# Tagged Questions

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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### Projection of a tensor

Consider the following tensor (abstract index notation, e.g. Wald's) $B_{ab}$ and timelike vector field $X^{a}$ such that $X^aX_a=-1$ and $$B_{ab}=\nabla_bX^a$$ Then one ...
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### Does $\partial_\mu =\frac{\partial }{\partial x^\mu}$ or $\partial_\mu =\frac{\partial }{\partial x_\mu}$? [migrated]

I am looking at the chain rule with covariant and contravariant vectors. I understand why we have: $$df=\frac{\partial f}{\partial x^\mu} dx^\mu$$ (Please correct me if I am wrong) since even though ...
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### Dirac notation - trace of product of (bipartite) density matrices

I'm getting confused by the Dirac notation. Suppose I have the following two objects. $$\rho = \sum_k p_k (\rho_A \otimes \rho_B) = \sum_k p_k |k \rangle \langle k | \otimes |k\rangle \langle k | ,$$...
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### What is the connection between the coordinate transformation properties and graphical representation of covariant and contravariant components?

So right now I am studying General Relativity (in particular tensor analysis), and I have a question regarding covariant and contravariant components of a vector. I was taught how to transform ...
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### Covariant Derivative of Kronecker Delta

I am reading Carroll's book on GR right now, and I ran into a little trouble in his chapter 3 on curvature. He is establishing the properties of the covariant derivative, and claims that the fact that ...
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### If $T_{kl}=\epsilon_{kil}m_i$, how to show $m_i=0.5\epsilon_{ilk}T_{lk}$? [closed]

In a book I am reading (about magnetic dipole), it is given that $T_{kl}=\epsilon_{kil}m_i$. Then, it says since $T_{kl}=-T_{lk}$, it can be shown that $m_i=0.5\epsilon_{ilk}T_{lk}$. I understand that ...
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### Rewriting Maxwell's equation in tensor form [closed]

Suppose $F_{ij}=\epsilon_{ijk}B_k$, how to prove the following: $\partial_iB_i=0$ becomes $\partial_iF_{jk}+\partial_jF_{ki}+\partial_kF_{ij}=0$ $B_iB_i$ becomes $F_{ij}F_{ij}/2$ I can see that ...
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### Usage of tensors in physics [closed]

As I understand it, tensors are multi-linear maps that map vectors (and dual vectors) to real (or complex) numbers, but I'm hoping to gain some intuition as to why they are useful in physics. Is it ...
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### Flat space Solution of Einstein Field Equation

Does a trace-free energy-momentum tensor $T_{\mu}^{\mu} = 0$ ensure that the Einstein's field equations have a flat space solution?
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### Manipulating tensors in relativistic quantum mechanics

I was doing a problem that involved showing a Heisenberg equation of motion was consistent with the Dirac equation. The question involved a lot of algebra which was generally fine but something done ...
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### Tensor notation of Maxwell's equations

Tensor notation of Maxwell's equation read So when we explicitly try to find the Maxwell's equation from the above tensor equation we only get gauss law and curl of B. The div.B=0 and curl of E are ...
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### How to show invariance using the Maxwell tensor?

I want to show the invariance of $E^2-c^2B^2$ under the Lorentz transformations. The obvious way to do this is to show that $$E^2-c^2B^2=E'^2-c^2B'^2,$$ where $E'$ and $B'$ are the Lorentz ...
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### Killing tensor and Riemann tensor identity

I know that if we have a Killing vector then it's straightforward to show the identity: $$\nabla_a \nabla_b K_c = R_{cba}^k K_d$$ I'm now trying to show the following identity for a $(0,2)$ Killing ...
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### Covariant derivative of a covariant derivative

I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$. This is something I've taken for granted a lot in calculations, namely I though that by the ...
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### Peculiarity about a system of three electrons

Consider three (or any number bigger than 2) electrons without spatial degrees of freedom, thus the only degree of freedom is the spins. The Hilbert space is then formed by the tensor product of the ...
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### Understanding basics of tensors

I am trying to understand tensors to learn General relativity. In the book that I am reading they claim that if the basis of a vector space undergoes a linear transformation $T$ then the components of ...
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### Dielectric permittivity tensor of rank 2

In special relativity, one knows the Euclidean and Minkowski metric to raise or lower the index of a covector or vector. The Euclidean metric is a (1,1) tensor and is represented by a 3x3 identity ...
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### Staggered Indices ($\Lambda^\mu{}_\nu$ vs. $\Lambda_\mu{}^\nu$) on Lorentz Transformations

I have some open-ended questions on the use of staggered indices in writing Lorentz transformations and their inverses and transposes. What are the respective meanings of $\Lambda^\mu{}_\nu$ as ...
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### How does the Einstein summation convention apply to the following equation?

This is the equation is in the "mathematical form" section of the following wikipedia article: http://en.wikipedia.org/wiki/Geodesics_in_general_relativity More specifically, the "Full geodesic ...
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### How do we determine if a certain physical quantity is a vector?

For instance in Newtonian physics we treat position of objects, displacements, velocities, forces, momenta, angular velocities etc all as vector quantities (little arrows in space which have a certain ...
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### Raising and lowering operators for a composite isospin $SU(2)$ system
Consider pion states composed of $q \bar q$ pairs where $q \in \left\{u,d \right\}$ transforms under an $SU(2)$ isospin flavour symmetry. These bound states transform in the tensor product \$R_1 \...