Questions tagged [tensors]
Use this tag for questions that involve tensors. Tensors are fundamental tools for linear computations, generalizing vectors and matrices to higher ranks. Mathematica 9 introduces powerful methods to algebraically manipulate tensors with any rank and symmetry.
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A code to calculate Einstein tensor [duplicate]
I use the following MA code to calculate Einstein’s tensor. I’m asking about the zero component of the Einstein’s tensor, is it correct?
Because I think $G_{00}$ should contains the terms in the zero ...
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Lagrangian density and Christoffel symbols [closed]
According to this article (equation 3.2) the Lagrangian density can be calculated by
$$ 2 \mathcal{L} = \left\{\begin{array}{c}
a \\
b c
\end{array}\right\}(g b c \sqrt{-g})_{, a}-\left\{\begin{array}{...
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Calculating and collecting the terms of the zero component of the Einstein’s tensor
I try to calculate the $G_{00}$ of the Einstein tensor
$G_{\mu\nu}= R_{\mu\nu} -\frac{1}{2} g_{\mu\nu} R$
for the metric:
$g_{00}=-a^2(\tau)\left( 1+2 \phi^{(n)}\right),$
$g_{0i} = a^2(\tau)\left( \...
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How to calculate Einstein tensor components for this metric?
I try to calculate the Einstein tensor compenents from the eqution:
$
G_{\alpha\beta} = \frac{\nabla_\beta (\partial_\alpha \phi)}{\phi} - \frac{1}{2\phi^2} \left[ \frac{\partial_4 \phi \partial_4 g_{\...
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How to create a vector as a tensor object for different euclidean bases?
The components of a tensor are always displayed with respect to one or multiple basis vectors.
For a tensor of rank 1, a vector, in 3D-euclidean space, we resort to three orthonormal basis vectors.
...
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Calculating the strength tensor of a vector field
I'm trying to calculate
$$T_{ab} = g_{ab}F_{gd}F^{gd} - F_a^g F_{bg},$$
where
$$F_{ab} = \partial_a A_b-\partial_b A_a$$
So I define $F_{ab}$ by:
...
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Is there a good package for solving the Einstein field equations given a metric tensor?
I have a very simple metric tensor. I've built a notebook to calculate the EFE solutions based on this metric. Does anyone know of a good package in the Mathematica library that takes a metric tensor ...
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Metric pertubation in xAct
I start to learn xAct. Following this thread:
expanding-the-riemann-tensor-perturbation I noticed that xAct set a default perturbation to the metric by:
...
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Which Slots Are Which? Tensors
I'm trying to verify a calculation in Mathematica and I'm confused about how Mathematica arranges tensor "slots" for contractions.
I have the matrix
$$J_1 = \begin{pmatrix} 0 & -ih & ...
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Simplifying the Einstein tensor in case of a perturbed FRW metric
I use the code in this thread's answer:
(Calculating Einstein tensor components in Kaluza-Klein model)
to get the Einstein tensor components of a four-dimensional Kaluza Klein model.
But instead of ...
- 209
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Cannot symmetrize products involving tensors with specified symmetry?
I am trying to check some manipulations with tensors with Mathematica (13.3 on Windows), but was stuck on an error from TensorTranspose. I isolated my problem to the following simple case:
...
- 387
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How would you find the metric tensor for this formula?
I have a metric formula that does some interesting things for me. It's excellent at predicting the luminosity of Sne 1a. I'd like to see what the EFE solutions are, but I need to convert it from ...
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Calculating Einstein tensor components in Kaluza-Klein model
I try to calculate the Einstein tensor of Kaluza-Klein model from this paper. It is given by Equation (55)
$
G_{\alpha\beta} = \frac{\nabla_\beta (\partial_\alpha \phi)}{\phi} - \frac{1}{2\phi^2} \...
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What are some Mathematica packages used for general relativity?
Could you suggest any Mathematica packages that are used for General relativity calculations? Id like to write code to solve the Schwarzchild Lagrangian equation.
