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.

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Transformation Properties of Connection Coefficients

This question is about pages 95 and 96 of Carroll's book: Spacetime and Geometry. We have the formula for the covariant derivate: $$\nabla _\mu V^\nu=\partial _\mu V^\nu + \Gamma _{\mu\lambda}^\nu V^\...
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Importance of Tracelessness of Tensor?

What makes the trace-free tensor (or part of it) so important? As in trace-free Ricci tensor or Weyl tensor.
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Raising and lowering indices: is it a convention? [duplicate]

We can raise and lower indices of any tensor with a non-zero rank by applying the metric tensor with indices properly located. My question is: why is the metric tensor the tool we use for such ...
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Lorentz invariance of the Lorentz force law

I'm self-studying Friedman and Susskind's book Special Relativity and Classical Field Theory. The following question popped up while reading section 6.3.4 Lorentz Invariant Equations. In this Lecture, ...
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Summation of a tensor product of two state functions

In the equation, where the $\vec{\Psi}$'s are particle states, $$ \sum_{l,m} C_{lm}\Big\{\Big[\frac{n^{2}}{c^{2}}\dfrac{\partial^2}{\partial t^2}\vec{\Psi}_l(\vec{r}_{1},t)-\nabla^{2}_{1}\vec{\Psi}_l(\...
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Raising and lowering indices in linearized gravity

In linearized general relativity indices are raised and lowerd by contracion with the flat space metric tensor $\eta_{\mu \nu}$. I don't really understand why we can do that. In the book gravitational ...
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How to be sure that a law is invariant under Lorentz's Transformation?

For starters let's talk about Maxwell's Equations; we know that Maxwell's Equations are invariant under Lorentz's Transformation, after all this is why all the relativity business got started. To ...
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Minkowski Inner Product's Shenanigans

In the context of Special Relativity, so in flat spacetime, and with the metric tensor $g_{\mu \nu}$ chosen with the signature: $(+,-,-,-)$, lets consider the following four vectors: $$x=(x_1,x_2,x_3,...
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If $ \partial_a F_{bc} + \partial_b F_{ca} + \partial_c F_{ab} = 0 $ then $ F_{ab} $ is the curl of a 4-vector

Any skew-symmetric tensor $ F_{\alpha\beta} $ which is the curl of a 4-vector $A_\mu$, that is each tensor having the form $ F_{\alpha\beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha $, ...
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How does the bi-vector $\mathbf{F}=\mathbf{E}+i\mathbf{B}$ generalize to curved space?

How does $\mathbf{F}=\mathbf{E}+i\mathbf{B}$ generalize to curved space? (where $\mathbf{F}$ is the bivector of electromagnetism). Here is what I am struggling with: On the one hand, I can expand $\...
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Proof of the Piola Transform

As I understand it, the relationship between the second order tensor $\bf T$ over a reference configuration and the same tensor in a deformed configuration $\bf T^\prime$ is given defined as follows: $...
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Inner product and projection in pseudo-riemannian manifolds

My question is about how to properly compute the projection of a tensor in a given direction using inner product product in a pseudo-riemannian manifold, since inner product isn't defined positive. ...
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Is Tensor in “TensorFlow” inapposite?

There is a library called TensorFlow largely used in machine learning/deep learning. They give this description on what a tensor is. I'm not prepared in physics so well to debate in this but in my ...
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Covariant derivate of constant vector

We know that $$\frac{dv}{d t}=\frac{d\left(v^{i} e_i\right)}{d t}=\partial_{j} v^{i} v^{j} e_{i}+v^{i} v^{j} \partial_{j} e_{i}$$ As $\partial_{j} e_{i}$ is another vector we can expand it in the same ...
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A doubt on calculation of tetrad basis vectos of a non-diagonal metric tensor

First of all, I will ask the patience of community because this is a "need-an-explicit-calculation answer". Pure abstract considerations will not, I think, help me very much. So, I would ...
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Interchange of indices [closed]

How to get the second equation, please? My result is wrong - I still have $(N-1)$. Thank you
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How can $F_{\mu\nu}F^{\mu\nu} = 2(B^2-E^2)$ be proved? [closed]

How does $F_{\mu\nu}F^{\mu\nu} = 2(B^2-E^2)$? $$ F_{\mu\nu}=\pmatrix{ 0&E_x&E_y&E_z\\ -E_x&0&-B_z&B_y\\ -E_y&B_z&0&-B_x\\ -E_z&-B_y&B_x&0 } $$ $$ F^{\mu\...
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General Relativity: change of coordinates in tangent space

