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.

Filter by
Sorted by
Tagged with
2
votes
0answers
134 views

How can I convert an action in terms of differential forms to tensors?

I have an action of a gravitational theory wich is in terms of differential forms. Now, I need to transform this action (including wedge product and exterior derivative of tetrad and metric ) to an ...
2
votes
2answers
220 views

Matrix multiplication and tensorial summation convention

I'm reading this introduction to tensors: https://arxiv.org/abs/math/0403252, specifically rules concerning summation convention (ref. page 13): Rule 1. In correctly written tensorial formulas free ...
2
votes
1answer
1k views

How does a vector field transform under an infinitesimal coordinate transformation?

If I have a vector $X^{\mu}(x)$, and then I consider an infinitesimal coordinate transformation of the form $x^{\mu} \to x^{\mu} + v^{\mu}(x)$, then how does my vector $X^{\mu}(x)$ transform? From ...
2
votes
1answer
421 views

relativistic addition of velocities using tensor notation? [closed]

I know the way of deriving the formula using usual lorentz transformation formulas,,but is there a way out of deriving it using 4-vector notation??please help
1
vote
0answers
392 views

Contraction of Kronecker delta = 4 [duplicate]

This suggests, as a shortcut notation, the concept of lowering indices; from any vector we can construct a (0, 1) tensor defined by contraction with the metric: $$A_\nu ≡ g_{\mu\nu}A^\mu$$ so that ...
1
vote
0answers
115 views

Equivalence of simple formulations of qubit entanglement

I'm reading some very elementary treatments of quantum computation and am unsure about the correspondence among "definitions" of qubit entanglement. One definition states that (1) the bits of a two-...
1
vote
2answers
457 views

What exactly is $T_{\mu\nu}$?

Continuous matter is described in special relativity by the matter tensor which is the so-called stress-energy-momentum tensor. I am finding a difficulty understanding how a tensorial tool (...
1
vote
1answer
198 views

Jacobi equation in the book The Large scale structure of space-time

On pp. 79, it is obvious that equation (4.2) \begin{equation} \frac{D}{\partial s}Z^a = {V^a}_{;\ b}Z^b \end{equation} holds, where $Z$ is the deviation vector and $V$ is the unit tangent vector along ...
1
vote
2answers
635 views

Understanding Tensor-operations, covariance, contravariance, … in the context of Special Relativity

I'm currently learning about special relativity but I'm having a really hard time grasping the Tensor-operations. Let's take the Minkowski scalar product of 2 four-vectors: $$\pmb U . \pmb V = U^0V^...
1
vote
0answers
78 views

Derivative with Respect to Symmetric Tensor [duplicate]

If you have a Lorentz tensor $T$ with components $T_{\mu\nu}$, it seems clear that $$ \frac{\partial T_{\mu\nu}}{\partial T_{\rho\sigma}} = \delta^\rho_\mu \delta^\sigma_\nu. \tag{1} $$ However, if $T$...
1
vote
2answers
57 views

Covariant surface vector

On pg 74 of Dalarsson's Tensors, Relativity and Cosmology (The Integral theorems for tensor field chapter), the covariant surface vector was defined as: $$dS_k=\frac{1}{2}\epsilon_{kmn}dx^mdx^n=\frac{...
1
vote
2answers
361 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
3answers
140 views

Intuition behind dual vectors ('Bongs of a bell' does not help)

Similar to the post here (How to visualize the gradient as a one-form?), I'm wondering about an intuition behind dual vectors and differential forms (and the link in that answer to Thorne's notes is ...
1
vote
0answers
149 views

What is the relationship between the formal definition of a tensor and the frequently discussed notion of a “higher order matrix”?

I've been doing some self study on the principles of tensors & manifolds in preparation for a first course in general relativity. I tend to learn better when presented with the full mathematical ...
1
vote
1answer
1k views

Levi-Civita symbol in Euclidean space

Suppose a component of tensor field is described by $B^k=\varepsilon^{kij} \phi_{ij}$. If we define $B^k$ in an Euclidean space then does the rising or lowering of the indices of the Levi-Civita ...
1
vote
3answers
406 views

Can I transform electromagnetic tensors by matrix multiplication?