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Tensor Algebra of Symmetric Algebra with Symoblic Tensors
Given
$V=\mathbb{R}^3,$
I would like to work symbolically with elements of
$\mathrm{T}^{(k_1,\dotsc,k_n)}(\mathrm{S} V),$
i.e. vectors of the form
$$\phi = \phi_1 \otimes \cdots \otimes \phi_n$$
with
$...
- 101
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2
answers
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Lorentz indices contraction
I have a complex expression involving second-rank tensors and Kronecker Deltas. How can I instruct Mathematica to utilize the Kronecker Deltas and perform proper index replacements? For example, ...
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1
answer
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Multiplying abstract tensors in Mathematica
I want to do an operation like
$$ AA \otimes BB \times AA' \otimes BB' $$
and have
$$AA \times AA' \otimes BB \times BB' $$
without specifying the elements of $AA$s and $BB$s (and so are their primes)....
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Variation of Lagrangian with respect to components of inverse of metric tensor
I am super new to Mathematica, so I apologize if the question is trivial.
I have defined metric tensor (4x4 matrix), inverse metric and Lagrangian:
...
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Symmetric expression not recognized
Why the following symmetric tensor is not recognized by TensorSymmetry?
...
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Operations on symbolic tensors
I have recently discovered the set of functions related to symbolic tensors, namely TensorProduct, TensorContract, ...
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3
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Product involving Kronecker Deltas
I am trying to do some products involving some objects made out of Kronecker deltas. For example, taking an object like $x_{abcd}=\delta_{ab}\delta_{cd}$, where all the indices run from $1$ to $N$, I ...
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How to define a Kerr-Schild metric in xAct?
A Kerr-Schild can be expressed as $g_{ab}=\eta_{ab}+\phi k_a k_b$, where $\eta_{ab}$ is Minkowski metric, $\phi$ is a scalar function and $k_a$ is a (co)vector field which is null and geodetic with ...
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Finding Petrov type of a metric
Is there a subpackage in xAct (or another Mathematica package for general relativity calculations) to determine the Petrov type of a metric?
I understand that xAct has a lot of subpackages for ...
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83
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How to write the metric tensor in n-dimensions using xAct?
I am trying to reproduce some calculations from a paper, where they express the angular part of the metric as a solid angle in (d-2) dimensions:
where
I know how to calculate the metric tensor using ...
2
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1
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169
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Efficient way to MapApply for Tensor
I need to Map a function with certain arguments $f(\alpha,\beta,\gamma,\delta)$ onto a whole tensor.
My current approach looks like
...
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3
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Implementing symmetry assumptions in FullSimplify
I want to symmetrise a long expression, M, that involves a function of 4 arguments, f[u1,u2,d1,d2], and its products (for ...
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answer
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Calculating the orthonormal frame of a metric in Mathematica
Let us have a given a general metric (like say Kerr metric) of which I want to find the orthonormal coordinates by developing a general code in Mathematica. One of the reliable method to do this (by ...
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What is the right way to simplify a tensor expression (with many indices, but no derivatives) in Mathematica?
After not finding the desired capabilities in base Mathematica, I am trying to use xAct`xTensor package. It appears that I do not need most of it capabilities ...
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Simplification of Momentums in FeynCalc
I have a little problem involving the following contraction in FeynCalc:
$$\frac{k^{\delta}k^{\sigma}}{k^2}$$
By hand one would lower $\sigma$ and contract with the other momentum to cancel $k^2$, ...
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How can I "multiply" nested lists?
Given two nested lists
alist={{a,b,c},{d,e,f}}
blist={{r,s,t},{x,y,z}}
How can I get
...
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How to multiply five tensors of different ranks (contravariant, covariant and mixed) in Mathematica?
How to compute the following multiplication of five different tensors of different ranks in Mathematica
...
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2
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Symbolic matrix tensor an identity without specifying the dimension?
I want to calculate an expression like $\left( M_1\otimes I+I\otimes M_2 \right) ^l$ with $M_i$'s symbolic matrices and $I$ the identity matrix with Mathematica. $M_i$'s are of the same dimension and ...
