Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
6
votes
5answers
176 views
In coordinate-free relativity, how do we define a vector?
Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR).
I would normally define a vector by its transformation properties: it's something whose components change ...
1
vote
1answer
81 views
Killing vector argument gone awry?
What has gone wrong with this argument?!
The original question
A space-time such that $$ds^2=-dt^2+t^2dx^2$$
has Killing vectors $(0,1),(-\exp(x),\frac{\exp(x)}{t}), ...
3
votes
2answers
110 views
Geodesic equations
I am having trouble understanding how the following statement (taken from some old notes) is true:
For a 2 dimensional space such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$
the timelike geodesics ...
3
votes
0answers
71 views
Curvature and spacetime
Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
1
vote
1answer
60 views
Evaluating the Ricci tensor effectively
If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
3
votes
1answer
66 views
Derivation of the volume element (which uses the metric tensor)?
I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be ...
0
votes
1answer
46 views
Parallel transport of a vector along a closed curve in curvilinear coordinates
There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor):
$$
\Delta A_{k} ...
3
votes
1answer
77 views
The most general form of the metric for a homogeneous, isotropic and static space-time
What is the most general form of the metric for a homogeneous, isotropic and static space-time?
For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
2
votes
1answer
43 views
“WLOG” re Schwarzschild geodesics
Why, when studying geodesics in the Schwarzschild metric, one can WLOG set
$$\theta=\frac{\pi}{2}$$
to be equatorial? I assume it is so because when digging around the internet, most references seem ...
1
vote
0answers
62 views
Do we expect that the universe is simply-connected? [duplicate]
I heard recently that the universe is expected to be essentially flat. If this is true, I believe this means (by the 3d Poincare conjecture) that the universe cannot be simply-connected, since the ...
1
vote
2answers
91 views
Ricci tensor for a 3-sphere without Math packets
Let's have the metric for a 3-sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
6
votes
2answers
128 views
First and second fundamental forms
I'm writing notes about the 3+1 formalism in general relativity, for myself. Inevitably I came across the notions of first and second fundamental forms. Mathematically, it is clear how these objects ...
0
votes
1answer
39 views
Contraction of the metric tensor
This is perhaps a simple tensor calculus problem -- but I just can't see why...
I have notes (in GR) that contains a proof of the statement
In space of constant sectional curvature, $K$ is ...
4
votes
2answers
78 views
Forces as One-Forms and Magnetism
Well, some time ago I've asked here if we should consider representing forces by one-forms. Indeed the idea as, we work with a manifold $M$ and we represent a force by some one-form $F \in ...
1
vote
3answers
66 views
Combining metric tensors/curvature tensors
I was thinking about the following scenario:
Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in ...
0
votes
0answers
82 views
Differential geometry/relativity question
Can anyone solve this? I'm having some difficulties solving these exercises - relevant to students of relativity and of differential geometry. I'm unsure of where to start and was hoping perhaps ...
0
votes
0answers
41 views
Ricci tensor question
Suppose the Ricci tensor $R_{a}^{\;\;b}$, considered as a linear map on $T_{m}M$ at an even $m$, has only real eigenvalues. What are the five possible Jordan forms of $R_{a}^{\;\;b}$? Show that the ...
1
vote
0answers
56 views
Newman-Penrose tetrad question
I have a question/exercise relevant to students of mathematical relativity:
Let $\left \{ l^{a},n^{a},m^{a},\bar{m}^{a} \right \}$ be a Newman-Penrose tetrad, where only the direction of $l^{a}$ is ...
2
votes
0answers
66 views
Why doesn't this metric cover all of de Sitter space?
This represents a confused attempt to work through a problem in Carroll's Spacetime and Geometry. Supposedly I should be able to use the geodesic equation,
...
0
votes
2answers
127 views
What is metric of spherical coordinates $(t,r,\theta,\phi)$?
In spherical coordinates the flat space-time metric takes:
$$ds^2=-c^2dt^2+dr^2+r^2d\Omega^2$$
where $r^2d\Omega^2$ come from when the signature of metric $g_{\mu\nu}$ is (-,+,+,+)?
what is ...
2
votes
1answer
90 views
Energy Functional
I am a graduate student in pure mathematics, during my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, ...
1
vote
2answers
65 views
What is path of light in the accelerating elevator?
Mathematically, (by mathematically I means by equations) what is path of light in the accelerating elevator?
What is the difference between an ordinary derivative and covariant derivative (which is ...
-5
votes
0answers
82 views
Connection and Christoffel symbols [closed]
Check that the Christoffel symbols transform like a connection. (Assume
that the metric transforms like a tensor.)
1
vote
0answers
49 views
Geometry for Physics [duplicate]
I am currently a high school student interested in a research career in physics. I have self taught myself single variable calculus and elementary physics upto the level of IPHO . And I am comfortable ...
4
votes
0answers
191 views
Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry
Background: Classical Mechanics is based on the Poincare-Cartan two-form
$$\omega_2=dx\wedge dp$$
where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, ...
