7
votes
Accepted
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
6
votes
Accepted
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 ...
- 1,014
6
votes
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 ...
- 469
4
votes
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)\...
- 1,775
3
votes
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).
- 207k
2
votes
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 ...
- 936
2
votes
Accepted
What is the determinant of the Wheeler-DeWitt metric tensor constructed from spatial metrics in ADM formalism?
The Wheeler-DeWitt metric
$$\begin{align} G~=~&G_{IJ}(\mathrm{d}y\odot\mathrm{d}y)^I\odot(\mathrm{d}y\odot\mathrm{d}y)^J\cr
~=~&G_{i_1i_2,j_1j_2}(\mathrm{d}y^{i_1}\odot\mathrm{d}y^{i_2})\odot(\...
- 207k
2
votes
Cone vs. small circle parallel transport
This is in a way a Riemann-geometric analogue of the Aharonov--Bohm effect in quantum mechanics.
Consider a manifold $M$ equipped with a linear connection $\nabla$. It is well-known that the parallel ...
- 11.3k
1
vote
How do I know if a motion is 1 dimensional or 2 dimensional?
the main thing take away: physics does not care a about your coordinates.
So if the thing is moving on:
$$ \vec r(t) = t\cos{\theta}\hat e_1 +t\sin{\theta}\hat e_2 $$
thats linear motion in a plane. ...
- 35.4k
1
vote
Boundary conditions on transition maps on general relativity
There's nothing fancy going on with transition maps. What a mathematician means by a "transition map", in the setting of general relativity, is nothing more or less than a coordinate change ...
- 289
1
vote
Accepted
What kind of object is a function in the context of gauge theory?
The terminology used in physics texts is often a bit imprecise, and thus the word "scalar" is somewhat ambiguous. Often if a function $\phi(x)$ of the coordinates is a "scalar", it ...
- 11.3k
1
vote
Accepted
Checking inverse metric and Christoffel symbols for the Kerr metric against references
UnkemptPanda wrote: "both sources seem to disagree by a factor of 2."
The factor of 2 is wrong in some sources where they forgot that the crossterms are added twice in the ds² of the line ...
- 12.4k
1
vote
What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$
The expression $\partial_\mu \mathbf{e}_\nu$, at face value, does not really make sense in curved space due to the vectors being in different tangent spaces. In order to make it work, vectors need to ...
- 12.1k
1
vote
Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
The law of local conservation of $T$ is consequence of the law of motion of the matter part of the total action matter + curvature
$$I[g]+ J[\phi, g]$$
These equations of motion for the matter arise ...
- 73.8k
1
vote
Homogeneous and Isotropic But not Maximally Symmetric Space
OP's question is closely related to their preceding one. I have actually provided a near-complete answer to the present question there, although it appears that OP's question predates the clarifying ...
- 11.3k
1
vote
Maximizing proper time with parabolic trajectory in uniform gravitational field
The uniform gravitational field is oriented in the $-z$-direction, with acceleration $g$. Clock A is at rest at $z=0$ and reads time $t_A$. Clock B is allowed to move in three-dimensional space $(x, y,...
- 1,900
1
vote
Trying to understand a visualization of contravariant and covariant bases
Given the metric tensor for the basis vectors
$$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 ...
- 902
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