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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
votes
0answers
92 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 ...
3
votes
0answers
362 views

I lost a factor of two in the electromagnetic field tensor

I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-) ...
2
votes
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
votes
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 ...
2
votes
1answer
137 views

Question from Schutz's

In q. 22 in page 141, I am asked to show that if $U^{\alpha}\nabla_{\alpha} V^{\beta} = W^{\beta}$, then $U^{\alpha}\nabla_{\alpha}V_{\beta}=W_{\beta}$. Here's what I have done: $V_{\beta}=g_{\beta ...
2
votes
2answers
224 views

Bracket Notation on Tensor Indices

I know about the () symmetrisation and [] anti-symmetrisation brackets on tensor indices so long as they appear on their own, such as : $$V_{[\alpha \beta ]}=\frac{1}{2}\left ( V_{\alpha \beta ...
2
votes
1answer
134 views

Hypersurface Normal

Could anyone explain why $$n^{a}n_{a}=\pm1$$ where $n^{a}$ is the normal to the hypersurface
2
votes
3answers
230 views

On Einstein notation with multiple indices

On Einstein notation with multiple indices: For example, consider the expression: $$a^{ij} b_{ij}.$$ Does the notation signify, $$a^{00} b_{00} + a^{01} b_{01} + a^{02} b_{02} + ... $$ i.e. you ...
2
votes
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 ...
2
votes
1answer
519 views

Deriving an equation involving Killing vectors

I'm currently studying Carroll's GR book Spacetime & Geometry, and ran into some trouble understanding the text. When discussing Killing vectors, Carroll mentions that one can derive ...
2
votes
1answer
137 views

Are there any clear and expressive plainword sense of metric tensor components?

Can the following groups of components of metric tensor can assigned a clear sense? https://docs.google.com/drawings/pub?id=1kVqkN1gT-a2fDy2S851l9iQKaMfaatCDo517OSZBHEo&w=467&h=228 I have ...
2
votes
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: ...
2
votes
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 ...
2
votes
1answer
561 views

'Easy way' of finding out the Killing vector fields?

Is there a way for calculating the Killing vector fields of a given metric in a quick way? Sure I can guess looking at the metric at the symmetries, and then guess some of them, but, for instance, in ...
2
votes
1answer
1k views

Stress energy tensor of a perfect fluid and four-velocity

In the following demonstration, there is an error, but I cannot find where. (I explicitely put the $c^2$ to keep track of units). We consider a metric $g_{\mu\nu}$ with a signature $(-, +, +, +)$ : ...
2
votes
1answer
297 views

Tensor Introduction

I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, ...
2
votes
1answer
54 views

Testing covariance of an expression?

This is something I've been unsure of for a while but still don't quite get. How does one tell whether an expression (e.g. the Dirac equation) is covariant or not? I get it for a single tensor, but ...
2
votes
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} ...
2
votes
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 $\omega \in \Omega^{r}(M)$ ...
2
votes
2answers
173 views

Recovering 4-vector Lorentz transformation from spinor formalism

I'm trying to recover the 4-vector transformation laws using spinors. I have defined $$v^{\dot{a}b} = v^{\nu} \sigma_{\nu}^{\dot{a}b}$$ as usual, with $\sigma_0=1$. Now with the rules for dotted ...
2
votes
1answer
269 views

Check it the Killing vectors satisfy Killing equation or not

I am going through Kerr/CFT correspondence paper again, and I am at the section where authors specify Killing vectors for near horizon extreme Kerr metric (shortly NHEK). The metric is ...
2
votes
1answer
477 views

Ricci identity/Riemann curvature tensor and covectors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
107 views

Non-diagonal elements when switching metric signature?

Considering a metric tensor with the signature $(-,+,+,+)$: $g_{\mu\nu}= \begin{pmatrix} -c^2 & g_{01} & g_{02} & g_{03}\\ g_{10} & a^2 & g_{12} & g_{13}\\ g_{20} & g_{21} ...
2
votes
1answer
245 views

Stress energy tensor and the covariant derivative of the 4-momentum

Another basic question. I have usually seen the stress energy tensor $T^{ij}$ described as the flow of the 4-momentum field $p^i$ along direction $x^j$ in spacetime with $p^0$ as energy and $x^0$ as ...
2
votes
1answer
265 views

Is there symmetry in 2d stress tensor in linear elastic fracture mechanics?

Assumptions: Cross terms in strain tensor are defined as equal $\varepsilon_{xy} = \varepsilon_{yx}$. pure mode I crack. Far from crack tip, material is purely elastic and we are way below yield ...
2
votes
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 ...
2
votes
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 $$\nabla_a R^a_{\space \space \space bcd}=0$$
2
votes
0answers
97 views

Can the two electromagnetic field tensors be combined into a more general tensor?

Given the electromagnetic field tensor $$\begin{align} F_{\mu\nu} = \begin{pmatrix} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & B_{z} & -B_{y} \\ E_{y} & -B_{z} & 0 ...
2
votes
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
votes
0answers
77 views

Stress calculations in a perforated paper

You have a sheet of paper (torn out of a good quality foolscap notebook) as shown above, and you start pulling it apart with both your hands (forces indicating by the blue arrows). Its difficult to ...
2
votes
0answers
127 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
2answers
312 views

Question with Einstein notation

Let’s consider this equation for a scalar quantity $f$ as a function of a 3D vector $a$ as: $$ f(\vec a) = S_{ijkk} a_i a_j $$ where $S$ is a tensor of rank 4. Now, I’m not sure what to make of the ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
1answer
306 views

Question about contraction with metric tensor

I just starting to study GR and I could not prove the following: if I have to tensors $T_{\mu\nu}$ and $Q_{\mu\nu}$ such that $T_{\mu\nu}=Q_{\mu\nu}$, why can I multiply both sides of the equation by ...
1
vote
2answers
365 views

What is the Riemann curvature tensor contracted with the metric tensor?

Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example $R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
1
vote
2answers
547 views

What kind of invariants are proper time and proper length?

Under the Lorentz transformations, quantities are classed as four-vectors, Lorentz scalars etc depending upon how their measurement in one coordinate system transforms as a measurement in another ...
1
vote
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): ...
1
vote
1answer
107 views

What is the difference between a skew-symmetric and an antisymmetric tensor?

What is the difference between a skew-symmetric and an anti-symmetric tensor? If they represent the same tensor, then why use different labeling.
1
vote
2answers
598 views

Kronecker delta confusion

I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same): ...
1
vote
1answer
329 views

What is the Lorentz tensor with a superscript and subscript index?

I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar ...
1
vote
1answer
677 views

metric signature explanation

Can anyone explain what metric signature is? I have a basic knowledge regarding tensors, btw. Also, how is it related to fundamental understanding of general relativity? Thanks.
1
vote
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 ...
1
vote
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 ...
1
vote
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) ...
1
vote
1answer
151 views

Transformation rule of a partial derivative

We know the following transformation rule: $$ \partial'_b = \frac{\partial}{\partial x'^b} = \frac{\partial x^c}{\partial x'^b} \, \frac{\partial}{\partial x^c} = \frac{\partial x^c}{\partial x'^b} ...
1
vote
1answer
106 views

S. Weinberg, “The Quantum theory of fields: Foundations” (1995), Eq. 2.4.8

Unfortunately I'm struggling to understand how do we get eq. (2.4.8) from eq. (2.4.7), p. 60; namely how $(\Lambda \omega \Lambda^{-1} a)_\mu P^\mu$ is transformed into ...