2
votes
1answer
112 views
Cartan equations versus Einstein equations in classical gravity
Are Cartan structural equations equivalent to Einstein's equations
$$G_{\mu\nu}=T_{\mu\nu}$$
and why (in the case of torsionless geometries, of course)? Does it also apply with a non-null ...
2
votes
1answer
117 views
Ricci identity/Riemann curvature tensor and covectors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
3
votes
2answers
97 views
Electric Field One-Form
I know for instance that we can interpret the electric field as the one-form that given a vector gives the change in potential in the direction of the vector, however I'm very unsure about how to ...
1
vote
0answers
89 views
Why are differential forms on a n-dimensional manifold a Grassmann algebra?
This is stated as an obvious example of a Grassmann algebra on page 32 in this tutorial I am trying to read, but to me it is unfortunately not so obvious.
So can somebody expand this comment a bit ...
3
votes
1answer
133 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 ...
0
votes
0answers
37 views
Curved space to flat space calculation
When changing the curved space co-ordinate into a flat space co-ordinate if a cone. I got the result transformation that i cannot get a transformation at the vertex(apex) why?
5
votes
3answers
221 views
What is a dual / cotangent space?
Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
6
votes
2answers
266 views
Dirac equation in curved space-time
I have seen the Dirac equation in curved space-time written as $$[i\bar{\gamma}^{\mu}\frac{\partial}{\partial x^{\mu}}-i\bar{\gamma}^{\mu}\Gamma_{\mu}-m]\psi=0 $$
This ...
1
vote
1answer
77 views
Where to read about Minkowski space [duplicate]
When I learned Special Relativity, it was taught in terms of basic linear algebra, without any mention of the Minkowski space, proper time as integration on the metric, etc.
However, when I am trying ...
5
votes
1answer
170 views
Fourier Transform on a Riemannian Manifold
The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold?
I've found that a similar question has been asked at Mathematics.SE but ...
2
votes
2answers
110 views
Differential Forms and Densities
I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a ...
2
votes
1answer
118 views
Newtonian Gravity on a Riemannian $3$-Manifold
To solve the Poisson equation for the Newton Potential, say $\phi$, one can use the divergence theorem, such that
$$\int_U \nabla^2 \phi \sqrt{g}~ dV= \int_{\partial U} <\nabla \phi,n> ...
9
votes
3answers
410 views
Physical and Geometrical interpretation of Differential Forms
I have a doubt about the physical and geometrical interpretation of differential forms. I've been studying differential forms on Spivak's Calculus on Manifolds, but my real intent is to use those ...
4
votes
2answers
108 views
Are Poisson brackets of second-class constraints independent of the canonical coordinates?
Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\{\chi_N(q,p),\chi_M(q,p)\}_P$ . Is ...
2
votes
1answer
175 views
Polyakov action: difference induced metric and dynamical metric
The Polyakov action is given by:
$$
S_p ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} ~=~ -\frac{T}{2}\int d^2\sigma ...
10
votes
3answers
324 views
Representing forces as one-forms
First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics.
This question ...
0
votes
1answer
52 views
Constraint on a metric
Given a metric of the form $$ds^2=dr^2+a^2\tanh^2(r/b)d\theta^2$$
why does it follow that $a=b$?
I can't quite spot a constraint condition...
0
votes
1answer
33 views
What is the Willmore energy of the Earth (or the geoid)?
Wikipedia defines the Willmore energy as:
$$e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A,$$
where $H$ stands for the mean curvature of the manifold $\mathcal{M}$.
What is the ...
2
votes
1answer
194 views
Covariant derivative
I would very much appreciate some help in The following:
What is 2nd order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric ...
1
vote
2answers
155 views
Null geodesic given metric
I (desperately) need help with the following:
What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$
I don't know how to transform a metric into a geodesic...! There is no need to ...
4
votes
1answer
97 views
Flat space metrics
This question concerns the metric of a flat space:
$$ds^2=dr^2+cr^2\,\,d\theta^2$$ where $c$ is a constant. Why is it necessary to set $c=1$ to avoid singularities and to restrict $r\ge 0$?
Thanks.
1
vote
1answer
122 views
Interpreting Vector fields as Derivations on Physics
I have a subtle doubt about the physical interpretation of the mathematical definition of vector field as a derivation. In basic physics we understand a vector quantity as a quantity that needs more ...
4
votes
2answers
174 views
Does Kaluza-Klein Theory Require an Additional Scalar Field?
I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2.
In one, there is a constant as well as an additional scalar field introduced:
$$\tilde{g}_{AB}=\begin{pmatrix}
...
0
votes
0answers
58 views
Is there a book that discusses General Relativity in terms of Modern Differential Geometry? [duplicate]
All of the physics books that I've seen which discuss General Relativity do so in terms of coordinates - the tensor calculus - even though the naturally relevant entities are invariant under general ...
3
votes
2answers
125 views
Metric coefficients in rotating coordinates
Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric
$$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$
I am trying to compute the metric coefficients under the change of ...