For starters, in the context of the tangent space of a manifold in GR, we can derive that: $$g'_{\mu \nu}=\frac{\partial x^\rho}{\partial x'^\mu}\frac{\partial x^\sigma}{\partial x'^\nu}g_{\rho \sigma}...
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What phenomena in physics cannot be expressed in terms of differential forms? [duplicate]

It seems all phenomena in physics can be expressed in terms of differential forms. There are textbooks dedicated to formulating the main theories of physics in terms of differential forms. The more ...
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Torsion tensor definition doubt

[Ref. Core Principles of Special and General Relativity by Luscombe, page 246] Let's say we have any two covariant derivative operators $\nabla$ and $\nabla'$. Then there exists a tensor $C^{\alpha}_{\...
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Simple derivation of the Maxwell's equations from the Electromagnetic Tensor

Lets start by considering the electromagnetic tensor $F^{\mu \nu}$: $$F^{\mu \nu}=\begin{bmatrix}0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & ...
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Why projection operator is not equal to zero, as we can write 1st term as 2nd term or vice versa via raising or lowering index with metric?

$$k^2g^{\mu\nu}-k^\mu k^\nu=k^2P^{\mu\nu}(k)$$ Here 1st term can be written as 2nd term via breaking square term and then raising index.
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Simple four-vector partial derivatives

I am a beginner in tensor calculus, and am finding it difficult finding the result to what I assume are basic identities. I am trying to compute the following : $$ \partial_{\mu} x_{\nu} \quad and \...
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How to prove $\operatorname{div} \mathbf{A}=\operatorname{Div} \mathbf{A} \mathbf{F}^{-\mathrm{T}}$?

I recently focus on solid mechanics and I am reading Nonlinear Solid Mechanics A Continuum Approach for Engineering by Gerhard A. Holzapfel. However, I was confused by a mathematical formula eq(2.49), ...
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Lorentz Transformation of a Tensor

If I have the electromagnetic field tensor, then, under a Lorentz transformation: $$F^{'}_{\mu\nu} = \Lambda_{\mu}^{\alpha} \Lambda_{\nu}^{\beta} F_{\alpha\beta} $$ I know that the Lorentz matrix is ...
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Covariant formulation of electrodynamics homogenous Maxwell eq

It is know that $$\epsilon{^\mu} {^\nu} {^\rho} {^\sigma} \partial_{\nu} F_{\rho} {_\sigma} = 0$$ How can one deduce from this equation that $$ \partial_{\mu}F_{\nu} {_\lambda} + \partial_{\lambda}F_{\...
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Poisson bracket properties for tensor densities

I am doing some constraint analysis in an extended theory of gravity, and I am confused about Poisson brackets. The standard PB relations are for example $\{ab,c\} = a\{b,c\} + \{a,c\}b$ etc. But I am ...
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Critique on tensor notation

I am studying tensor algebra for an introductory course on General Relativity and I have stumbled upon an ambiguity in tensor notation that I truly dislike. But I am not sure if I am understanding the ...
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How is tensor calculus applied to Einstein's field equations? [closed]

What is the relation between tensor calculus and Einstein's field equations? or What is the contribution of tensor calculus to Einstein’s field equations?
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Matrix “dimensional analysis” of Lagrangians in QFT

Since the important things in the QFT Lagrangian are vectors and matrices, I wanted to do a "matrix dimensional analysis" of each term. The electromagnetic Lagrangian (ignoring all constants ...
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Why can I use the Covariant Derivative in the Lie Derivative?

The Lie derivative is the change in the components of a tensor under an infinitesimal diffeomorphism. It seems that this definition does not depend on the metric: $$ \mathcal{L}_X T^{\mu_1...\mu_p}_{\...
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How do slanted indices work in special relativity? [duplicate]

What is the difference between $T^{\mu}{}_{\nu}$ and $T_{\nu}{}^{\mu}$ where $T$ is a tensor?
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Doubt on Tetrads, Energy-momentum tensors and Einstein's equations

Given, for instance, the perfect fluid energy-momentum tensor: $$T_{\mu\nu} = (\rho+p)u_{\mu}u_{\nu} - pg_{\mu\nu}\tag{1}$$ We can put (due to diagonalization procedure) into the diagonal for as: $$...
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Differential forms or Tensors for modern theoretical physics?

There many proponents to teaching differential forms and others teach with tensors. This is true for both mathematics and physics education. It seems mathematicians prefer to teach differential ...
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Physically measure the covariant and contravariant components of a vector?

