Questions related to the mathematical aspects of Einstein's theory of relativity. For the physics and its interpretations, please ask at the physics.SE. You may also consider the tags (differential-geometry) and (pde).
Mathematical topics within the theory or relativity include:
Look at the wikipedia article (paragraph: Spherical wavefronts of light) about the derivation of Lorentz transformation between two reference frames $O(x,y,z,t)$ $O'(x',y',z',t')$ in standard position. At a certain point one reads :
" The relation…
Closed trapped surfaces are important in general relativity in large part due to Penrose's incompleteness theorem.
The theorem states that: if a spacetime $(M, g)$ is globally hyperbolic, with a noncompact Cauchy hypersurface $H$ and a closed…
In ADM numerical relativity, I take it that initial data is given as a 3-dimensional $g_{ij}$ configuration slice frozen in time (eg. BL, Misner solutions). The time evolution is to be calculated step by step via
$$ \frac{d}{dt}g_{ij} =…
Let $(M,g)$ be a Lorentzian manifold. Denote $g_{ij}$ to be the metric components and $(g^{-1})^{ij}$ to be the metric components of its inverse. Let $u$ and $\underline{u}$ satisfy eikonal equations, i.e.…
Consider a 2d metric given by:
$$g = dx^2 - x^2 dt^2$$
Then my notes mention that null geodesics are governed by the following equation:
$$\dot{x}^2 - x^2 \dot{t}^2$$
where the dot indicates differentiation with respect to some affine paramter…
https://en.m.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity
https://en.m.wikipedia.org/wiki/Geodesics_in_general_relativity
In his 1915 paper Einstein derives the equations of geodedics, he uses the principle of least…
I am currently a 3rd year undergraduate electronic engineering student and I am very interested in physics. I have completed a course in dynamics, calculus I, calculus II and calculus III.
I have often read articles on general relativity and in the…
If someone has studied calculus, what "instruments" or what fields does one still need to understand the formulas behind the 2 theories of relativity (special and general)? By understand I mean more that the general concept.
Thanks in advance.
1. The problem statement, all variables and given/known data
Consider $\mathbb{R}^3$ in standard Cartesian co-ordinates, and the surface $S^2$ embedded within it defined by $(x^2+y^2+z^2)|_{S^2}=1$. A particular set of co-ords on $S^2$ are defined…
If we have, expression (1) with the $\star$ sign used for Hodge star $$\star(d(\alpha))$$ where $\alpha$ is a complex function. We are speaking in 3 dimensions (x,y,z) that is
expression(1) can be written as
$$\star (\partial_x(\alpha)dx +…
I've been told that Maxwell’s equations in the curved space-time $(\mathscr{M},g)$ take the form
$$\nabla^a F_{ab} =0 \, \,(*), \quad \nabla_a F_{bc} + \nabla_b F_{ca} + \nabla_c F_{ab} = 0 \, \,(**)$$
where $F_{ab} = -F_{ba}$ are components of a…
I'm trying to find the angle subtended by the unit hyperbola through the point $(ct,x)=(1,0)$. I think that I should be integrating something, but I'm not sure how to set it up. I've been trying to think of this as it would be related to a unit…
I would like to prove that the multilinearity condition for a tensor field is equivalent to the transformation rule for its components in a coordinate basis. One way is straightforward, but I haven't been able to find a proof of the converse. For…
The Enstein field equations are $R_{\mu\nu} - \frac{R} {2} g_{\mu\nu} = - \frac{8\pi G} {c^4} T_{\mu\nu} $
In vacuum, it's assumed that $R_{\mu\nu} = 0$.
The energy momentum tensor has to be zero in vacuum. Now, I'm wondering, why it's…
