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How to calculate Christoffel symbols and Curvature for the Alcubierre metric

I am trying to calculate the the christoffel symbols and a few other quantities for the following metric using the code from https://web.physics.ucsb.edu/~gravitybook/mathematica.html Using the code ...
Hans's user avatar
  • 111
0 votes
0 answers
37 views

Covariant derivative of a Riemann tensor [migrated]

I'm trying to calculate the covariant derivative of a Riemann tensor, and I'm using the following way, but there is some problem in my calculations because my calculations do not match with the ...
MMS's user avatar
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0 answers
49 views

performance of the NDEigensystem and DEigensystem

I can't find any result and I can't understand what is going wrong. Any help please? My code is: ...
HarrisModel's user avatar
2 votes
1 answer
61 views

Handling different kind of indices in single tensor equation

I want to solve the following differential equation, $$\partial_a (\sqrt{h} \, h^{a b} \partial_b X^{i}) = 0$$ where $$X^i \equiv \{X^1(\sigma^1, \sigma^2), X^2(\sigma^1, \sigma^2), X^3(\sigma^1, \...
Physics Moron's user avatar
13 votes
1 answer
546 views

Curve shortening flow

I'd like to use mma to recreate this curve shortening flow effect. I have something that works for simple shapes (LHS), but not for more complex curves, which it causes to self-intersect (RHS gif): <...
martin's user avatar
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3 votes
2 answers
77 views

Dsolve gives back argument for non-constant coefficients

I wrote code solving the frenet equations in 2D analytically (so only curvature and no torsion) and plotting the curve with local coordinates afterwards. For constant curvature kappa it functions just ...
builtdifferential's user avatar
0 votes
1 answer
115 views

How to calculate this covariant derivative?

I try to calculate the covariant derivative: $ \nabla_\beta \partial_\alpha~ \phi = \partial_\beta \partial_\alpha~ \phi + \Gamma^\sigma_{\beta\alpha} ~\partial_\sigma~ \phi $ Where $\phi$ is a ...
Dr. phy's user avatar
  • 287
0 votes
1 answer
133 views

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 ...
Dr. phy's user avatar
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0 votes
0 answers
53 views

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( \...
Dr. phy's user avatar
  • 287
0 votes
0 answers
75 views

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_{\...
Dr. phy's user avatar
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7 votes
4 answers
344 views

Minimal surface bounded between turns of helix

I'm curious to find the shape of a surface bounded between the rungs of a helix, ie the shape of the cloth stretched between the rungs of this child's play tunnel. I'm wondering if we could find it ...
Ariana Fenris's user avatar
2 votes
0 answers
69 views

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: ...
Dr. phy's user avatar
  • 287
0 votes
0 answers
113 views

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: ...
Dr. phy's user avatar
  • 287
0 votes
0 answers
153 views

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 ...
Dr. phy's user avatar
  • 287
0 votes
1 answer
128 views

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 ...
The Shepard's user avatar
1 vote
1 answer
203 views

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} \...
Dr. phy's user avatar
  • 287
0 votes
0 answers
88 views

Solving the clothing problem with Mathematica

Given a surface $\mathbf{r}:\mathbb{R}\rightarrow \mathbb{R}^3$, the Chebyshev clothing problem consists in finding a parametrization $(u,v)$ such that \begin{align} \left|\frac{\partial \mathbf{r}}{\...
Daniel Castro's user avatar
0 votes
1 answer
101 views

How to force Wolfram solve the ODE with respect to h[s]?

I have a problem with the DSolve operator. It just gives me the initial ODE as an answer, however, obviously, I need to find the answer h[s] as a function of f[s] and its powers (or whatever else). It ...
Denis D. Bavrin's user avatar
0 votes
1 answer
83 views

How to reduct the terms of differential equation?

How can one bring the terms of a differential equation below to a standard form with coefficients before h[s], h'[s], h''[s] and their powers? Or, better, can someone help me with solving it with ...
Denis D. Bavrin's user avatar
1 vote
3 answers
407 views

Solving Geodesics from Christoffel Symbols

I am somewhat new to using Mathematica and I am facing difficulties with a specific problem related to the geodesics of Einstein's field equation in a vacuum. The metric I am working with is derived ...
HMZ's user avatar
  • 11
1 vote
0 answers
79 views

NoncommutativeMultiply, wedge product and exterior algebra

I would like want to automate some calculations involving wedge products of differential form of different order. Is it possible to define a NoncommutativeMultiply function that has the properties of ...
Spinoro's user avatar
  • 11
3 votes
2 answers
150 views

How can I calculate exponential map for cylinder?

