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2answers
83 views

Tensor product postulate [duplicate]

Non relativistic quantum mechanics assumes that a composite system should be described with the tensor product of the component systems. This is the tensor product postulate of quantum mechanics. I ...
0
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2answers
58 views

Interference of two non-entangled photons, calculation using tensor product of Hilbert spaces

I'm trying to calculate the interference of two non-entangled photons, like in a double-slit experiment with two photon sources, one behind each slit (follow-up on this question). The individual ...
3
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3answers
116 views

Interference between two photons, tensor product of individual wave functions?

I have learned that the wave function cannot be visualized as a real physical wave like for example the EM field, because for multi-particle systems, it is not a wave in $\mathbb{R}^3$ but in ...
1
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0answers
24 views

LinAlg based physics textbooks [duplicate]

I'm in my second year studying physics, and ever since I took LinAlg, I've been noticing LinAlg-related concepts pop up all over the place, but it has never been presented directly as matrices, bases, ...
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1answer
137 views

Gradient, divergence and curl with covariant derivatives

I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives. ...
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1answer
67 views

Two Point Correlator

I have a problem to reproduce the following identity: \begin{equation} \Pi_{\mu\nu}(q^2) = i \int d^Dx e^{iqx} \langle 0 | T \{j_\mu(x) j_\nu(0) \} | 0 \rangle = (q_\mu q_\nu - g_{\mu\nu} q^2 ) ...
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1answer
68 views

Lowering/raising metric indexes

So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing. Consider Minkowski spacetime. The trace of a matrix $A$ can be ...
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0answers
61 views

Transformation of Christoffel symbols [closed]

Friends I have little problem with transformations:) In General relativity is Christoffel symbol of second kind defined as: $$ \Gamma^{l}_{ij}=g^{lk}\left(\frac{\partial g_{ki}}{\partial ...
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2answers
50 views

From tensor seen as linear maps on vector spaces to index contraction

I am considering a second-order tensor $\mathbf T$ seen as a linear maps on vector spaces, acting on a vector $\mathbf v$ to create a new vector $\mathbf u$, thus, $\mathbf u= \mathbf{T}(\mathbf{v})$. ...
0
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2answers
180 views

What is a Christoffel symbol?

What is a Christoffel symbol? I often see that Christoffel symbols describe gravitational field and at other times that they describe gravitational accelerations. Then, on some blogs and forums, ...
0
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2answers
232 views

What exactly is $T_{\mu\nu}$?

Continuous matter is described in special relativity by the matter tensor which is the so-called stress-energy-momentum tensor. I am finding a difficulty understanding how a tensorial tool ...
0
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0answers
28 views

How do you solve for flux density when you are given the E(x,y,z) and relative permittivity as a tensor?

This given material is not isotropic therefore the relative permittivity is represented as a tensor given by the matrix: $$\left(\begin{matrix} 3 & 1 & 2 \\ 2 & 3 & 3 \\ 2 & 2 ...
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0answers
75 views

Minkowski metric and Null tetrad metric

I'm starting with the Newman-Penrose formalism and have a very basic question that I'm very confused about. The standard Minkoswki metric is $\eta_{ab}=\mathrm{diag}(-1,1,1,1)$. Is then the null ...
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1answer
63 views

The Ricci tensor and its relation to volume

From Wikipedia's entry on Ricci tensor, In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in ...
1
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1answer
47 views

Tensor as outer product

This is a problem I am trying to solve and need help with. Given a $ \left( \begin{array}{} 0 \\ 2 \end{array} \right)$ tensor h such that h$(\quad ;A)=\alpha $h$(\quad ;B)$ for any two vectors ...
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0answers
40 views

Show that $M^{\mu\nu}$ describes the angular momentum of the system

Define $M^{\mu\nu}$ = $\int d^3x(x^\mu T^{0 \nu}-x^{\nu}T^{0 \mu})$ describes the angular momentum of the system. I don't want you to solve it but I'm not really sure what kind of criterion it ...
4
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1answer
86 views

Gradient one-form [duplicate]

I am trying to understand what gradient one-form means actually. In the book that I'm following (A first course on General Relativity by Schutz) it's told that gradient is a one-form and it's ...
3
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0answers
54 views

What kind of math do I need got general relativity? [duplicate]

