All Questions
23
questions
2
votes
1
answer
109
views
Boundary conditions on transition maps on general relativity
On the initial courses of topology and differential geometry, we learn again and again about charts, and atlas, and transition maps. I feel that transition maps are a very powerful idea, because they ...
- 157
2
votes
2
answers
305
views
Derivation of the Schwarzschild metric: why are $g_{22}$ and $g_{33}$ the same as for flat spacetime?
I'm trying to understand the derivation of the Schwarzschild metric from Wikipedia, but I simply do not understand why, therein, $g_{22}$ and $g_{33}$ must be those of the flat spacetime.
Couldn't $g_{...
- 93
0
votes
1
answer
42
views
How is the time defined with paths consistent with the idea that we assign time to frames?
In 49:34 of this lecture by Frederic Schuller, it is explained that time is a derived quantity defined through this integral:
$$\tau = \int_{\lambda_o}^{\lambda_1} \sqrt{ g(v_{\gamma}, v_{\gamma}) }$$...
- 7,980
0
votes
1
answer
611
views
Why is the second-order (covariant) derivative of metric tensor NOT zero while the first-order is?
If the first-order differentiation has given off zero, wouldn't the second-order be the result of differentiating that ZERO, just leading to zero?
1
vote
0
answers
61
views
The Relationship between Coordinates and SpaceTime
I was reading a paper describing the contributions integral mathematicians and physicists have made in the advancement of physics by Michael Atiyah (https://www.jstor.org/stable/24111066), but have ...
- 11
1
vote
1
answer
121
views
Does the spacetime curvature in the vicinity of a massive body increase, decrease or remain unchanged with the increasing velocity of an observer?
Does the spacetime curvature in the vicinity of a massive body such as the sun increase, decrease or remain unchanged with respect to an observer's increasing velocity relative to that massive body?
- 79
2
votes
1
answer
242
views
A coordinate-free understanding of the space-time manifold
I study dynamics and continuum mechanics. Over the years I've gotten used to the coordinate-free, or geometric, way of thinking. A velocity vector, for example, is a tensor. It is the same object when ...
- 1,261
4
votes
1
answer
209
views
Special relativity and general relativity: are the local charts affine spaces?
Well, when you study special relativity in a "covariant way" (using the formalism of Lorentzian geometry and so on...) you realize that the structure of the spacetime isn't a mere vector ...
- 3,085
2
votes
2
answers
623
views
Why don't we assume a vector space structure for spacetime?
At the outset I'll state that I understand completely why, physically speaking, there's no preferred frame - that's not what I'm asking in the question.
I'm not sure why we don't give spacetime a ...
- 1,250
0
votes
1
answer
255
views
The meaning and use of $dx^\mu$ in the metric of General Relativity
Inspired by this answer, I start toying with the general equation of the metric tensor
$$ ds^2 = g_{\mu\nu}dx^\mu dx^\nu .$$
Let $g$ be diagonal, i.e. $g_{\mu\nu}=0$ for $\mu\neq\nu$ and let $x^0=ct$...
- 749
0
votes
2
answers
123
views
Coordinate independence in spacetime
As far as I know, the $n$ coordinates $(x_1, x_2, ..., x_n)$ chosen to describe an $n-$manifold have to be mutually independent $\to$ the mutual derivatives must equal $0$ (for example, $\frac{dx_1}{...
- 719
1
vote
2
answers
1k
views
Metric tensor: Why relate it to Cartesian/Minkowski coordinates?
Why does the metric tensor always relate to cartesian coordinates?
Let's take the simple case for the metric tensor in 3D-space without a time dimension,
$g_{ij}=
\begin{bmatrix}
1 & 0 &...
- 1,662
1
vote
1
answer
245
views
Chart(s) of space-time as a smooth manifold
So we all know that space-time in general relativity is modeled as a smooth (pseudoRiemannian) manifold.
Each point (event) on space-time is labeled with a unique coordinate $(t,x,y,z)$ in a specific ...
- 61
1
vote
0
answers
54
views
Physical Complex Space [closed]
I think if there is a system of coordinates consisted of real number, there is also a system of coordinates consisted of imaginary number. But this system is not a complex coordinate. It is only ...
- 11
3
votes
2
answers
4k
views
Assumed symmetry of Christoffel Symbols
With reference to the discussion in an earlier question on the independence of metric and Christoffel symbols, it was discussed that the symmetry of the Christoffel symbols ($\Gamma_{\mu\nu}^{\alpha} =...
- 301