Tagged Questions
2
votes
1answer
88 views
Energy Functional
I am a graduate student in pure mathematics, during my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, ...
3
votes
1answer
130 views
Change of coordinates from an arbitrary frame to a locally inertial frame in General Relativity
If I have the following metric:
$$ds^2=(1-2\phi)c^2 dt^2 - (1-2 \phi)(dx^2+dy^2+dz^2)$$
$\phi$ being the gravitational potential with $|\phi| << 1$ everywhere.
How do I find a coordinate ...
3
votes
1answer
151 views
Calculating Riemann Tensor Using Tetrad Formalism
I was trying to calculate the Riemann Tensor for a spherically symmetric metric:
$ds^2=e^{2a(r)}dt^2-[e^{2b(r)}dr^2+r^2d\Omega^2]$
I chose the to use the tetrad basis:
$u^t=e^{a(r)}dt;\, ...
0
votes
0answers
58 views
Is there a book that discusses General Relativity in terms of Modern Differential Geometry? [duplicate]
All of the physics books that I've seen which discuss General Relativity do so in terms of coordinates - the tensor calculus - even though the naturally relevant entities are invariant under general ...
6
votes
1answer
128 views
Are group representations possible when the solution space is not a vector space?
As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector ...
6
votes
1answer
114 views
Einstein's equations as a Dirichlet boundary problem
Can Einstein's equations in vacuum $R_{ab} - \frac{1}{2}Rg_{ab} + \Lambda g_{ab}= 0$ be treated as a Dirichlet problem?
I am thinking of something along those lines:
Consider a compact manifold $M$ ...
7
votes
1answer
250 views
Diffeomorphisms, Isometries And General Relativity
Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while.
Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
3
votes
1answer
97 views
An issue about the compactness and the existence of CTCs
There is a well known fact that a compact spacetime necessarily contains a closed timelike curve (CTC). Proof can be found in several books on GR (e.g. Hawking, Ellis, Proposition 6.4.2), and in ...
0
votes
0answers
47 views
Cauchy Problem in Convex Neighborhood
While reading the reference
Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available here,
there is something that I don't quite understand.
...
0
votes
0answers
76 views
Searching for clues on “spacelike”-ness by “impossible figures”
Remark on version (3):
In the first version of my question main text the already lengthy "logic statement" -- let's call it "paradigm statement" in the following -- involved conditions on only 24 ...
1
vote
1answer
101 views
If it is given which intervals are spacelike, can be determined which intervals are lightlike?
Provided that the notion of "$\mbox{spacelike}$"-ness (of an interval) is symmetric: $$\text{spacelike}( \, x - y \, ) \Longleftrightarrow \text{spacelike}( \, y - x \, ),$$
then for any set $X$ (of ...
5
votes
3answers
245 views
Could general relativity and gauge theories in principle be covered in one course?
It's always nice to point out the structural similarieties between (semi-)Riemannian geometry and gauge field theories alla Classical yang Mills theories. Nevertheless, I feel the relation between the ...
1
vote
0answers
115 views
How is the poincare conjecture(and perelman proof) helpful in studying the properties of the universe?
Can someone tell me how the poincare's famous conjecture or its proof by perelmen can be helpful in deciding some properties like the shape of the universe?
7
votes
1answer
87 views
Fourier Methods in General Relativity
I am looking for some references which discuss Fourier transform methods in GR. Specifically supposing you have a metric $g_{\mu \nu}(x)$ and its Fourier transform $\tilde{g}_{\mu \nu}(k)$, what does ...
9
votes
3answers
884 views
Why can't General Relativity be written in terms of physical variables?
I am aware that the field in General Relativity (the metric, $g_{\mu\nu}$) is not completely physical, as two metrics which are related by a diffeomorphism (~ a change in coordinates) are physically ...
2
votes
1answer
212 views
Vanishing Ricci flow on a curved manifold
If I understand this right the Ricci flow on a compact manifold given by
$\partial g_{\mu \nu} = - 2R_{\mu \nu} + \frac{2}{n}\!R_{\alpha}^{\alpha} \,g_{\mu \nu}$
tends to expand negatively curved ...
2
votes
1answer
165 views
What determine whether the dynamical equations are tensor equations or vector equations?
Newton's 2nd law which is central to Newtonian dynamics, is a vector equation
$\sum\textbf{F}_{external}=m\textbf{a}$
Same with Maxwell's equations in the covariant form.
On the other hand, ...
9
votes
5answers
1k views
Mathematically-oriented Treatment of General Relativity
Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would
Prove all theorems used
Use modern "mathematical notation" as ...
4
votes
0answers
99 views
K3 gravitational instanton
Could you please recommend a sufficiently elementary introduction to K3 gravitational instanton in general relativity and the problem of finding its explicit form?
Under 'sufficiently elementary' I ...
4
votes
2answers
394 views
Positive Mass Theorem and Geodesic Deviation
This is a thought I had a while ago, and I was wondering if it was satisfactory as a physicist's proof of the positive mass theorem.
The positive mass theorem was proven by Schoen and Yau using ...
8
votes
4answers
863 views
What does a frame of reference mean in terms of manifolds?
Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects.
In (say special) relativity, we have a Lorentzian manifold, ...
2
votes
2answers
68 views
In a gas of particles, how is the displacement vector related to the number density?
Suppose I have a gas of particles that is initially uniformly distributed so that the number density is $n_0$ (number of particles per unit volume), and then I displace the particles by the vector ...
3
votes
1answer
318 views
Math and Wormholes
Hopefully this is the correct forum for this. I felt that Physics Overflow may not be the correct place. I had a student approach me ask me what kinds of mathematics goes into the study of wormholes. ...
0
votes
1answer
146 views
Meaning and types of singularity in case of string or any cosmological model (Mathematical description)
What is actual meaning of singularity can we use this term for conclusion in any research paper( related to cosmological models ).what r the types .
2
votes
0answers
210 views
Singularities in Bianchi models in general relativity ( physical science)
what are the conditions to check point type singularity in a bianchi type model ?
bianchi type model are of Type I,II,III,IX,IV or u can say we use different Bianchi type models having some specific ...
3
votes
1answer
370 views
What is a maximal analytic extension?
Can someone explain (as rigorously as possible) what is involved in analytically continuing, say, the Schwarzschild solution to the Kruskal manifold? I understand the two metrics separately but I'm ...
2
votes
5answers
434 views
Black Hole Singularities
If two black holes collide and then evaporate, do they leave behind two naked sigularities ore? If there are two, can we know how they interact?