I'm 15 this year and have a passion in physics What kind of math do I need to tackle general relativity? Also what year in uni do we learn about general relativity?
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3answers
153 views

4-velocities in different frames

We have an observer in an inertial frame $S$ who measures a particle's 4-velocity as $U$. We then have another inertial frame $S'$ with $X'=\Lambda{X}$, where $\Lambda$ is a matrix representing a ...
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1answer
39 views

Metric Tensor and Strain Rate Tensor- Comparison of Units

Is there any way the metric tensor can have a dimension in general relativity? I ask because there is an equation where the strain rate tensor of continuum mechanics is expressed as a difference of ...
0
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1answer
66 views

Proving the invariance of the inner product

If we define the inner product as ${\textbf{u}\cdot\textbf{v}=g_{ij}u^{i}v^{j}}$, where ${g_{ij}}$ is the metric tensor, ${S}$ and ${T}$ are transformation matrices, ${S}$-for covariant indices and ...
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votes
2answers
106 views

Is $X^{\mu\nu} \equiv A^{\mu}+B^{\nu}$ a tensor? [closed]

Consider $X^{\mu \nu}$ an "object" with two indices, defined as $X^{\mu\nu} = A^{\mu}+B^{\nu}$. Is $X^{\mu\nu}$ a tensor? Exists some transformation law to carry $X$ to a new coordinate system ? What ...
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1answer
66 views

Work out components $F^{01}$ and $F^{ij}$ of the antisymmetric tensor $F^{\mu\nu}$ under the Lorentz Transform [closed]

Work out explicitly how the components $F^{0i}$ and $F^{ij}$ of the antysymmetric tensor $F^{\mu\nu}$ introduced in chapter I.6 transform under a Lorentz transformation This problem is from Zee, ...
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3answers
431 views

How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ ...
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votes
1answer
87 views

How can you have $\frac{DA^\mu}{d\tau}$?

If a covariant derivative is given by: $$D_\nu A^\mu=\partial_\nu A^\mu +\Gamma^\mu_{\nu \lambda} A^{\lambda}$$ Then how does $\frac{DA^\mu}{d\tau}$ make any sense? Since there are no 'differentials' ...
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1answer
65 views

Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?

The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is: ...
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2answers
221 views

Tensors, indices and matrix notation - is there a common convention?

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as ...
0
votes
1answer
80 views

In field theory, why are some symmetry transformations applied to the field values while other act on the space that the fields are defined on?

My basic understanding is that a field theory consists of symmetry groups, a space $S$ that the symmetry groups act on and of fields defined on that space $S$. In other words, the space $S$ is the ...
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2answers
230 views

Variation of square root of determinant of metric, $\delta g$ [closed]

I am trying to calculate $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}},$$ where $g = \text{det} g_{\mu \nu}$. We have $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}} = - \frac{1}{2 ...
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2answers
112 views

Transformation of four-velocity in special relativity

I am revising special relativity introducing more matrix form in the equation. Currently I am reading book in which transformation matrix is defined as $${\Lambda= \begin{bmatrix} \gamma & ...
-1
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1answer
62 views

The Riemannian Curvature in Deformations

Is there a direct correlation between the Riemannian Curvature tensor and the deformation gradient tensor in continuum mechanics?
0
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1answer
179 views

Jaumann deviatoric stress rate

Background about terms in this question: Hookes law and objective stress rates From my understading, the Jaumann rate of deviatoric stress is written as: $$dS/dt = \overset{\bigtriangleup}{{S}} = ...
1
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1answer
76 views

Covariant derivative commutator on spinors [closed]

What is this object $[\nabla_{\mu},\nabla_{\nu}]\epsilon$ in terms of curvature tensor $R_{\mu\nu}$? Where $\nabla_{\mu}$ is the covariant derivative on a four sphere and $\epsilon$ is spinor. PS: I ...
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0answers
60 views

Prove that $T_{00}$, $T_{10}$, $T_{01}$, and $T_{11}$ are all $L/(4\pi x^2)$ at $(ct, x, 0, 0)$ for star of constant luminosity $L$

We have a star of constant luminosity $L$. We want to prove that the components $T_{00}$, $T_{10}$, $T_{01}$ and $T_{11}$ are all the same for the event $(ct,x,0,0)$ and they are all $L/(4\pi x^2)$. ...
0
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1answer
84 views

When is Einstein summation implied by Lorentz indices?

