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Question 1

1vote

1answer

123views

Image relating the geometric notions of tensors, covectors, vectors and metric

Does anyone have a clear image relating the concepts of vectors, covectors, tensors and metric? Sure I can do computations with them, but I still have limited image of how all of these relate to each ...

Aurora Borealis

435

askedNov 17 at 17:10

Question 2

0votes

1answer

136views

Do values of the metric tensor elements depend on the coordinate system of our choosing, but stress-energy tensor elements don't?

If that's the case, how is the Einstein field equation satisfied for different coordinate systems of our choosing? I'm asking in context of my previous question.

Kajak

33

askedNov 17 at 0:50

Question 3

2votes

0answers

75views

Is angular momentum generator a tensor in the Poincare group?

According to Weinberg' QFT vol I, (2.4.8), the generator of angular momentum $J^{\rho\sigma}$ transform as$$U(\Lambda, a) J^{\rho \sigma} U^{-1}(\Lambda,a) =\Lambda_\mu^{\,\,\rho} \Lambda_\nu^{\,\,\...

orange

63

askedNov 16 at 11:11

Question 4

0votes

1answer

141views

Ricci and Einstein tensor

How from these two equations:$$\nabla^\mu R_{\rho\mu}=\frac{1}{2}\nabla_\rho R$$and$$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$$follows$$\nabla^\mu G_{\mu\nu}=0~?$$Also, what is $\nabla^\mu$ ...

user1766349

79

askedOct 24 at 10:58

Question 5

1vote

0answers

131views

How does the warped-product curvature identity determine the 5-D Ricci scalar and the sign of the spherical-curvature term? [closed]

In a 5-dimensional spacetime written as a warped product,$$ds^2 = e^{2A(\rho)}\,dt^2 - d\rho^2 - e^{2B(\rho)}\,d\Omega_3^2,$$where $d\Omega_3^2$ is the metric on the unit 3-sphere $(R[S^3]=6)$ and ...

nonymous

187

askedOct 21 at 8:00

Question 6

3votes

2answers

168views

General Relativity and Differential Forms

I’m learning general relativity from "Covariant Physics: From Classical Mechanics to General Relativity and Beyond" by Moataz H. Emam.He uses pseudo-Riemannian manifolds and index notation, ...

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Gabriel Hiroshi Nakanishi

Question 7

-1votes

1answer

84views

Why does $\vec v = e_i v^i $ remain a vector if the RHS is invariant under coordinate transformations?

In tensor notation, a vector $\vec v$ can be expanded as $ \vec v = e_i\, v^i$,where $e_i$ are the basis vectors and $v^i$ are the components.Under a coordinate transformation $x^i \to x^{i'}$, the ...

Jo Carlo

21

askedOct 16 at 6:48

Question 8

-3votes

1answer

112views

How to prove that the $T^{\mu\nu}$ is a tensor? [closed]

When we consider no variations of the field configuration, we have:$$T^{\mu\nu}= \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\partial^\nu\phi - g^{\mu\nu}\mathcal{L}.$$How can one prove ...

imbAF

2,048

askedOct 10 at 14:42

Question 9

1vote

2answers

117views

If scalar is a magnitude, vector is a magnitude and direction, then what tensor is about? [duplicate]

Well, from precisely mathematicians, i am not sure if scalar and vector can be view as i said. But i notice thatScalar describes as 1 aspect quantity, whether it is magnitude or amount of something....

user222803

askedOct 2 at 13:11

Question 10

2votes

0answers

183views

Algebraic Bianchi identity for dual Riemann tensor

The algebraic Bianchi identity for the Riemann tensor is$R_{a[bcd]}=0. \tag{1}$A dual Riemann tensor, defined as $*R_{abcd}=\tfrac{1}{2}\epsilon_{ab}{}^{ef}R_{efcd}$, obeys a corresponding algebraic ...

Susan

77

askedSep 23 at 14:50

Question 11

8votes

5answers

2kviews

Scalars, Vectors, Tensors: are they all?

In many books on Physics, I read this:(i) Some quantities are represented in terms of numbers; they are called scalars.(ii) Some quantities are expressed in terms of numbers and directions; they are ...

Maths Rahul

191

askedSep 17 at 10:29

Question 12

1vote

0answers

101views

Expression for the Levi-Civita tensor in terms of the tensor density [closed]

If $|g|$ is the absolute value of the determinant of the metric, the Levi-Civita Tensor $\epsilon$ in terms of its tensor density $\epsilon'$is defined as:$$\epsilon_{\mu_1..\mu_n}=\sqrt{|g|}\epsilon'...

Ambica Govind

1,905

askedSep 10 at 2:17

Question 13

4votes

3answers

346views

Understanding $k$ in Fourier’s Law: scalar or tensor?

I'm trying to understand the connection between the general (vector) form of Fourier’s law, related, for example to this YouTube videofor heat conduction:$$\mathbf{J} = -k \boldsymbol{\nabla} T \...

Sebastiano

2,907

askedAug 29 at 11:41

Question 14

1vote

0answers

62views

A basic trick $k^\mu k^\nu=\frac14g^{\mu\nu}k^2$ when calculating Feynman integrals [duplicate]

I am reading Schwarz's book "Quantum Field Theory and Standard Model", chap 17, anomalous magnetic moment. In 17.2, page 319, when simplifying the integral, the book says "Using $k^\mu ...

go-getter

333

askedAug 23 at 5:02

Question 15

0votes

0answers

120views

Few doubts in Tensor Densities (Levi-Civita Tensor) section of Adler, Bazin, Schiffer General Relativity book

I am reding Introduction to General Relativity Book by Maurice Bazin, Menahem Max Schiffer, and Ronald Adler.1st page$$\Im_{\alpha \beta}^\gamma=T_{\alpha \beta}^\gamma \sqrt{-g}$$is a tensor ...

Sagar K. Biswal

395

askedAug 22 at 14:28

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Questions tagged [tensor-calculus]

Image relating the geometric notions of tensors, covectors, vectors and metric

Do values of the metric tensor elements depend on the coordinate system of our choosing, but stress-energy tensor elements don't?

Is angular momentum generator a tensor in the Poincare group?

Ricci and Einstein tensor

How does the warped-product curvature identity determine the 5-D Ricci scalar and the sign of the spherical-curvature term? [closed]

General Relativity and Differential Forms

Why does $\vec v = e_i v^i $ remain a vector if the RHS is invariant under coordinate transformations?

How to prove that the $T^{\mu\nu}$ is a tensor? [closed]

If scalar is a magnitude, vector is a magnitude and direction, then what tensor is about? [duplicate]

Algebraic Bianchi identity for dual Riemann tensor

Scalars, Vectors, Tensors: are they all?

Expression for the Levi-Civita tensor in terms of the tensor density [closed]

Understanding $k$ in Fourier’s Law: scalar or tensor?

A basic trick $k^\mu k^\nu=\frac14g^{\mu\nu}k^2$ when calculating Feynman integrals [duplicate]

Few doubts in Tensor Densities (Levi-Civita Tensor) section of Adler, Bazin, Schiffer General Relativity book

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