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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Inner product of Riemannian metric tensor and Minkowski metric tensor
I am a little bit confused about whether the inner product between the Riemannian metric tensor and Minkowski metric tensor is equal to the Kronecker delta function:
$${\eta}^{{\mu}{\nu}}{g}_{{\mu}{\...
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Dimensional regularisation and Wick theorem [duplicate]
Consider an integral:
$$I^{ij}=\int\frac{d^d\textbf{p}}{(2\pi)^d}p^i p^j f(\textbf{p}^2).$$
How can we show that this is equal to:
$$I^{ij}=\frac{\delta^{ij}}{d}\int\frac{d^d\textbf{p}}{(2\pi)^d}\...
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What is the connection between the Ricci tensor and the metric flatness?
Actually, this question was answered by Lawrence B. Crowell,
but I would like to explore this topic further.
Can anyone give me please references on where I can find it?
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Variation vs. derivative wrt a symmetric and traceless tensor
Consider a Lagrangian, $L$, which is a function of, as well as other fields $\psi_i$, a traceless and symmetric tensor denoted by $f^{uv}$, so that $L=L(f^{uv})$, the associated action is $\int L(f^{...
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Question on transformation law of spinors and the law $\xi^{\mu '} = D(L)^{\mu '}_{\nu}\xi^{\nu}$ where $D(L)$ is a representation of Lorentz group
In the reference $[1]$, the author presents the tensor quantities via its transformation laws. I'm pretty confortable with Pseudo-Riemannian geometry and tensors. But, when group theory enters the ...
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What do the different elements in strain tensors tell us?
I'm working with strain tensors of all sorts at the moment, and I think I've understood how they're derived. However, I'd like to get more intuition of what they're actually telling us.
More ...
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Transformation of basis vector in Kruskal coordinate
For Schwarzschild solution:
$\mathrm{d} s^2=\left(1-\frac{2m}{r}\right)dt^2 - \left(1-\frac{2m}{r}\right)^{-1}\mathrm{d} r^2-r^2(\mathrm{d}\theta^2+sin^2\theta \mathrm{d}\phi^2),$
Introduce tortoise ...
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Why is rank-3 tensor in 3D with two antisymmetric indices equivalent to rank-2 tensor?
I'd like to know how many irreducible representations of $SO(n)$ when it comes to rank 3 tensor. Here $n=3$. Among the rank 3 tensor components, there might be antisymmetric parts and symmetric parts ...
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Problem with understanding contravariant component transform
I am reading Susskinds book General Relativity: The theoretical minimum and I got a bit stuck on the transformation rule of contravariant components. The book defines the components of a vector $(V^{’}...
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Breaking product of three vectors into symmetric and anti-symmetric vectors [closed]
Let's consider we have three arbitrary vectors A, B and C. We have the quantity $A_{\mu}B_{\nu}C_{\rho}$. Is it possible to break the above quantity into sum of symmetric and anti-symmetric vectors in ...
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Confusion regarding Riemann Tensor and Ricci Tensor
Ricci Tensor is the contraction of the Riemann Tensor. Even if all the components of the Ricci Tensor is zero, that doesn't mean that the spacetime is flat. If all the components of the Riemann Tensor ...
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How to express symmetry of a mixed (1,1) tensor with upper and lower index?
In the context of general relativity, I am working with the energy-momentum tensor $T$, which is a rank-2 tensor whose components are usually denoted by $T^\mu_{\ \ \ \ \nu}$. However, I am unsure of ...
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Dimension of a vector space of all tensors of rank $(k,l)$ in 4D
Dual vector space is the set of all linear functionals defined on a given vector space. The vector space and dual vector space is isomorphic and hence have the same dimension. A rank $(k,l)$ tensor is ...
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Taking the derivative of the integral of a tensor [migrated]
This is a bit of a technical question. Say I had a tensor of some arbitrary rank, for the sake of this example I'll use a vector $V_i$. If I were to take the following integral:
$$\int V_i dx^i$$
And ...
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Is there a Lorentz invariant electromagnetic quadrupole moment tensor?
I'm familiar with the electric and magnetic quadrupole moment tensors. However, I'm bothered that these objects are tensors only in the sense of spatial rotations. After all, Maxwell's equations and ...
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How to derive the energy tensor invariantly?
On a (pseudo-)Riemannian manifold $(M, g)$ I can define the following action for any $\phi \in C ^{\infty}(M)$:
$$
\mathcal{S}(\phi) = \int_M g(\text{grad }\phi, \text{grad }\phi)
\mathrm{d} V.
$$
...
