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1
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0answers
27 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 ...
2
votes
1answer
43 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
votes
0answers
76 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 ...
2
votes
4answers
168 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 ...
1
vote
2answers
112 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
votes
2answers
65 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: ...
0
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0answers
42 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, ...
3
votes
1answer
31 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
votes
1answer
92 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 ...
2
votes
1answer
45 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} ...
5
votes
3answers
236 views

Error in books of conformal field theory?

If you look at the book Conformal Field Theory (by Philippe Francesco, Pierre Mathieu and David Senechal) or the lecture notes Applied Conformal Field Theory (by Paul Ginsparg), and many other places: ...
1
vote
1answer
72 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
votes
1answer
52 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 $f(x^2)$ ...
1
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0answers
34 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 ...
12
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6answers
2k views

What is a tensor?

I have a pretty good knowledge of physics but couldn't understand what a tensor is. I just couldn't understand it, and the wiki page is very hard to understand as well. Can someone refer me to a good ...
3
votes
3answers
133 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 ...
3
votes
1answer
55 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 ...
1
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0answers
53 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 ...
2
votes
1answer
47 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
votes
1answer
93 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 ...
5
votes
1answer
86 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
votes
1answer
105 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
vote
1answer
43 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 ...
6
votes
2answers
472 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 ...
1
vote
2answers
102 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
votes
4answers
137 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 ...
1
vote
2answers
94 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} + ...
11
votes
2answers
157 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 ...
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 ...
3
votes
2answers
120 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 ...
4
votes
1answer
82 views

Invariants of Connection Form

I am somewhat going out "on a limb" here, since I am much more grounded in the physics side of things than I am in mathematics. Nonetheless, I am wondering if someone is able to comment on the ...
2
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0answers
87 views

Questions about closed forms and cycles

I read the section closed forms and cycles in Arnold's Mathematical Methods of Classical Mechanics (page 196-200), but the problems in this section is too difficult to solve in the way following the ...
2
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0answers
90 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 $$\nabla_a R^a_{\space \space \space bcd}=0$$
3
votes
0answers
88 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 $ ...
1
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2answers
220 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. ...
0
votes
1answer
91 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
vote
1answer
53 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 ...
0
votes
3answers
64 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 ...
4
votes
1answer
66 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 ...
6
votes
1answer
410 views

energy momentum tensor and covariant derivative

In field theory, the energy momentum defined as the functional derivative wrt the metric $T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$ (up to a sign depending on ...
3
votes
1answer
120 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 ...
1
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0answers
55 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
votes
1answer
104 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
votes
0answers
60 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
1answer
68 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} = ...
3
votes
1answer
178 views

Proper time along path in Minkowski Space

Consider the path $x^\mu(u)$ in Minkowski space; such that: $$t = \frac{a}{c} \sinh(u) , \quad x = a \cosh(u) ,\quad y = 0 ,\quad z = 0 $$ where $a$ is a positive constant and $u$ is a parameter ...
3
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1answer
172 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 ...
16
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8answers
969 views

Is it foolish to distinguish between covariant and contravariant vectors?

A vector space is a set whose elements satisfy certain axioms. Now there are physical entities that satisfy these properties, which may not be arrows. A co-ordinate transformation is linear map from a ...
1
vote
1answer
80 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 ...
6
votes
3answers
224 views

Resources showing how to use differential forms in Physics

I've been learning for a while about multivectors and forms and how they simplify many things that in simple vector calculus seems to be complicated. The only problem until now is that differently ...