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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67 views

Spacing in indices, and relation to matrices, in special relativity notation

I have some general confusion regarding notation on tensors in special relativity, and how indices correspond to the matrix representation of second-rank tensors. When one has a second-rank tensor $...
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How do you write $A A^T$ in Einstein notation?

In index notation it makes sense as $$\sum_j {A_{ij} A_{jk}^T} = \sum_j {A_{ij} A_{kj}}.\tag{1}$$ But this doesn't make sense for Einstein notation where in $$A^\mu_\sigma (A^\sigma_\nu)^T = A^\...
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Geodesic equation and spatial variation of time

I am trying understand the interpretation of geodesic equations. For simplicity, let us take a metric $$ds^2 = g_{00}(x)dt^2 + a(x,y,z)(dx^2 + dy^2 + dz^2).$$ I interpret the metric to be a spacetime,...
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Raising and lowering the indices of a perturbed metric

I am looking at a metric which is defined as (Eq 2.4 Glampedakis & Babak) $$ g_{\mu \nu} = g_{\mu \nu}^K + \epsilon h_{\mu \nu}$$ where $g_{\mu \nu}^K$ is the original unperturbed metric (Kerr) ...
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81 views

Electromagnetic tensor [closed]

How to prove the equality in this? $$F^{\mu\nu}F_{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}\left(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\right)\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\...
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46 views

Projection tensor in General Relativity

In MTW "Gravitation", the projection tensor is defined as $$\boldsymbol{P} = \boldsymbol{g} + \boldsymbol{u}\otimes\boldsymbol{u}$$ And one exercise asks to prove that a tangent vector $\boldsymbol{...
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What's the most common convention for torsion and contorsion tensor index position?

In Einstein-Cartan theory, the torsion tensor is usually defined as the antisymetric part of the connection: \begin{gather} \nabla_{\mu} \, A^{\lambda} = \partial_{\mu} \, A^{\lambda} + \Gamma_{\mu \...
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Raising and lowering indices and tensor contraction

I'm really confused by the notation of raising and lower indices in tensors when mixed with einstein summation notation and referencing the metric tensor. I need help separating several conflicting ...
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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^...
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Transforming a tensor from Crystal to Laboratory frame of Reference

I want to transform the stiffness tensor of a rhombohedral crystal from crystallographic frame of reference to laboratory fame of reference, how to do it ? For crystal structures having orthogonal ...
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Is my thinking correct for partial derivatives and tensors?

So I was transforming the affine connection and I ended up with a term like this: $$ \frac{\partial^2 x'^a}{\partial x'^b \partial x^p} $$ where $x$ and $x'$ are two different coordinate systems ...
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Integral of the parallel transport equation

The parallel transport of a vector $v_0^\alpha$ along the curve $\gamma$ is given by a vector field $v^\alpha$ which satisfies the equation $$ \frac{\mathrm d x^\mu}{\mathrm d \lambda}\frac{\partial v^...
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Why are isotropic tensors not considered scalars?

In introductory textbooks (Griffiths, Shankar, Boas) a tensor is introduced as a mathematical objects which transform in a specific manner under changes of basis (i.e. changes of the coordinate system)...
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What's the covariant derivative of a normalized, timelike Killing vector?

I'm reading The large scale structure of spacetime and in page 72 the author says: A static metric admits a timelike killing vector $K$. We define the timelike unit vector $V$ as $V=K/f$, where $f^...
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The spinor metric, basic spinor calculations and spinor indices

I'm currently reading the textbook "Finite Quantum Electrodynamics" by Günter Scharf, but I find myself stuck already on page 24. Background Scharf introduces the index-raising symbol (spinor metric)...
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Coordinate-free proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic

I would like to know if the following proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic is correct. I already found a coordinate-...
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Moment of Inertia Tensor Terminology

I've learned about the moment of inertia tensor as a matrix that can be used to compute angular momentum, moment of inertia, etc. for a system. But why is it often described as a tensor instead of a ...
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Christoffel symbols in general coordinates

In order to understand the meaning of covariant derivative, I have seen the following argument. Let us consider a covariant vector $V_\mu$. We would like to understand whether $$T_{\mu\nu} = \frac{\...
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Silly doubt about relationship between Levi-Civita Connections and Koszul Form