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TensoriaCalc does not display the correct output
I am trying to use TensoriaCalc to calculate the components of the Ricci and the Riemann tensor of the following metric: $R^{2} \left(d\theta^{2} + \sin^{2}\left(\theta \right)d\phi^{2} \right)$;
...
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Mathematica, ML and TensorFlow
I am currently studying a specialization on Coursera in Machine Learning and am investigating various tools to help me out with the maths and with visualisations and so on.
Although I have many ...
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4
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Quickly summing matrix elements
I have a pair of rank-4 tensors, (T,V), where each index takes four values. I want to quickly contract these with the rank-4 antisymmetric tensor using the following operation:
...
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Polar to Cartesian coordinate transformations [duplicate]
I'm trying to plot the energy boundary function EBminus[r, \theta, LL, S, BB] using Polar to Cartesian coordinate transformations. The plot should be a closed curve....
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2
votes
1
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120
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Pathological expression for field strength contractions in a curved background
I am trying to define in Mathematica the quantity $\star F^{\mu}=\frac{1}{2}\epsilon^{\mu\alpha\beta}F_{\alpha\beta}$, where $F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu+\left[A_\mu,A_\nu\right]$. ...
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Simulation of Quantum pseudo-telepathy
I would like to mathematically simulate Quantum pseudo-telepathy on The magic square game from Wikipedia.
In section Pseudo-telepathic strategies we can read:
The trick is for Alice and Bob to share ...
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Why is TensorContract[x, {}] not always x?
If I use TensorTranspose on an undefined symbol, nothing happens unless the permutation is the identity. For instance, ...
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Tensor equation problem (xact package)
So i want to solve the tensor equation P1[-μ,-ν]=0 with respect to A[r] and then B[r] but the problem is when hitting shift-enter the tensor given is way too long (if i press "show all" my ...
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Fast Sparse Tensor Addition
How can one do a fast sparse tensor addition?
Below, I have the following code: We first generate 3 random sparse 1000x1000x1000 tensors with 10^6 entries each.
Then, I want to add them. But the usual ...
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How to define a traceless tensor with xAct
I would like to define a tensor $A_i^j$ which is traceless ($A_i^i = 0$) and to obtain $A_i^j \delta^i_j = 0$ with xAct / xTensor / xCoba.
I first tried defining an antisymmetric tensor, since they ...
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2
answers
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Code for Tensor decomposition
I would like to know if there is a package or some MMA code to perform tensor decomposition as e.g. defined in a paper by Robeva "Orthogonal decomposition of symmetric tensors" or some ...
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2
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"Symmetrize" shall symmetrize only tensor values $\ne 0$
Using Symmetrize I want to transform a tensor to be symmetric under all index permutations for all original entries $\ne 0$. However each entry is normalized by the ...
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Is there a way of calculating Expectation Values of tensor operators in Mathematica?
This Wikipedia article in on Bell's Theorem lists a whole bunch of expectation values for Bell states:
$$\langle A_0 \otimes B_0 \rangle = \frac{1}{\sqrt{2}}, \langle A_0 \otimes B_1 \rangle = \frac{1}...
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2
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468
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Torch-like permutation for arrays?
Update: edited to clarify my confusion.
In Mathematica Transpose[] operates like this:
...
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2
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Symmetric product
Let define the full symmetrization of a tensor $e_1 \otimes ... \otimes e_N$ by
$$ Sym : e_1 \otimes ... \otimes e_N \rightarrow \frac{1}{N!} \sum_{\pi \in S_N} e_{\pi^{-1} (1)} \otimes ... \otimes e_{...
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1
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Certain block matrix
A block matrix like
$$m_{(ij),(kl)}=\delta_{ik}\delta_{jl}$$
can be constructed as
L=3;
id=IdentityMatrix[L];
m=KroneckerProduct[id, id];
But how to construct
$$m_{...
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0
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Error message installing Ricci
I just started using Mathematica. With the goal of solving
a Loop integrals over a scalar gravity interaction. For the tensor calculus necessary i wanted to use, Ricci. https://sites.math.washington....
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