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Notational meaning of $\nabla_{\lambda}V^{\rho}$ and $\nabla_{\mu}\nabla_{\nu}V^{\rho}$
This question is related to Reconciling different expressions for Riemann curvature tensor, but it's different since it asks for some notational clarification arising out of calculations I did. To not ...
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Bilinear covariants of Dirac field
In the book "Advanced quantum mechanics" by Sakurai there is a section (3.5) about bilinear covariants, however i can't really find a definition of these objects, neither in the book nor ...
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Calculations with tensors give two different results from seemingly equivalent paths:
$\require{cancel}$
I want to calculate some operators in the context of Metric-Affine-Gravity: specifically spinors. I will work in tangent space, where all greek ($\mu,\nu,..$) indices are cast into ...
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Exterior derivatives Leibniz rule
I want to prove Sean Carroll's "spacetime and geometry"'s eq.(2.78):
$$
\mathrm{d}(\omega \wedge \eta)=(\mathrm{d} \omega) \wedge \eta+(-1)^p \omega \wedge(\mathrm{d} \eta) \tag{2.78}
$$
...
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Contraction of Lorentz indices in gluon propagator of QCD
In QED, the photon propagator has a factor of $g^{\mu \nu}$, and both $\mu$ and $\nu$ contract with the $\gamma$ matrix indices, which come from the fermion antifermion photon vertices on either end ...
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Lie derivative in terms of covariant derivative and the symmetry of Christoffel symbols
I want to verify that if a manifold is torsion-free with a metric compatible derivative operator $\nabla_a$, the Lie derivative of a vector $W^a$ along $V^a$ can be written as
$$L_V W^a = V^\nu\nabla_\...
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Divergence of the Oseen tensor for the Zimm model?
While studying the Zimm model on the Doi & Edwards book "The Theory of Polymer Dynamics", I faced equation 4.41 which states that:
$$
\frac{\partial}{\partial\boldsymbol{R}_j}\cdot\...
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What is the connection between a mathematician and physicist's definition of a tensor?
I study mathematics but I have a deep interest in physics as well. I have taken a course in smooth manifolds where a tensor is defined as an alternating multilinear function. Recently I have learned ...
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Reconciling different expressions for Riemann curvature tensor
[Note: I think this question is more suited to Physics SE rather than Math, since it refers to Carroll's notes and some equations might have inherent Physics-related assumptions
EDIT: This question ...
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Identities regarding covariant derivative with tetrads
While doing some index gymnastics in tetradic general relativity, I encountered these expressions:
\begin{align}
\qquad\qquad
\eta^{ab} \nabla_{e_a}\kern-0.25em \nabla_{e_b} e_c
\,,
\qquad\...
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How many independent equations are contained in $R_{rsmn}=0$ in consideration of the Bianchi identity?
In $d$ dimensions, how many independent equations are contained in $R_{rsmn}=0$ in consideration of the Bianchi identity $\nabla_{[a}R_{bc]de}=0$?
This discussion reveals the independent equations ...
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Is there a physical interpretation of why Christoffel symbols do not transform like a tensor? [duplicate]
I understand mathematically why they don’t, but I was hoping someone could provide a physical interpretation to this. Is there a physical consequence of this fact?
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Can either the covariant or the contravariant version of a physical tensor be more fundamental?
This question may be too subjective, but here goes:
Essentially any physically interesting quantity can be represented by a tensor on an inner product space or by a tensor field on a pseudo-Reimannian ...
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How does this derivation of the proper time derivative of a covariant vector work?
Define the operator $\frac{D}{D\tau}$ by its action on an arbitrary contravariant vector $A^\lambda$:
$$\frac{DA^\lambda}{D\tau} = \frac{dA^\lambda}{d\tau} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{d\...
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Why is the Lie derivative of a differential 1-form tensorial?
It says in Appendix B of Sean Carroll's "Spacetime and Geometry" that the Lie derivative of a differential 1-form, defined by
$$
\mathcal{L}_{V} \omega _{\mu} = V^{\nu} \partial _{\nu} \...
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Continuity equation for the conservation of energy from the conservation of the energy-momentum tensor
I am working through the book Cosmology by Daniel Baumann, and in the subsection that covers the continuity equation (part of section 2.3.1 on perfect fluids) the author makes a claim that confuses me....
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Transpose of a bilinear in Einstein notation
In Einstein notation we can take generic 1-vectors $x, y$ and (1,1) tensor $M$.
As we know $x_{\mu}$ represents $x^{T}$, i.e. row vector (a co-vector), while $x^{\mu}$ is a column vector.
So we can ...