I want to calculate the exp map and log map for cylinder. But as shown figure, I only know the geodesic equation of cylinder is helix. And I search it in some books and website like the google scholar ...
sy shen's user avatar
  • 31
1 vote
2 answers
168 views

Discontinuity in an embedded diagram

I am trying to plot the embedment diagram of a two-dimensional section along the equatorial plane $t =$ constant, $\theta = \pi / 2$ of a Morris-Thorne wormhole with the embed function: $$ z(r) = \pm ...
Soliton-104's user avatar
2 votes
1 answer
226 views

Plotting / Animating a test planet around a star

I am trying to plot/animate the motion of a test planet around a star using Mathematica in the framework of general relativity. In fact, I want to see the perihelion shift. I am using as inspiration ...
kevin Tah N.'s user avatar
1 vote
1 answer
80 views

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 ...
SCh's user avatar
  • 175
7 votes
2 answers
329 views

How to use MMA to solve the minimal surface?

There is a great old post, but since MMA greatly improves the ability of solving differential equations, especially the Region can be used to define the range of variables. So I ask it again. As the ...
yode's user avatar
  • 26.8k
0 votes
2 answers
124 views

How to get the normal vector of a planar curve pointing "outside" (its circle of curvature)?

I have the following implicit function, for which I need to pick up a point $(x0, y0)$ on it and determine the expression of its normal vector pointing outside. "Outside" here means that the ...
larry's user avatar
  • 735
1 vote
1 answer
125 views

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: $R^{2} \left(d\theta^{2} + \sin^{2}\left(\theta \right)d\phi^{2} \right)$; ...
RKerr's user avatar
  • 113
1 vote
0 answers
70 views

Vector Potential Function of Vector Field with DSolve?

(This is a redux of this.) Why doesn't this work? I have a vector field F1 whose div is 0 with a known vector potential function A2, and I try to get DSolve to solve the differential equation for A2 ...
Jeffrey Rolland's user avatar
1 vote
0 answers
81 views

Integral of (Elementary) Volume Form over Orientable Surface of Genus 2?

Per this post on math.stackexchange.com, I am told it is impossible to find an elementary formula for a coordinate patch covering of the orientable surface of genus 2 (or higher genus) (or the non-...
Jeffrey Rolland's user avatar
3 votes
3 answers
281 views

Length of a toroidal helix

For the toroidal helix defined by ...
Michał Kuczynski's user avatar
1 vote
0 answers
45 views

Integrate Differential Form with Parameter with DifferentialForms.m?

In this post, it is shown how to integrate a function with parameter(s) and output a function of the parameter(s). I am trying to do something similar with the DifferentialForms.m package. I have ...
Jeffrey Rolland's user avatar
-2 votes
2 answers
165 views

Plotting surface [closed]

Does anyone know how to plot this image? I am writing an example for Kenmotsu's representation theorem , for $\varphi :% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion -\{0\}\...
user981656's user avatar
2 votes
1 answer
149 views

Calculating distance between a set of random points in hyperbolic space

Given the metric of the Poincaré upper half-plane model $$(ds)^2 = \frac{(dx)^2 + (dy)^2}{y^2}$$ and two known points $(x_1, y_1)$ and $(x_2, y_2)$ in the corresponding hyperbolic space $\mathbb{H} = \...
apg's user avatar
  • 2,155
0 votes
0 answers
45 views

Output of expressions unable to have operations be performed on the expression

I am currently trying to use the code linked in the catalog of spacetimes pdf to calculate chirstoffel symbols for a metric which follows as n := 4 ...
aygx's user avatar
  • 119
4 votes
1 answer
271 views

Curvature and torsion of this curve

i was trying to get curvature and torsion of curve in mathematica. of this curve ...
shamim riten's user avatar
0 votes
1 answer
451 views

Covariant Derivative

I'm trying to create a function which receives as inputs a coordinate system $x^\mu$, metric tensor $g_{\mu\nu}$, a general tensor $T$ of rank $n$, and a list $I$ of length $n$ that looks like this: $$...
Amit Zach's user avatar
  • 123
0 votes
2 answers
138 views

Is it how you do a projection? Is it a Projection? I don’t know what this is

I don't really know, if this is a projection. Maybe you can tell me. . . From the parametric equations ...
J.Doe's user avatar
  • 27
1 vote
1 answer
167 views

How to compute the divergence of a four-vector?

