All Questions
Tagged with terminology differential-geometry
52
questions
0
votes
1
answer
89
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GR and Riemann Surfaces -- does the complex plane have anything to do with it?
I have only the vaguest understanding of Riemann Surfaces -- my sense is that Einstein used them in General Relativity because of their shape.
But Riemann Surfaces I think are not just deformations of ...
- 2,238
4
votes
4
answers
826
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Why do we call the Riemann curvature tensor the curvature of spacetime rather than the curvature tensor of its tangent bundle?
I was studying the mathematical description of gauge theories (in terms of bundle, connection, curvature,...) and something bothers me in the terminology when I compare it with general relativity.
In ...
- 123
0
votes
3
answers
206
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What is the correct term for $\nabla\phi$? Co-vector or 1-form or both?
In the olden days, $\nabla\phi$ was used to be called a covariant vector (Weinberg used this language in his book Gravitation & Cosmology). But this terminology is considered bad for several ...
- 11.7k
1
vote
0
answers
101
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What is a hypersurface?
What is the concept of hypersurface in general relativity? I know it could be characterized into three categories but how do we define hypersurface (in general) in physics? I didn't get what thing it ...
- 59
1
vote
1
answer
178
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Dust solutions in general relativity
What is the precise definition of a dust solution in general relativity?
If the Einstein tensor of a metric has only the first diagonal term non-zero, it that sufficient for that solution to be called ...
- 97
1
vote
1
answer
82
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What is a "timelike half-curve"?
I know what a timelike curve is. But what is a time-like half-curve, as in the definition of a Malament-Hogarth spacetime (below), which appears in this paper?
Definition: A spacetime $(M,g)$ is ...
- 121
3
votes
1
answer
218
views
Why do we call it "Euclidean Quantum Gravity" instead of "Riemannian Quantum Gravity"?
Euclidean quantum gravity is an approach to quantum gravity based on working with Riemannian-signature manifolds and eventually relating the results to our Lorentzian spacetime by means of analytic ...
- 22.2k
1
vote
1
answer
305
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Precise definition of a string worldsheet as a manifold in string theory
I've spent some time studying some definition in smooth manifolds theory in order to give a proper definition of a worldsheet in classical string theory at least. My attempt is the following:
...
- 1,290
0
votes
1
answer
137
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Hawking & Ellis: typo on page 16?
On page 16 of The Large Scale Structure of Space-Time (1973) by Hawking and Ellis, they describe the basics of tangent spaces. This line appears near the top of the page:
Thus the tangent vectors at $...
- 824
1
vote
1
answer
96
views
Well-behaved metric
What is a well-behaved metric in general relativity (GR)? Should every metric be well-behaved in GR? And what is the mathematical description for this kind of metric and what does it mean physically ...
0
votes
1
answer
105
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What is "antipodenpunkte" (this is German)? [closed]
What does "antipodenpunkte" mean?
If you can't find a word about Einstein on the Internet, who should know it?
"In addition, the question arises: can we see stars very close to our ...
- 39
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votes
1
answer
120
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Confusion on two tensors constructed from Riemann curvature tensor and its dual
Assuming the metric signature is $(-+++)$ and solves vacuum Einstein equation, we start from Riemann curvature tensor $R_{\mu \nu \rho \sigma}$ and its dual ${}^*\!R_{\mu \nu \rho \sigma}$ and ...
- 995
3
votes
1
answer
243
views
Carroll's Spacetime and Geometry - Notion of open subset of a manifold
In Sean Carroll's Spacetime and Geometry, an introductory section on manifolds contains the following:
A chart or coordinate system consists of a subset $U$ of a set $M$ along with
a one-to-one map $\...
- 1,215
0
votes
2
answers
146
views
Confusion about word "vector" in "basis vectors" of General Relativity
If a vector is defined as a rank-1 tensor, it should be invariant under change of basis, while its components transform. So why a basis vector $\boldsymbol{e}_\alpha$ is called "vector" even ...
- 882
0
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3
answers
161
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Need clarity about the definition and notation of $p$-forms used in physics
Consider the objects $$A_\mu, ~~F_{\mu\nu}:=\partial_\mu A_\nu-\partial_\nu A_\mu,$$ and the objects $$A:=A_\mu dx^\mu,~~F:=\frac{1}{2!}F_{\mu\nu} dx^\mu\wedge dx^\nu.$$ While reading it from Zee's ...
- 11.7k