Questions tagged [differential-geometry]
The differential-geometry tag has no usage guidance.
162
questions
0
votes
1
answer
62
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 ...
- 213
0
votes
0
answers
49
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( \...
- 213
0
votes
0
answers
60
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_{\...
- 213
7
votes
4
answers
278
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 ...
- 155
2
votes
0
answers
48
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:
...
- 213
0
votes
0
answers
75
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:
...
- 213
0
votes
0
answers
131
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 ...
- 213
0
votes
1
answer
91
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 ...
- 15
1
vote
1
answer
130
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} \...
- 213
0
votes
0
answers
84
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}}{\...
- 739
0
votes
1
answer
78
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 ...
0
votes
1
answer
69
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 ...
1
vote
3
answers
257
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 ...
- 11
0
votes
0
answers
65
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 ...
- 1
3
votes
2
answers
127
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 ...
- 31
1
vote
2
answers
149
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 ...
- 95
2
votes
1
answer
174
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 ...
- 165
1
vote
1
answer
69
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 ...
- 175
7
votes
2
answers
317
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 ...
- 26.3k
0
votes
2
answers
107
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 ...
- 695
1
vote
1
answer
108
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)$;
...
- 113
1
vote
0
answers
66
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 ...
- 263
0
votes
0
answers
145
views
Find Vector Potential from Vector Field with Div = 0?
This works to compute the scalar potential function of a vector field whose curl is 0 (using the DifferentialForms.m package):
...
- 263
1
vote
0
answers
78
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-...
- 263
3
votes
3
answers
253
views
1
vote
0
answers
39
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 ...
- 263
-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\}\...
- 47
2
votes
1
answer
126
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}$$
and two known points $(x_1, y_1)$ and $(x_2, y_2)$ in the corresponding hyperbolic space $\mathbb{H} = \...
- 2,027
0
votes
0
answers
45
views
Output of expressions unable to have operations be performed on the expression
I am currently trying to use the code linked in the catalog of spacetimes pdf to calculate chirstoffel symbols for a metric which follows as
n := 4
...
- 119
4
votes
1
answer
252
views
Curvature and torsion of this curve
i was trying to get curvature and torsion of curve in mathematica. of this curve
...
- 41
0
votes
1
answer
349
views
Covariant Derivative
I'm trying to create a function which receives as inputs a coordinate system $x^\mu$, metric tensor $g_{\mu\nu}$, a general tensor $T$ of rank $n$, and a list $I$ of length $n$ that looks like this:
$$...
- 123
0
votes
2
answers
135
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
...
- 27
1
vote
1
answer
151
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
<...
0
votes
1
answer
491
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?
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 ...
- 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
$$g=\tau^{-2}\left(-\left(\Lambda / 3-\tau^{2}\right)^{-1} d \tau^{2}+\left(\Lambda / 3-\tau^{2}\right) d t^{2}+g_{\mathbb{S}^{2}}\right)$$
...
- 103
2
votes
3
answers
216
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 ...
- 1,560
0
votes
0
answers
104
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 ...
- 119
0
votes
0
answers
91
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{...
- 83
2
votes
1
answer
105
views
8
votes
0
answers
301
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(...
- 407
0
votes
1
answer
92
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:
...
- 1
2
votes
2
answers
327
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 ...
- 21
0
votes
0
answers
144
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 ...
- 131
1
vote
1
answer
66
views
8
votes
2
answers
340
views
What is the best source to learn how to use tensor operations (exterior algebra) in Mathematica?
I'm specifically interested in the TensorProduct,TensorWedge, HodgeDual and certain build in functions to do tensor arithmetic like TensorReduce, TensorExpand.
I would like to do exterior algebra ...
- 147
1
vote
1
answer
214
views
Einstein field equations for Bondi-Sachs formalism
I'm trying to re-derive the results of Bondi-Sachs formalism. The metric is given in the form
\begin{array}{c}g_{a b} d x^{a} d x^{b}=-\frac{V}{r} e^{2 \beta} d u^{2}-2 e^{2 \beta} d u d r+r^{2} h_{A ...
- 13
3
votes
2
answers
294
views
Calculate surface normals at the boundary of a Graphics3D object
How do I go about calculating and plotting the surface normals at the boundary of a Graphics3D object?
For example, consider this custom-defined ...
- 543
4
votes
1
answer
199
views
Differentiation by indexed variable in equation of Christoffel Symbols
I am very new to Mathematica, and am trying to use it to compute Christoffel symbols for a certain manifold. All this requires is taking some indexed sums of derivatives, but it builds up on several ...
- 43
5
votes
1
answer
286
views
Finding unit tangent, normal, and binormal vectors for interpolated function
As an extension of this question, is it possible to find the unit tangent, normal, and binormal vectors for an interpolated function? eg
...
- 8,152