I know that the eletromagnetic field tensor $F^{\mu\nu}$, can be transfomed to another reference frame by $$F^{\alpha\beta} = \varLambda^{\alpha}_{\mu}\varLambda^{\beta}_{\nu}F^{\mu\nu}$$ Since ...
0
votes
1answer
132 views

Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?

See the bold text for my question. This question regards Dirac's General Theory of Relativity page 13. In his demonstration that the length of a vector is unchanged by parallel displacement Dirac ...
0
votes
3answers
391 views

Are contravariant basis vectors and basis 1-forms identical?

The reason I'm asking this is because I am trying to develop a set of notes from my reading of MTW (and Wrede, Menzel, Bergman, etc.). I represent covariant basis vectors with $\mathfrak{e}_{i}$, ...
0
votes
2answers
538 views

The matrix of the Lorentz transformation is or isn't a tensor?

The matrix of the Lorentz transformation isn't a tensor, because it switches the sign of the non-diagonal components during the inverse transformation, right? So it isn't 'basis independent', but the ...
0
votes
1answer
251 views

Why do we always need quantities to be Lorentz Invariant (LI) in relativity?

In particle physics, for example, we add gauge fields, look for the covariant derivative and so on. All to find the LI form of the Lagrangian. Why do we need the LI form? My impression is that when ...
0
votes
0answers
112 views

What is not a tensor? [duplicate]

I've been taking GR, and all of a sudden I am not sure that I know the necessary and sufficient requirements of the tensor coefficients. This is because the lecturer asked me to prove that that "there ...
0
votes
0answers
72 views

A specific derivation of Yang-Mills equations of motion

I am not happy about the derivation of Yang-Mills equations of motion (YM eom) given here @Prahar https://physics.stackexchange.com/a/312681/42982: @Prahar said: Yang-Mills action is $$ S = \int ...
6
votes
3answers
2k views

Tensor Product of Hilbert spaces

This question is regarding a definition of Tensor product of Hilbert spaces that I found in Wald's book on QFT in curved space time. Let's first get some notation straight. Let $(V,+,*)$ denote a set ...
4
votes
1answer
2k views

Derivation of the volume element (which uses the metric tensor)?

I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be ...
4
votes
1answer
749 views

Invariant tensors in a general representation and their physical meaning

I'm trying to use tensor methods to find invariant elements of representations. Specifically I'm looking at representations of $SU(5)$. I can show that the invariant element in $5\otimes\bar{5}$ (or ...
4
votes
0answers
208 views

Cauchy stress tensor for a spherically symmetric problem [closed]

Given a sperically symmetric problem, I am asked to show that its Cauchy stress tensor, in spherical coordinates will assume the form: $${\overline{\overline{\sigma}}}=\sigma_{RR}(r)\overline{e_r}\...
3
votes
1answer
1k views

Hookes law and objective stress rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
3
votes
3answers
265 views

Rotation in the x-t plane

I am currently studying special relativity using tensors. My lecture notes (which happen to be publicly accessible, see top of page 99) say that the standard configuration can be viewed as a rotation ...
3
votes
1answer
96 views

Difference between these two tensors? (help with index notation)

What is the difference between the these two tensors? $$A_{~~i}^{j} \text{ and } A_{i}^{~~j} $$ In my lecturers notes he states that $A^{~~i}_{j}=(A^T)^{i}_{~~j}$. Why is it this and not $A^{~~i}_{j}=...
3
votes
0answers
205 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
2answers
3k views

Tensor product notation [closed]

In the image there is a tensor product: $$F_{\mu\nu}F^{\mu\nu}=2(B^2-\frac{E^2}{c^2})$$ It's about how this operation on the co- and contravariant field strength tensors can give one of the ...
2
votes
1answer
225 views

If $\Lambda$ isn't a tensor, what is the meaning of $\Lambda ^\mu _{~~~\nu}$ and $\Lambda _{\mu \nu} $ and so on?