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Four-vector and Notation significance [closed]
As the title suggest, this has to do, on the most part, with four vector notation. I have a series of questions, the majority, related to this topic:
1- If we assume a lorentz boost in the x direction ...
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Trace and index manipulation
Imagine that I have a quantity $F_{ab}$ multiplying the stress tensor $T^{ab}$:
\begin{equation}
F_{ab} T^{ab}.
\end{equation}
There is also a metric, say $h_{ab}$. If I want to write the above ...
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Problem with proving the invariance of dot product of two four vectors
I am having a spot of trouble with index manipulation (its not that I am very unfamiliar with this, but I keep losing touch). This is from an electrodynamics course - we're just getting started with 4 ...
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What is the symbol to differentiate between 3D and 4D tensors?
I am writing a computer program and in there I need to differentiate 3D tensors (metric tensor, Riemann tensor, Ricci scalar, Christoffel Symbols, etc.) from 4D ones.
I wanted to write something like $...
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Calculating Conformal Killing Vectors
I'm trying to understand how to calculate conformal killing vectors. The equation is:
$K_{a;b}+K_{b;a}=\frac{2}{n}g_{ab}K^c_{;c}$
Where $;$ means covariant differentiation and $n$ is the number of ...
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Killing field and vector field type
The Killing field condition can be defined as;
$$\mathcal{L}_{X}g=0$$
for a metric $g$ and vector field $X$. In this case does it matter the type ($X^{\alpha}$ or $X_{\alpha}$) of the $X$ used in this ...
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Help with Index Notation in Fourth Order Tensor
I am working on a continuum mechanics problem. I have the following relationship:
$$\frac{\partial\Psi}{\partial \xi_{ik}}=-R_{ikjl}\dot{\xi}_{jl}$$
where $\xi$ is a viscous deformation gradient ...
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Identity with Riemann tensor
Is there a fast way to derive the identity
$$(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha})T_{\gamma\delta}={R_{\alpha\beta\gamma}}^{\lambda}T_{\lambda\delta}+{R_{\alpha\beta\delta}}^{\...
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Not so trivial indeces in isometries of special relativity
I am trying to understand isometries and how to work with tensors.
I know that in special relativity metric transforms as follows
$$
g_{\alpha^{\prime} \beta^{\prime}}=g_{\alpha \beta} \Lambda_{\...
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Inverting the Propagator [closed]
I would like to know how I can invert the following expression:
$$
S^{\mu\nu}=(\!\!\not{p} -m)\eta^{\mu\nu}+\gamma^{\mu}\!\!\!\not{p}\gamma^{\nu}+m\gamma^{\mu}\gamma^{\nu} \ ,
$$
to get $(S^{−1})^{\mu\...
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Is a perturbation of a tensor field a tensor field?
Let say I take some $2$-tensor field $T_{\mu\nu}$ on some pseudo-Riemannian manifold. Now, often, we are interested in its linearization, which means that we take a family of tensor fields $T_{\mu\nu}(...
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How is the full Riemann curvature tensor determined if the Einstein field equations only present the Ricci curvature tensor? [duplicate]
I'm currently learning general relativity following Leonard Susskind's lectures, and I was very surprised by the fact that the components of the full Riemann curvature tensor are relevant even though ...
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Closure of Lie brackets associated to Brown-Henneaux boundary conditions
When we impose Brown-Henneaux boundary conditions to the metric field on AdS$_3$,
\begin{align}
\begin{split}
g_{tt}&=-\frac{r^2}{\ell^2}+\mathcal{O}(r^0)\,,\\
g_{t\phi}&=\mathcal{O}\...
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Covariant divergence and derivatives of coordinate basis vectors
I've been trying to derive the covariant divergence of a vector field and I've ran into 2 problems.
Basically I have $\nabla_aA^a=\partial _{a}A^a+\Gamma^{a}{}_{ab}A^b$, and then I found $\Gamma^{\mu}{...
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answers
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Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?
On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
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Can the Lagrangian density of vacuum Maxwell equation be written into tensor contraction without a basis?
The Lagrangian density of the Maxwell equations in vacuum is
$$
\mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} . \tag{1}
$$
My question is, $F$ is a tensor, namely
$$
F = \frac{1}{2}F_{\mu\nu} dx^{\...
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How does a general operator in a tensor product space act on operators explicitly written as the tensor product of operators in each subspace?
Suppose I model an atom as a two-level system with states $|g \rangle$ and $|e\rangle$, with eigenvalue equations $\hat{H_1}|g\rangle = g|g \rangle$ and $\hat{H_1}|e\rangle = e|e \rangle$, and an ...
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