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
0 votes
0 answers
58 views

Could Spacetime have a more General Geometry than just a Metric Field? [closed]

In standard general relativity the fundamental structure of spacetime is given through the six tuple $({M},\mathcal{O},\mathcal{A},{g},\nabla, {W}[\Phi,{g}])$, where ${M}$ is a smooth four-dimensional ...
user avatar
0 votes
1 answer
27 views

Contravariant Vector Component Transformation from Polar to Cartesian

I am new to tensors and I have just learned that the contravarient components of a vector transforms in the following way (using Einstein summation convention) $$A^{'i}=\frac {\partial x^{'i}}{\...
user avatar
2 votes
0 answers
51 views

Spherical harmonics expansion: from scalars to tensors

It is well known that a scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any ...
user avatar
  • 2,885
1 vote
1 answer
45 views

How do I expand: $\langle u , \nabla\rangle u$?

I'm studying the Landau (VI) and when he "introduces" the material derivative (he is building up the continuity equation), something like this appears: $(u,\operatorname{grad})u$ (sometimes ...
user avatar
4 votes
1 answer
91 views

Generalizing Fermi-Walker Derivative/Transport to General Vector Bundles

The usual definition of the Fermi derivative (eg as given in Hawking and Ellis) is to consider a Lorentzian manifold $(M,g)$ and a unit timelike curve $\gamma$, a smooth vector field $X$ along $\gamma$...
user avatar
0 votes
1 answer
41 views

Riemann curvature tensor in an inertial frame

My understanding is that the mathematical definition of an inertial frame at $x_0$ is a choice of coordinates s.t: $g_{\mu\nu}(x_0) = \eta_{\mu\nu}(x_0)$ $\partial_\rho g_{\mu\nu}(x_0) = 0$ I've ...
user avatar
2 votes
1 answer
40 views

From the point of view of physics, why is it useful to know the irreps of rotation group?

In 3D, the rank two tensorial physical quantities, for example, the electric susceptibility, the conductivity, the stress tensor etc, are in general, not irreducible representations i.e. neither ...
user avatar
0 votes
2 answers
61 views

Are tensors constructed such that one forms "act" on some complex vector field?

I have some confusion understanding the motivation in constructing tensors (or tensor fields). On a differentiable manifold $\mathcal{M}$ consider a vector field $X$. At any point $p\in \mathcal{M}$, ...
user avatar
  • 582
1 vote
0 answers
60 views

Trouble with the Ricci tensor formula giving alternate answer than the answer given by the computer

Suppose we have a metric $g_{\mu\nu}$ and want to find the Ricci tensor components. The formula for the Ricci tensor is $$R_{\mu\nu}=\partial_{\alpha}\Gamma^{\alpha}_{\mu\nu}-\partial_{\nu}\Gamma^{\...
user avatar
  • 313
2 votes
2 answers
135 views

Expressing Maxwell's equations in tensor form using Electromagnetic field strength tensor [closed]

I have yet another derivation question from Carroll's General Relativity textbook. Given the electromagnetic field strength tensor is of the form: $$ F_{\mu\upsilon} = \left( \begin{matrix} 0 & -...
user avatar
  • 105
2 votes
1 answer
74 views

Expressing Maxwell's equations in tensor notation

I've been teaching myself relativity by reading Sean Carroll's intro to General Relativity textbook, and in the first chapter he discusses special relativity and introduces the concept of tensors, ...
user avatar
  • 105
0 votes
0 answers
13 views

Generalization of the Impact Depth Equation

Newtons Impact Depth Equation. $L = l *\dfrac{p1} {p2}$ $L$ is the impact depth $l$ is the length of the projectile $p1$ is the density of the projectile $p2$ is the density of the Target has many ...
user avatar
  • 101
0 votes
1 answer
53 views

Kerr Solution metric ansatz for EFEs

The Schwarzschild metric ansatz is given by $$ds^2=-A(r)dt^2+B(r)dr^2+r^2d \Omega^2$$ where upon applying the Einstein Field Equations $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we obtain the normal ...
user avatar
  • 313
0 votes
2 answers
43 views

Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$

We can show that the contraction of some arbitrary $2\times2$ matrix $A_{\mu}^{\ \lambda}$ with the Levi-Civita symbol is once again antisymmetric \begin{align*} \varepsilon^{\mu\nu}A_\mu^{\ \lambda} ...
user avatar
  • 1,448
0 votes
1 answer
66 views

Show that the contraction of a covector and a vector is Lorentz invariant

I just got Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity a couple of weeks ago, and I have resolved to go through the entire book. In the first chapter, he prompts the ...
user avatar
  • 105
0 votes
0 answers
21 views

Reference which explains Penrose Diagramatic notation in simple way

In both Penrose's Road to reality and Spinor's and space-time, the following notation is shown: With a lot other examples for doing calculation with Tensors. Could someone give another reference ...
user avatar
5 votes
2 answers
196 views

What is the idea behind 2-spinor calculus?