Following this question that asserts that $\Lambda$ (the transformation matrix in Lorentz group) is not a tensor, then if $\Lambda^\mu_{~~~\nu}$ is THE Lorentz transformation matrix, what is the ...
2
votes
1answer
332 views

Spin tensor and Lorentz group operator in bispinor case

For infinisesimal bispinor transformations we have $$ \delta \Psi = \frac{1}{2}\omega^{\mu \nu}\eta_{\mu \nu}\Psi , \quad \delta \bar {\Psi} = -\frac{1}{2}\omega^{\mu \nu}\bar {\Psi}\eta_{\mu \nu}, \...
2
votes
2answers
480 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: ...
1
vote
1answer
5k views

Calculating the determinant of a metric tensor

Suppose the line element is $$ds^2 = -A(t,r)^2dt^2+B^2(t,r)dr^2+C^2(t,r)d\theta^2+C^2\sin^2\theta d\phi^2.$$ Since the metric is diagonal, to find the determinant I can multiply the diagonal entries, $...
1
vote
1answer
72 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
3answers
135 views

Functional derivative of metric

To do functional derivative of some actions, we need to know a functional differential of metrics $g_{\mu \nu}(x)$. One of the formulae about that is: $$g_{\mu\nu}\delta g^{\mu\nu} = - g^{\mu\nu} \...
1
vote
1answer
777 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 $R$...
1
vote
3answers
137 views

Doubt about the vacua equations of General Relativity

I'm facing a quite annoying conceptual problem concerning the Einstein Field Equations (EFE) in so called "vacuum". This problem is both physical and mathematical. So, in a elementary point of view, ...
1
vote
3answers
195 views

Learn about tensors for physics

Can I have suggestions for some good book regarding tensors for physics.
1
vote
2answers
131 views

Transferring from vectors and tensors in $(-+++)$ signature to $(+---)$ signature?

Minkowski space with the signature $(+---)$ can be described by $\Bbb{C}^{1,3}$ whilst with the signature $(-+++)$ by $\Bbb{C}^{3,1}$ (I am using $\Bbb{C}$ instead of $\Bbb{R}$ to allow for Wick ...
1
vote
1answer
673 views

Calculating electromagnetic invariant in matrix form

I'm kind of confused. I want to calculate the electromagnetic invariant $I := F^{\mu\nu}F_{\mu\nu} $, but I'm not sure what is the easiest way to do so. So, I was trying to do it in matrix form, i.e. ...
1
vote
2answers
206 views

What does it mean to go from a co-variant vector to a contravariant vector?

In most presentations of general-relativity I see the following statement, We can change from a covariant vector to a contravariant vector by using the metric as follows, ${ A }^{ \mu }={ g }^{ \...
1
vote
1answer
53 views

What exactly do raised indices mean in the context of 2-dimensional tensors?

I was reading Sean Carroll's Introduction to General Relativity On Page 12 there is an equation given for defining the Lorentz group as a collection of $4\times 4$ matrices that satisfy $$ \Lambda^...
1
vote
0answers
327 views

Is potential energy a scalar operator?

If a scalar operator $\hat{S}$ is defined as an operator that is invariant under rotations, i.e $$U^\dagger S U = S,\,\,\,\,\,\,\, U=e^{-i\theta\hat{\mathbf{J}}\cdot{\mathbf{n}}}$$ which is ...
1
vote
1answer
737 views

Lorentz Transformation: Index Notation [duplicate]

I am having some fundamental issues with manipulating Lorentz transformation matrices using index notation. However I'm struggling to pin down what the actual issue is. Hopefully my misunderstanding ...
1
vote
1answer
141 views

Tensor Analysis and SR [duplicate]

I'm looking for a textbook that covers, at least for a large part, special relativity with tensors/a geometrical approach. Most textbooks I have found develop tensors for the purposes of GR; I'd like ...
1
vote
0answers
301 views

Riemann tensors in 3 dimensions [closed]

So in 3 dimensions, Riemann tensor has 6 independent terms. So we can fully describe it in terms of the Ricci tensor. How do I show that $R_{abcd}=T_{ac}g_{bd}+T_{bd}g_{ac}-T_{ad}g_{bc}-T_{bc}g_{ad}...
1
vote
1answer
317 views

Covariant derivative ordering

I was working on a problem involving Bianchi identities, in a particular case I have to take the covariant derivative of the following, which indeed is the Ricci tensor in linearised limit $$r^{\mu}_{\...
1
vote
2answers
363 views

The meaning of covariant but not manifestly covariant

What is the most general meaning of the expression covariant, but not manifestly covariant? Suppose I have a general (local) change of coordinates, $x^{\prime} = f(x)$, on an $(n+1)$-dimensional ...