Questions tagged [differential-geometry]
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169
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Calculate and transfrom the metric to the orthonormal frame
Let's suppose, the line element is like this:
$$ds^2 = -a^2(1+2\psi)d\tau^2 - 2a^2 B_idx^i d\tau + a^2(1-2\phi)(dx^2 + dy^2 + dz^2)$$
I like to get the output of the metric components from it, ...
- 149
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39
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Package for calculating ADM mass for a given asymptotically flat metric tensor
I am asking for Mathematica package, such that for the input of an asymptotically flat Lorentzian metric tensor $g_{\mu\nu}$ it will give the ADM mass of the object that creates the corresponding ...
1
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1
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83
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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 ...
- 111
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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:
...
- 11
2
votes
1
answer
62
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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, \...
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13
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1
answer
557
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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):
<...
- 8,778
3
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2
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92
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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 ...
0
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1
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120
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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 ...
- 287
0
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1
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162
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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 ...
- 287
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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( \...
- 287
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77
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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_{\...
- 287
7
votes
4
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350
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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 ...
- 165
2
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72
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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:
...
- 287
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131
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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:
...
- 287
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0
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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 ...
- 287