New answers tagged differential-geometry
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How do I know if a motion is 1 dimensional or 2 dimensional?
the main thing take away: physics does not care a about your coordinates.
So if the thing is moving on:
$$ \vec r(t) = t\cos{\theta}\hat e_1 +t\sin{\theta}\hat e_2 $$
thats linear motion in a plane. ...
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How do I know if a motion is 1 dimensional or 2 dimensional?
Both you and your teacher can be correct depending on what you mean by "dimension."
In everyday experience, we generally consider dimensionality to be the number of independent parameters ...
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How do I know if a motion is 1 dimensional or 2 dimensional?
Let's say you have the following motion (the red arrow)
It looks 2D on this plot, since its motion changes both the $x$- and $y$-coordinates. However, if you redefine the axes like so:
Then the ...
- 21.4k
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How do I know if a motion is 1 dimensional or 2 dimensional?
One dimensional motion is any kind of motion that happens on a line. There are many ways to define this. For example, you could say that the position vector of the particle is always $\vec{r}(t)=r(t)\...
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Boundary conditions on transition maps on general relativity
There's nothing fancy going on with transition maps. What a mathematician means by a "transition map", in the setting of general relativity, is nothing more or less than a coordinate change ...
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Trying to understand a visualization of contravariant and covariant bases
Given the metric tensor for the basis vectors
$$A = \begin{bmatrix}\mathbf e_1 \cdot \mathbf e_1 & \mathbf e_1 \cdot \mathbf e_2 \\
\mathbf e_2 \cdot \mathbf e_1 & \mathbf e_2 \cdot \mathbf ...
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Accepted
What kind of object is a function in the context of gauge theory?
The terminology used in physics texts is often a bit imprecise, and thus the word "scalar" is somewhat ambiguous. Often if a function $\phi(x)$ of the coordinates is a "scalar", it ...
- 11.3k
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Cone vs. small circle parallel transport
This is in a way a Riemann-geometric analogue of the Aharonov--Bohm effect in quantum mechanics.
Consider a manifold $M$ equipped with a linear connection $\nabla$. It is well-known that the parallel ...
- 11.3k
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GR and Riemann Surfaces -- does the complex plane have anything to do with it?
This question seems mainly spurred by a conflation of a Riemann surface (which is a 1-dimensional complex manifold), and a (pseudo)Riemannian manifold (which is the underlying geometry of GR).
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What actually is Boyer-Lindquist coordinates?
For me, the main difference between spherical and BL coordinates is related to the Cartesian coordinates which are as follows:
spherical coordinates
$$ x = r\sin\theta\cos\varphi $$
$$ y = r\sin\...
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Cone vs. small circle parallel transport
The path the vector you are describing takes on the cone is a closed circle while the cone is rolled, but it is no longer a closed circle once the cone is sliced and flattened!
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Is it possible to understand in simple terms what a Symplectic Structure is?
At the most rough level possible, a symplectic structure (geometrically) is an even-dimensional manifold together with a preferred choice of two-dimensional planes which, taken together, span the ...
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General Relativity in a Differential Geometry concept
Definition: We say that a subset $\mathcal{X}$ of $\mathcal{A}$ is an elementary surface if there exists a parametrization $\gamma$: $\mathcal{U}$ $\longrightarrow$ $\mathcal{A}$ (*) (called ...
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Geometrical interpretation of gauge fields of spin other than 2
``Geometry'' is a very vague term. The usual gauge theories (Yang-Mills theory) is about connections on vector bundles, see https://en.wikipedia.org/wiki/Gauge_theory_(mathematics). If you include ...
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Checking inverse metric and Christoffel symbols for the Kerr metric against references
UnkemptPanda wrote: "both sources seem to disagree by a factor of 2."
The factor of 2 is wrong in some sources where they forgot that the crossterms are added twice in the ds² of the line ...
- 12.4k
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Confusion over what constitutes a uniform gravitational field in relativity
a. The Rindler coordinates are sometimes referred to as a "uniform gravitational field" because they have the property of not exhibiting any tidal-forces, like a classical uniform ...
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What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$
The expression $\partial_\mu \mathbf{e}_\nu$, at face value, does not really make sense in curved space due to the vectors being in different tangent spaces. In order to make it work, vectors need to ...
- 12.1k
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Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
${G^{\alpha\beta}}_{;\beta} = 0$ is an identity, but ${T^{\alpha\beta}}_{;\beta} = 0$ is valid only `on-shell'.
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Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
The law of local conservation of $T$ is consequence of the law of motion of the matter part of the total action matter + curvature
$$I[g]+ J[\phi, g]$$
These equations of motion for the matter arise ...
- 73.8k
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Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
I haven't read Dirac's book specifically, but usually authors prove the Einstein equations are consistent in part by showing that the divergence on both sides is 0. The construction of stress-energy ...
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Homogeneous and Isotropic But not Maximally Symmetric Space
OP's question is closely related to their preceding one. I have actually provided a near-complete answer to the present question there, although it appears that OP's question predates the clarifying ...
- 11.3k
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Homogeneous and Isotropic But not Maximally Symmetric Space
You are correct, a spacetime need not be maximally symmetric to be homogenous and isotropic. Isotropy and homogeneity are restrictions on the spatial structure of the universe, which lead to spatial ...
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Question about Young-Laplace equation proof
The observed physical property of surface tension, as tension acting on bodies in the interface, is that this is due to a set of forces that act on all elements of the body that are on the boundary ...
- 39k
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What is the determinant of the Wheeler-DeWitt metric tensor constructed from spatial metrics in ADM formalism?
The Wheeler-DeWitt metric
$$\begin{align} G~=~&G_{IJ}(\mathrm{d}y\odot\mathrm{d}y)^I\odot(\mathrm{d}y\odot\mathrm{d}y)^J\cr
~=~&G_{i_1i_2,j_1j_2}(\mathrm{d}y^{i_1}\odot\mathrm{d}y^{i_2})\odot(\...
- 207k
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Maximizing proper time with parabolic trajectory in uniform gravitational field
The uniform gravitational field is oriented in the $-z$-direction, with acceleration $g$. Clock A is at rest at $z=0$ and reads time $t_A$. Clock B is allowed to move in three-dimensional space $(x, y,...
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