In this paper [1] the author wrote and interresting relationship between differential geometry objects (the Levi-civita connection and Koszul form) by means of a musical isomorphism [2] (roughly ...
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Difference between position of indexes in tensor notation (SR) [duplicate]

I am learning SR, and don't understand the difference between the following notations of a Lorentz transformation $\Lambda$ $$\Lambda_{\mu\nu} , \Lambda_{\mu}\ ^\nu , \Lambda^{\mu\nu}$$ I know that ...
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1answer
164 views

Wedge product, tensor product, and Levi-Civita tensor/symbol

Source: Pages 89 and 90 of Sean Carroll's Spacetime and Geometry Quite a confusion in two steps of this quantity: $$ \begin{eqnarray} \sqrt{|g|}d^n x &=& \sqrt{|g|}dx^0 \land ... \land dx^{...
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Are there physical quantities constitute of magnitude, direction and rotation along that direction?

There are scalar quantities(magnitude) and vector quantities(magnitude and direction), but are there fundamental quantities that also depends on how it's oriented/rotated along the direction(magnitude,...
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Yang-Mills Bianchi identity in tensor notation vs form notation

I've seen the Yang-Mills Bianchi identity written as both $$0 = dF^a + f^{abc} A^b \wedge F^c$$ and, in tensor notation, as $$\epsilon^{\mu\nu\lambda\sigma}D_{\nu} F^a_{\lambda\sigma} = 0.$$ Here ...
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Trace of second-order tensor and its invariance under coordinate transformation

Let's consider an arbitrary scalar field. If I act twice on the scalar field with a gradient operator, I will obtain second-order tensor. If I will take a trace of this tensor, I will obtain another ...
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Upper bound for norm of 1-body correlation tensor of qubit

Any $n$-qubit state can be expressed as $$\rho=\frac{1}{2^{n}} \sum_{\mu_{1}, \ldots, \mu_{n}=0,1,2,3} T_{\mu_{1}, \ldots, \mu_{n}} \sigma_{\mu_{1}} \otimes \ldots \otimes \sigma_{\mu_{n}}$$ where $...
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What is the magnitude of a tensor property in a fixed direction?

If I have a physical property represented by a $3 \times 3$ tensor, how can I find its magnitude in a particular direction, say $(\phi, \theta)$ in spherical coordinate system?
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Dirac matrices for generalized metric tensors

The Dirac matrices are defined by the relations $$\left [\gamma^{i},\gamma^{j}\right]_{+}=2\eta^{i,j}\mathbb{1}$$ where $[\cdot,\cdot]_{+}$ is the anti-commutator. What happens if I replace $\eta^{i,...
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Vector Calculus Recommendations [duplicate]

Is there any book which teaches multivariable calculus from a physics perspective? I understand math a lot better when it is applied to physics, and I was wondering if there is a "physical" approach ...
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103 views

Varying covariant derivatives

If we take a variation of a covariant derivative, we must take into account the connections, so we get: $$ \delta (\nabla_\beta T_{\mu \nu}) = \nabla_\beta \delta(T_{\mu \nu}) -\delta (\Gamma_{\beta ...
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Riemann Dual Tensor and Scalar Field Theory

I'm trying to find the component equation of motion for the action in a paper. The action for the system is, $$S=\frac{m_P^2}{8\pi}\int d^4x\sqrt{-g}\bigg(\frac{R}{2}-\frac{1}{2}\partial_\mu\phi\...
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1answer
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Tensors time derivative in moving frames

I know that the following relation exists between the time derivative of a proper vector "v" in an "absolute" frame A and the time derivative of the same vector in a "relatively moving" frame B: $$ \...
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How do you work out the coefficients of the metric tensor?

The definitions of covariant and contravariant tensor quantities are that they transform as $A' ^i=\frac{\partial x_j}{\partial x'_i} A^j$ and $A'_i=\frac{\partial x'_i}{\partial x_j}A_j$ respectively,...
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Identity for the inverse metric tensor using its determinant

I would like to prove this relation: $$g^{\mu\nu} = \frac{1}{3!} \frac{1}{g} \epsilon^{\mu\rho\sigma\kappa}\epsilon^{\nu\alpha\beta\gamma} g_{\rho\alpha} g_{\sigma\beta} g_{\kappa\gamma}, \tag{1}$$ ...
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How does the stress energy tensor change in different reference frames?