In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
user avatar
0 votes
1 answer
78 views

A particular contraction of Levi-Civita symbols and tetrads

Consider a four-dimensional spacetime. Consider the following contraction between Levi-Civita symbols and tetrads $$\epsilon_{\alpha \beta i j}\,{\epsilon^{ij}}_k\, e^\alpha\!\wedge e^\beta\!\wedge e^...
user avatar
0 votes
0 answers
9 views

About Volterra's displacement equation on dislocation: cancellation of a surface integral on stress

In the theory of dislocations, the displacement induced by a dislocation in an anisotropic solid media can be expressed by Volterra's displacement equation as follows: $$ u_j(\mathbf{x}) = \int\int\...
user avatar
  • 11
0 votes
1 answer
35 views

Interpreting stress at the ends of a bar

Consider a bar loaded in tension by distributed loads applied on its ends as shown in the figure. The stress at any cross section of this bar will be $$\sigma = \frac{P}{A}$$ From what I know about ...
user avatar
0 votes
2 answers
51 views

How to calculate the rank of a tensor?

I was studying a little of tensorial calculus and came up with this problem: Given a tensor with a rank of (0,2), $T_{\alpha\beta}$. Calculate the rank of this tensor $T_{\alpha\beta}T_{\gamma}^{\...
user avatar
0 votes
1 answer
40 views

Components of the fully contravariant Kronecker Delta in Schwarzschild Metric

I thought the kronecker delta $\delta^{\mu\nu}$ should always be of value $1$ if both indices are equal and $0$ if they are different. However it seems that the delta has components different from $1$ ...
user avatar
  • 327
4 votes
3 answers
194 views

Ricci Identity with Torsion Proof

In the notes I'm using for General Relativity, the author begins their proof of the Ricci identity with torsion by writing $$\nabla_{[\mu}\nabla_{\nu]}Z^{\sigma}=\partial_{[\mu}(\nabla_{\nu]}Z^{\sigma}...
user avatar
2 votes
0 answers
74 views

Choosing diffeomorphisms for the pullback metric in the Weak Field approximation

In the weak field approximation of the EFEs $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we take $g_{\mu\nu}\approx \eta_{\mu\nu}+h_{\mu\nu}$. The $\eta_{\mu\nu}$ term is just the flat space Minkowski metric and $...
user avatar
  • 313
0 votes
0 answers
35 views

Symmetries of Riemann tensor

Is there a way to show that the symmetries of Riemann tensor are preserved even if the indices are raised or lowered in general. I know how to do it individually for each symmetry but am not sure how ...
user avatar
4 votes
1 answer
86 views

Argument of a scalar function to be invariant under Lorentz transformations

I'm trying to prove that a Lorentz scalar object $\rho(k)$ which is a function of a cuadri-vector $k^{\mu}$ can only have a $k^2$ dependency in the argument. I can imagine that this object has to ...
user avatar
1 vote
1 answer
43 views

Preservation of symmetries of Tensors under lowering and raising indices

How do you go about showing that symmetry properties of tensors are preserved during lowering and raising indices in a metric space? I know how do do it for individual tensors with given symmetries ...
user avatar
1 vote
0 answers
29 views

Correlation function of 4-currents on a general QFT

Given $V^\mu(x)$ a 4-current of a general unitary, Poincaré invariant QFT, I need to show that the correlation function: $$iC_{\mu\nu}(x-y) = \langle {\tilde{0}}|T\left[V_\mu(x)V_{\nu}^\dagger(y)\...
user avatar
  • 21
5 votes
1 answer
77 views

Why is a Lorentzian metric still Lorentzian after a general coordinate transformation?

In my GR course, we define a lorentzian metric $g_{\mu\nu}(x)$ as a symmetric $(0,2)$ tensor field having 3 positive and 1 negative eigenvalue. Now given a general coordinate transformation described ...
user avatar
  • 149
0 votes
1 answer
66 views

Software recommendation for a tensor calculation

What is the best software/package to calculate $$2R_{\alpha\mu\beta\nu}R^{\mu\nu}-\nabla_\alpha\nabla_\beta R + \Box R_{\alpha\beta}-\frac12g_{\alpha\beta}\Big(R_{\mu\nu}R^{\mu\nu}-\Box R\Big)$$ for a ...
0 votes
2 answers
41 views

Maxwell's stress tensor and pressure

I am studying Electromagnetism from Griffiths and in the book it is stated that diagonal elements of Maxwell's tensor represent pressure. I want to calculate pressure on the wirings of an infinitely ...
user avatar
1 vote
0 answers
28 views

What are the significances of the contravariant and covariant four-momentum and their corresponding four-forces in General Relativity?

In General Relativity, the four-momentum is defined as $$p^r=\frac{dr}{d\tau}$$ where $\tau$ is the proper time. Here, we find that the contravariant index is used. However, I am confused on the ...
user avatar
  • 1,612
2 votes
1 answer
73 views

How is it that adding a random field to the partial derivative results in a tensorial operation?