I would like to ask if it is possible to find out whether Einstein summation is used in an equation. For example, $$A^{\mu \nu} = 1$$ can either mean $\sum_{\mu\nu} A^{\mu \nu}=1$ or $A^{\mu \nu}=1$ ...
0
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4answers
227 views

Is this a Lorentz-scalar? How do I tell?

I'm struggling to identify whether a scalar is a Lorentz-scalar. E.g: $$\partial_i A^i \quad i \in {1,2,3}.$$ How do I determine if this is a Lorentz-scalar or not? If got the same problem with ...
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0answers
39 views

What are the dimensions of the stress energy tensor in relativity? [duplicate]

Can anyone tell me what the dimension of the stress energy tensor is? Also, if it represents energy density, will calling it kinetic or potential energy be appropriate?
3
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1answer
127 views

On the Lorentz Group representation [closed]

I am going through the notes on QFT by Srednicki (which is certainly a worth reading on the subject, and can be found online, see http://web.physics.ucsb.edu/~mark/qft.html). When describing ...
0
votes
1answer
120 views

Stress-energy tensor on spacetime satisfying Klein-Gordon equation

Consider the stress-energy-momentum tensor $$T_{\alpha \beta}=(\nabla_\alpha \phi )\nabla_\beta \phi -\frac{1}{2}g_{\alpha \beta}((\nabla^\nu \phi ) \nabla_{\nu} \phi +m^2 \phi^2$$ where the ...
0
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0answers
41 views

Is there a physical interpretation of the alternating property?

A map from a vector-space to its base field is called "alternating" if each vector with repeated elements is mapped to zero. I've read that symplectic geometry is an important representation of ...
0
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1answer
104 views

Matrices as second order tensors proof?

I am trying to proof that all matrices are tensors. I have got to a stage where I need to proof that: $$\gamma_{li} \gamma_{kj}= \frac{\partial q_j}{\partial q_k'} \frac{\partial q'_l}{\partial ...
2
votes
1answer
369 views

Variation of Christoffel symbol and Lie derivative

I've also asked this question on Math Overflow; I hope that asking in two separate fora is not a solecism. Under an infinitesimal diffeomorphism the Riemann metric changes by the Lie derivative $$ ...
0
votes
1answer
96 views

How to write the Lagrangian in terms of a projection

We know that $$ L=\frac{1}{2}\left(\partial_{\mu} A_{\nu} \partial^{\mu} A^{\nu}-\partial_{\mu} A_{\nu} \partial^{\nu} A^{\mu}\right) $$ But how do we write the Lagrangian in the following way: ...
4
votes
1answer
188 views

Why are densities not fields?

I have read (in Statistical mechanics of lattice system 2: exact, series and renormalization group methods by D.A. Lavis and G.M. Bell pg 2 ), that intrinsic variables are either fields or densities. ...
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0answers
35 views

How is the electromagnetic tensor expanded?

The electromagnetic tensor is given by $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$, and it appears in the Lagrangian as $L = -\frac{1}{4}F_{\mu\nu}^2 - A_{\mu}J_{\mu}$. The text I'm ...
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0answers
76 views

Non-local gravitational energy tensor

The well-known derivation of the Landau-Lifshitz gravitational energy pseudotensor, relies on several requirements: 1) that it be constructed entirely from the metric tensor 2) that it be index ...
0
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1answer
86 views

How do you take the derivative with respect to a rank two tensor?

I am learning classical field theory and am trying to find the momentum density of the electromagnetic lagrangian as part of an example of Noether's Theorem. The derivative I am encountering is: $$ ...
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2answers
88 views

Clarification on meaning of scalar in math and scalar in physics

When a mathematician says something is a scalar, say on the plane, they mean that it associates to points on the plane real numbers. When a physicist says something is a scalar, they mean that if we ...
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0answers
93 views

Nabla or semicolon notation for covariant derivative? [closed]

$$A_{\,;\alpha}^{\mu}=\nabla_{\alpha}A^{\mu}$$ Are there any pros and cons regarding these two notations for denoting the covariant derivative?
3
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1answer
303 views

How to write a generic density matrix for multi qubit system

I was reading the paper device independent outlook on quantum mechanics. The author defines a generic two qubit density matrix as $$ \rho=\frac{1}{4}\left( I \otimes I + \vec{r_{\rho}} \cdot ...