two tensors, Kronecker delta and Levi-Civita epsilon tensor. However, why isn't it advantageous to define some totally symmetric tensor ...
2
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0answers
93 views

Simple General Relativity Relation [closed]

Given the identity $$\nabla_a(R^{ab}-\frac{1}{2}R g^{ab})=0,$$ how do I then show that $R_{ab}=0$ implies $$\nabla_a R^a_{\space \space \space bcd}=0$$
4
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0answers
112 views

Why is the $\theta$ term of QCD violating charge and parity (CP) symmetries?

From the non-trivial nature of the QCD vacuum, the Lagrangian is augmented with a term like \begin{equation} \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu} \end{equation} where $ ...
0
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1answer
94 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 ...
1
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2answers
222 views

Tensor algebra doubt

Is it possible to take a tensor to the other side of the equation, and the tensor becomes its inverse(i.e contravariant becomes covariant and vice versa)? It is a stupid question, but It confuses me. ...
1
vote
1answer
56 views

Proof that 4-potential exists from Gauss-Faraday field equation

This is a problem concerning covariant formulation of electromagnetism. Given $$\partial^{[\alpha} F^{\beta\gamma]}= 0 $$ how does one prove that $F$ can be obtained from a 4-potential $A$ such ...
4
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1answer
72 views

Toroid moments tensor decomposition

I am currently working on my bachelor's thesis on the anapole / toroidal moment and it seems that I am stuck with a tensor decomposition problem. I have actually never had a course about tensors, so ...
1
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0answers
68 views

Index Notation Double Curl

My question is about Einstein notation. It does not matter the specifics of this example (the del operator could be another random vector), I just want to know if my assumption about notation is ...
3
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1answer
107 views

What is the difference between $\nabla _{\sigma} $ and $ \nabla^{\sigma}$?

What is the difference between: $\nabla _{\sigma} $ and $ \nabla^{\sigma}$? I've been told that the first is the covariant derivative, however I'm just starting a course on spacetime geometry and ...
0
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0answers
63 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
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1answer
75 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
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3answers
81 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 ...
3
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1answer
195 views

Question on index notation and metric tensor

I found this expression in my SR notes: $$ (\Lambda^{-1})^{\lambda}_{\ \ \ \sigma} = g^{\lambda\mu}~\Lambda^{\rho}_{\ \ \ \mu} ~g_{\rho\sigma} = \Lambda_\sigma^{\ \ \ \lambda}$$ I know where it ...
3
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1answer
133 views

Epsilon Tensor in FeynCalc

A few days ago I started to use the Mathematica package FeynCalc and one thing confuses me: Assume we have a four-vector $p_\mu$ and we contract it with the epsilon tensor. FeynCalc produces ...