We know that the partial derivative of a tensor is not a tensor. But how is this problem fixed by adding to the partial derivatives, a field of Christoffel symbols? Christoffel symbols are a ...
user avatar
  • 459
0 votes
1 answer
70 views

Inverse of a metric under variation

Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as: $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as: $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the ...
user avatar
  • 327
1 vote
1 answer
63 views

General relativity algebraic manipulation help

I'm having difficulty understanding a lot of the fundamentals behind the algebra of general relativity. I have a specific question I'm trying to understand but any pointers about how any of it works ...
user avatar
0 votes
1 answer
61 views

Tensor products in Howard Georgi's "Lie Algebras in Particle Physics"

My question is regarding eq.(3.39) in the second edition of Georgi's book (for those who have the book:)). The section deals with tensor product states where the states comprising the product ...
user avatar
0 votes
0 answers
26 views

Vanishing of axial torsion vector

Suppose that one is working in some theory of gravity based on a torsionfull connection, e.g. Einstein-Cartan theory. I know that the torsion tensor can be decomposed into the tentor, trator and ...
user avatar
  • 23
0 votes
1 answer
30 views

Relationship between three dimensional spinorial tensor

In a homework assignment, I am asked to show that the components of three dimensional spinorial tensor obey the following relationship $$\Psi_{12}=-\Psi_{1}^{1}=-\Psi^{21}, \Psi_{11}=\Psi_{1}^{2}=\Psi^...
user avatar
  • 19
2 votes
0 answers
46 views

Which of these is the logical way to establish tensors on a manifold?

You start by defining a vector space at each point of the manifold. The defining feature being the vector transformation law under change of co-ordinates. Then you define dual vectors as linear ...
user avatar
  • 917
0 votes
1 answer
41 views

Product of Covariant and Contravariant rank 2 Tensors

By the definition, Contravariant tensor transforms like $ (A')^{ij}=\sum_{k,l}{\frac{\partial (x')^i}{\partial x^k}\frac{\partial (x')^j}{\partial x^l}A^{kl}}$ Covariant tensor transforms like $ (A')...
user avatar
1 vote
1 answer
37 views

Matrix index notation and Einstein summation

Can I ask why these two expressions are not equal? $$\begin{align}A_{ij}V^j&\ne V^kA_{ki}\\A_{ij}B^{ij}&\ne A^i{}_jB^j{}_i\end{align}$$
user avatar
0 votes
1 answer
56 views

Transformation matrix from Cartesian to Polar of a Covariant vector

Suppose that there exists some contravariant 2-vector $V^{\alpha}$ . Now, let $V^{\alpha}$ be an element of the Cartesian coordinate system (x,y) such that it can be transformed into polar coordinates ...
user avatar
  • 313
2 votes
1 answer
89 views

Metric tensor from hyperbolic PDE

It is clear that when a differential equation is composed of the second partial derivatives only, it could be written in the form $$ g^{\mu\nu} \frac{\partial^2 \psi}{\partial x^\mu \partial x^\nu} = ...
user avatar
  • 2,677
2 votes
1 answer
73 views

How to find covariant derivative of a contravariant tensor?

Let I have a contravariant tensor $A^\alpha$, I want to find covariant derivative of the contravariant tensor, From the transformation of the contravariant tensor ($A^\alpha=\partial_\gamma x^\alpha A^...
user avatar
1 vote
2 answers
48 views

How to find inertia tensor of a circular ring from angular momentum and velocity?

Consider a thin circular ring with radius $R$ and axis of rotation as shown in the figure. If $\vec{L}$ denotes angular momentum and $\vec{w}$ is the angular velocity then $$\vec{L}=\begin{bmatrix} I_{...
user avatar
  • 390
0 votes
1 answer
79 views

What's exactly is moment of inertia?

I know that angular momentum can be expressed in terms of moment of inertia tensor as follows, $$\vec{L}= I_{\text{tensor}}\vec{w}$$ Where $I_{\text{tensor}}$ is tensor for moment of inertia. It can ...
user avatar
  • 390
0 votes
1 answer
43 views

How to write the dot product of 3rd order tensor (piezoelectric constant) with 1st order tensor (Electric field vector) in the matrix form?

In the Stress-Charge form of Piezoelectric constitutive equations: $T$ $=$ $c$ : $S$ $-$ $(e^T) . E$ ( The symbols are explained in the picture below) My question is, why do we take the transpose of $...
user avatar
2 votes
1 answer
47 views

Why is viscous stress a tensorial quantity?

In an incompressible fluid, the viscous stress (in Cartesian) is defined by \begin{align} \tau_{ij} = \eta(\partial_i v_j + \partial_j v_i) \end{align} for dynamic viscosity $\eta$ and velocity field $...
user avatar
  • 2,670
0 votes
1 answer
113 views

Ways to represent the metric tensor using a vector field

I was wondering if there were any ways of representing the metric tensor with a vector or scalar field and started calculating some potential ways. I recently stumbled across the equation $$\widetilde{...
user avatar
  • 741
-4 votes
1 answer
102 views

What do the symbols $\mu$ and $\nu$ mean in General Relativity?

I'm not an expert on general relativity, and below the tensors in the Einstein Field Equations, there are two confusing symbols: $\mu$ and $\nu$ below them. What do they mean? Any equations are ...
user avatar

1
2 3 4 5
44