Is the Stress-Energy tensor invariant in all RFs? If not (which is highly probable) how does it change? EDIT: does the Einstein equation help? Since (without $\Lambda$) $$ R_{\alpha\beta} -\frac{1}{2}...
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Can we generalize matrix model theory?

As in the question, can matrix model theory be generalized to a tensor model theory? Will the results be different or useful in describing real world phenomena? Details: in matrix model theory we ...
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75 views

Is the inertia tensor a tensor field?

The inertia tensor seems like it cannot depend in any way on position, but every other tensor in physics is a tensor field (stress tensor, electromagnetic tensor...) so, which is it?
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Looking for physical intuition into the Electromagnetic Tensor:

I have done some work with the electromagnetic tensor and I'm fairly good at manipulating it and using it to transform the Maxwell Equations into tensored forms. Admittedly though, I have no physical ...
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Do the Christoffel symbols $\Gamma_{rn}^w\partial_sV_w = \Gamma_{sn}^w\partial_rV_w$?

In lecture 3 (about 97 min into the lecture) of Leonard Susskind's general relativity course, he suggests finding the Riemann curvature tensor in terms of the Christoffel symbols as an exercise. I ...
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298 views

How to prove that the covariant derivative obeys the product rule [closed]

In General Relativity the covariant derivative of contravariant vectors $A^\mu$ is: \begin{equation} \nabla_\mu A^\nu=\partial_\mu A^\nu+\Gamma^\nu_{\mu\alpha}A^\alpha \end{equation} where $\Gamma^\...
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Contracting Riemann Tensor Troubles

It has been several years since I looked at General relativity, and I am trying to brush up on it because it was always interesting and I am in need of it for my research. Specifically, I am looking ...
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1answer
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General relativity: Principle of minimal coupling computations

I have a question about computations in general relativity and transition from a Lorentz frame to a general fame just by substituting the flat metric with a general one and ordinary derivatives with ...
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1answer
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Thinking of the Faraday field-strength tensor as a 2-form

Background I'm familiar enough with the Faraday tensor $F_{\alpha\beta}$ to know that it's is a 2-form. Hence, at each point $P$ in spacetime $V$, it's a multilinear map $$F: T_PV\times T_pV\to\mathbb{...
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Are Lagrange's equations physical laws?

Well, I studied that a physical law is an equation between tensors that are function of position and time because when the frame is changed tensors change in order to leave the equation true: $$T_1(\...
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Still confuse about tensor

In special relativity, a four-vector $\mathbf{x}$ in an inertial frame is related to $\mathbf{\overline{x}}$ through a Lorentz transformation $\mathbf{\Lambda}$: \begin{align} \overline{\mathbf{x}}...
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Timelike and spacelike projections in General Relativity and associated conservation laws

For any timelike curve $p_\mu$ in General Relativity (section 3 of this review), we can project this into its timelike and spacelike components. Further, these projections are associated with ...
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Scalar coupled to Gauss-Bonnet invariant vs Horndeski theory

So here it is a somewhat tormenting question. The first statement will be a little specific but then I will make clear what the jargon indicates. How can we show that a Lagrangian made of a scalar ...
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1answer
336 views

Indices misprint in Sean Carroll's Spacetime and Geometry?

To my knowledge, 3 or more indices may not appear in a given term, as I've found in a video produced by "Faculty of Khan": However, on page 30, Sean Carroll writes: As obvious, the indices 0 and 1 ...
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Is $X\otimes X$ not the simultaneous position operator?

I had thought that $X\otimes X$ would be the operator on $H_1\otimes H_2$ to simultaneously measure the x-positions of two particles. But there seems to be something wrong with this -- for a given ...
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Orientation and sign convention in 2D electrodynamics using differential forms

I've been following this paper for a treatment of electrodynamics using differential forms. In particular, they demonstrate that Maxwell's equations expressed using differential forms are form-...
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244 views

Compact expression of Maxwell's equations: is there a missing minus sign?

The compact form of Maxwell's equations: $$\boxed{\square\, \boldsymbol{\mathsf{F}}=\mu_0 \boldsymbol{\mathcal{J}}} \tag{1}$$ where the current density quadrivector is given by the relation $\...