I'm just wondering if there is a way to physically measure the covariant and contravariant components of a vector without prior knowledge of the metric. Suppose I have a speedometer of some sort to ...
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What is a coordinate-free formulation of deformation theory?

For example how are stress, strain and shear tensors described invariantly, without any coordinates, purely in a geometric manner? A formulation that avoids indices coordinates and matrices, even in ...
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Commutator of derivatives with torsion

I am currently looking at "Physical aspect of space-time torsion" by IL Shapiro. There in eq. 2.10, it mentions that in space-time with torsion the commutator of covariant derivatives acting on the ...
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1answer
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Constructing the supertraceless portion of a connection over a supermanifold

Consider a tensor, $T$ of rank $(r,s)$ over a supermanifold, $M$ and take the supertrace over its indices $p$ and $q$ (DeWitt, p. 77, eq. 2.4.33): $$(-1)^{a_q(1+a_{p+1}+...+a_{q-1})}T^{a_1...a_{p-1}...
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How to calculate the number of independent components/degrees of freedom for symmetric tensors? [closed]

I was studying about the cosmological perturbation theory and came across this: ""Being symmetric, the two perturbed tensors contain ten degrees of freedom each, describing different aspects of ...
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Electromagnetic four-potential manipulation

I was looking at my special relativity notes and when covering EM, I was wondering if the following is true? Namely, given the four-potential $A_{\mu}$, by definition it is that: $F_{\mu \nu} = -A_{\...
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Inertia tensor in non-cartesian coordinates

given a rigid body $K$, I always had seen the formula \begin{equation} I_{ij} = \int_K[\mathbf{x}^2\delta_{ij} - x_ix_j]\rho(\mathbf{x})\mathrm{d}^3\mathbf{x} \end{equation} for the inertia tensor. ...
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Parameter Question In General Relativity

I am a math student taking a course in General Relativity. I haven't taken many physics/applied maths courses before, so I am not sure if I can describe this question well, but I am slightly confused ...
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Doubt on the precise definition of a general stationary rotating metric: the metric coefficients have which form?

Considering the following metric tensor $[1]$, with signature $(-,+,+,+)$, coordinates $(x^{0},x^{1},x^{2},x^{3})\equiv (t,r,\theta,\phi)$ and $c=1$: $$ds^{2} = g_{00}dt^{2} + g_{11}dr^{2} + g_{22}d\...
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Commutator of covariant derivatives acting on a vector density

Let $\mathfrak n^\alpha$ be a vector density of weight 1. Define the covariant derivative $\nabla$ such that under a coordinate transformation $x^\mu \to \bar x^\mu$ $$ \nabla_\rho \mathfrak n^\alpha ...
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Deriving the general formula of $\mathcal{\epsilon_{ijk}} \mathcal{\epsilon^{ijk}}$

As stated in the title, with $i,j,k=1,...,N$. I expanded $\mathcal{\epsilon}_{ijk}\mathcal{\epsilon}^{ijk}$ as follows: $$\mathcal{\epsilon}_{ijk}\mathcal{\epsilon}^{ijk}=\underbrace{\mathcal{\...
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How do I tensor differentiate a factor without tensors?

How do I tensor differentiate a factor without tensor, such as: $$\partial_\mu e^{i\Lambda(x)}\tag{1}$$ Should it be zero or should I differentiate it twice changing the order of the tensors follows:...
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Efficient method to evaluate the Christoffel symbols and Riemann tensor in Bondi-Sachs coordinates

In General Relativity we may employ the so-called Bondi-Sachs coordinates $(u,r,x^A)$ adapted to a null foliation. The level sets of $u$ are null hypersurfaces and $(r,x^A)$ are coordinates on the ...
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Tensor derivative in special relativity and fluid mechanics

I’m working through Special Relativity by V. Faraoni, and am puzzled by something in his chapters on tensors. He tells us that the partial derivative of a tensor field, e.g. $T_{\alpha, \gamma}$, is ...
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Doubt on proper time explicit integration

I have a doubt on explicit calculation of proper time. Considering that the metric is given by: $$ds^{2} = -Adt^{2} + B^{-1}dr^{2}+Cd\Omega^{2} -2Ddtd\phi \tag{1}$$ where $d\Omega^{2}$ is the solid ...
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How to express a rank-2 tensor as a spherical tensor?

A common example how to write a rank-2 tensor in the spherical basis is an outer product of two vectors, $$ T_{ij} = a_i b_j $$ such that $$ T_{ij} = \frac{\textbf{a}\cdot\textbf{b}}{3}\delta_{ij} + ...

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