# 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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### Poisson bracket properties for tensor densities

I am doing some constraint analysis in an extended theory of gravity, and I am confused about Poisson brackets. The standard PB relations are for example $\{ab,c\} = a\{b,c\} + \{a,c\}b$ etc. But I am ...
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### Critique on tensor notation

I am studying tensor algebra for an introductory course on General Relativity and I have stumbled upon an ambiguity in tensor notation that I truly dislike. But I am not sure if I am understanding the ...
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### How is tensor calculus applied to Einstein's field equations? [closed]

What is the relation between tensor calculus and Einstein's field equations? or What is the contribution of tensor calculus to Einstein’s field equations?
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### Matrix “dimensional analysis” of Lagrangians in QFT

Since the important things in the QFT Lagrangian are vectors and matrices, I wanted to do a "matrix dimensional analysis" of each term. The electromagnetic Lagrangian (ignoring all constants ...
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### Differential forms or Tensors for modern theoretical physics?

There many proponents to teaching differential forms and others teach with tensors. This is true for both mathematics and physics education. It seems mathematicians prefer to teach differential ...
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### Physically measure the covariant and contravariant components of a vector?

I'm just wondering if there is a way to physically measure the covariant and contravariant components of a vector without prior knowledge of the metric. Suppose I have a speedometer of some sort to ...
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### What is a coordinate-free formulation of deformation theory?

For example how are stress, strain and shear tensors described invariantly, without any coordinates, purely in a geometric manner? A formulation that avoids indices coordinates and matrices, even in ...
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### Commutator of derivatives with torsion

I am currently looking at "Physical aspect of space-time torsion" by IL Shapiro. There in eq. 2.10, it mentions that in space-time with torsion the commutator of covariant derivatives acting on the ...
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### How do I tensor differentiate a factor without tensors?

How do I tensor differentiate a factor without tensor, such as: $$\partial_\mu e^{i\Lambda(x)}\tag{1}$$ Should it be zero or should I differentiate it twice changing the order of the tensors follows:...
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### Efficient method to evaluate the Christoffel symbols and Riemann tensor in Bondi-Sachs coordinates

In General Relativity we may employ the so-called Bondi-Sachs coordinates $(u,r,x^A)$ adapted to a null foliation. The level sets of $u$ are null hypersurfaces and $(r,x^A)$ are coordinates on the ...
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### Tensor derivative in special relativity and fluid mechanics

I’m working through Special Relativity by V. Faraoni, and am puzzled by something in his chapters on tensors. He tells us that the partial derivative of a tensor field, e.g. $T_{\alpha, \gamma}$, is ...
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### Doubt on proper time explicit integration

I have a doubt on explicit calculation of proper time. Considering that the metric is given by: $$ds^{2} = -Adt^{2} + B^{-1}dr^{2}+Cd\Omega^{2} -2Ddtd\phi \tag{1}$$ where $d\Omega^{2}$ is the solid ...
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### Can we do better than “a spinor is something that transforms like a spinor”?

It's common for students to be introduced to tensors as "things that transform like tensors" - that is, their components must transform in a certain way when we change coordinates. However, we can do ...
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### Energy and spacetime: a doubt on notation

In the reference $$ I saw a very neat formula, given by: $$\mathcal{E} =: \int_{\Sigma} d^{3}x T_{00} = \frac{1}{8\pi G}\int_{\Sigma} d^{3}x G_{00}. \tag{1}$$ The author stated that this is the &...
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### How to calculate the field tensor from a metric?

Given a metric, for example $$ds^2 = -A(r)dt^2 + B(r)dr^2 + C(r)d\theta^2 + D(r) d\phi^2,$$ and assuming that the fields go as $$\textbf{E} = E(r)\hat{r} \quad \text{and} \quad \textbf{B}=0,$$ ...
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### Is Ricci's theorem can be simply deduced using covariant derivatives of fundamental tensors? [duplicate]

Well Ricci's theorem is given by: $$\mathrm{D}g_{ij}=\mathrm{D}g^{ij}=0$$ I was wondering that if the theorem can be proved using covariant derivatives of $\delta_i^k$, $g_{ij}$ and $g^{ik}$. I ...
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### Ordering of Tensor indices: a specific example

As an example of construcing tensors out of the product of other tensors I found the following example. $$T_{i}{}^j{}_{kl}=A_i{}^jB_{kl}+C_m{}^{mj}D_{ikl}$$ But I am a bit confused about the ...
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### Matrix form of second order contravariant and covariant tensors

In Schaum Tensor Calculus Solved Problems 3.10 and 3.14 the following matrix equivalent of second order contravariant and covariant tensors are made, but without the derivation being explained (\$\...
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### Can a 1-form be the sum of two 1-forms?

I am interested in creating a Lie algebra-valued 1-form (gauge field/connection) for the Poincare algebra. Note that this algebra has two species of generators. In SU(2) Yang-Mills, the connection ...