I have a quadri-vector which is given by u = {(E^(-φ0[r]))*(1 - ε δφ[t, r]), (E^(-φ0[r])) D[ε ξ[t, r], t], 0, 0} and a quantity n which is given by <...
Isabella Nunes's user avatar
0 votes
1 answer
662 views

How to solve the einstein field equation symbolically? [closed]

How can I use Mathematica to solve the disturbed Einstein field equation? Is there any notebook that introduces this subject or package like diffgeo?
Isabella Nunes's user avatar
1 vote
2 answers
1k views

How do I get the Schwarzschild solution from the Einstein Equations?

Consider the Schwartzchild-like metric $$ds^2=-A(r)dt^2+B(r)dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2).$$ The Einstein field equations for this metric reduce to $$R_{\mu\nu}=0,$$ which is also known as ...
aygx's user avatar
  • 119
0 votes
1 answer
1k views

How to do Einstein Summation in Mathematica?

I am working with the de Sitter metric which takes the form $$g=\tau^{-2}\left(-\left(\Lambda / 3-\tau^{2}\right)^{-1} d \tau^{2}+\left(\Lambda / 3-\tau^{2}\right) d t^{2}+g_{\mathbb{S}^{2}}\right)$$ ...
Student's user avatar
  • 113
2 votes
3 answers
300 views

What is the most efficient way to turn a metric formula into a metric tensor?

I have a metric formula: ds=(-dt^2)*(c3+a3*t)^2+(dxC^2*(t0^2+t1^2)^2)/(4*t0^4)+ (dxM^2*(t0^2+t1^2)^2)/(4*t0^4)+(dxY^2*(t0^2+t1^2)^2)/(4*t0^4) How do I turn this ...
Quark Soup's user avatar
  • 1,610
0 votes
0 answers
110 views

Christoffel symbol of the first kind [duplicate]

Suppose that we are given a metric $$ds^2=-\left(1-\frac{r_s}{r}\right)c^2dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2d \theta^2+r^2\sin^2(\theta)d \phi^2.$$ Given the Catalogs of Spacetimes pdf we ...
aygx's user avatar
  • 119
0 votes
0 answers
102 views

Finding transformation matrix for a coordinate transform

I have a matrix in two different bases. Suppose they are called $g_{\mu \nu}$ in one basis and $\eta_{ab}$ in the other basis. If the transformation between the bases is represented by $T_{\mu}^{\text{...
newtothis's user avatar
2 votes
1 answer
111 views

Problem with ArcCurvature

When comparing the outputs of ...
user57467's user avatar
  • 2,728
8 votes
0 answers
322 views

Shortest Distance between two points on a 2D surface

I am working on an interesting problem, that consists of trying to compute the shortest distance between two points on a 2D surface. This 2D surface can be represented in a 3D space by the function $f(...
Matt's user avatar
  • 427
0 votes
1 answer
100 views

Define Matrix Function in a For loop

Let $g\in\mathcal R^{3\times 3}$ be a given known rank 2 tensor function, $gInv$ its inverse, and $dg\in\mathcal R^{3\times 3\times 3}$ its gradiant: ...
Tom's user avatar
  • 1
2 votes
2 answers
397 views

Plot or draw a differential geometry graph in Mathematica?

I want to draw a differential geometry diagram, but I don’t know how to draw it. I can only draw graphics like the first picture. There are many differential geometry diagrams in textbooks. What tools ...
King.Max's user avatar
1 vote
0 answers
159 views

Computations on differential geometry

I have to do some really long computations of basic riemannian geometry. I am trying to use Mathematica to check them. This is the kind of computations I want to do. Let $M$ be a riemannian manifold ...
davidivadful's user avatar