Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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3
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49 views

Angle sum of triangle in Schwarzschild solution

Curvature of space is often intuitively explained as angles of a triangle not adding up to 180 degrees. My questions concerns that. Suppose you have a perfectly spherical star of uniform density - so ...
4
votes
1answer
87 views

Where does the “Supersymmetry” in Witten's proof of the Morse inequalities come from?

Where does the "Supersymmetry" in Witten's proof of the Morse inequalities (original paper and outline of proof for mathematicians) come from? Hopefully someone can provide an intuitive understanding? ...
2
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1answer
74 views

Maxwell's equations in integral form using differential geometry

So I've been trying to convert from Maxwell's equations in terms of differential forms to the integral versions of Maxwell's equations that we know from vector calculus. We have, in vector calculus ...
1
vote
1answer
36 views

Covariant derivative commutator on spinors [on hold]

What is this object $[\nabla_{\mu},\nabla_{\nu}]\epsilon$ in terms of curvature tensor $R_{\mu\nu}$? Where $\nabla_{\mu}$ is the covariant derivative on a four sphere and $\epsilon$ is spinor. PS: I ...
0
votes
1answer
35 views

Elementary question about distributive property of variation operator on an exterior product

I am trying to work out the equations of motion of a 11-dimensional supergravity action $$S = \frac{1}{2\kappa^2}\left(\gamma\int d^{11}x\sqrt{|g|}\mathcal{R} - \frac{\alpha}{2}\int G \wedge \star G ...
2
votes
0answers
67 views

Euclidean metric on a Riemannian manifold [migrated]

Lets say we have a Euclidean configurations space $\mathbb E^n$ equipped with a smooth inner product $\langle \cdot ,\cdot \rangle$ with positive signature in the tangent space above each point. We ...
1
vote
1answer
59 views

Is EM interpreted in a principal or vector bundle?

I've read in a few places that EM is a $U(1)$-principal bundle; but is this correct? Isn't it rather an associated vector bundle using the adjoint representation of $U(1)$?
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0answers
24 views

Topologically non trivial cycles [migrated]

I am studying a Stiefel manifold $X$ which is topologically an $S^3$ bundle over an $S^4$ but is not a product space. I am not able to understand that why is it not the product space $S^3 \times S^4$? ...
5
votes
2answers
302 views

How does one write Newtons 2nd Law using the language of forms?

Newton's second law says that $F=ma$. Supposing that the force is conservative and can thus be expressed in terms of a potential $V$ we have that $F=-dV$. We have that $V$, being a function, can ...
3
votes
2answers
83 views

Telling if geometry is curved without Riemann tensor

If you're only interested in telling whether a certain geometry is flat or curved, and you do not need to know in which way it is curved, do you still need the Riemann tensor? When I try to visualize ...
1
vote
0answers
35 views

Normal of a null surface and null junction conditions in general relativity

I am trying to use the null junction formalism in general relativity (as explained in eg http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.3763&rep=rep1&type=pdf, "Junctions and thin ...
2
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0answers
168 views

What are the equations of motion for the scalar field in the tetrad formalism?

The action of a massless scalar field in curved spacetime is given by: \begin{equation} S(\phi)=\int d^{4}x \sqrt{-g}\left(g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right) \end{equation} Now the action can ...
0
votes
0answers
38 views

Gradient in cylindrical coordinates [migrated]

This is more of a maths question, but several sources point at different expressions for the gradient in cylindrical coordiantes. Sometimes I see the radial component for the gradient of a scalar ...
1
vote
0answers
82 views

Master's Thesis in General Relativity [closed]

just throwing a query out to the Physics community. I'm about to embark on a master's in Gravitation, Cosmology and General Relativity and was looking for possible subjects to start researching. My ...
3
votes
1answer
46 views

Gradient in the Frenet-Serret coordinate

I was simply thinking that the gradient in the Frenet-Serret coordinate at a particular point is similar to the gradient in the Cartesian coordinate. I simply assumed that Frenet space is an ...
6
votes
3answers
1k views

Does spacetime position not form a four-vector?

When one starts learning about physics, vectors are presented as mathematical quantities in space which have a direction and a magnitude. This geometric point of view has encoded in it the idea that ...
1
vote
0answers
96 views

Calculating euler number of disk [migrated]

I'm trying to do exercise 3.1 from Polchinski, which should be a rather easy differential geometry problem. I have to calculate the euler number defined by $$\chi = \frac{1}{4\pi}\int_{M}d^{2}\sigma ...
0
votes
0answers
48 views

Is a ball noncompact? [migrated]

A compact manifold usually refers to "a manifold without a boundary", for example the usual 2-sphere $S^2$. What about a manifold with a boundary? Intuitively, I think such an example, e.g. a ball ...
2
votes
1answer
93 views

Variation of Christoffel symbol and Lie derivative

I've also asked this question on Math Overflow; I hope that asking in two separate fora is not a solecism. Under an infinitesimal diffeomorphism the Riemann metric changes by the Lie derivative $$ ...
0
votes
0answers
42 views

Extension of vector field and fluid velocity

I've been studying locomotion at low Reynolds number with gauge theories reading this paper and on pages 567 and 568 we find the explanation on how to compute the field strength tensor. For simplicity ...
0
votes
0answers
20 views

The null geodesic for given geodesic [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
0
votes
0answers
57 views

What is the null geodesic equation? [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
2
votes
2answers
134 views

Gaussian integral on a Riemannian manifold

How do I estimate the Gaussian integral $\int d^nx \sqrt{g(x)}~e^{-x^2} $ on a Riemannian manifold $(M,g=det~g_{\mu\nu})$? I've tried to consider $\sqrt{g(x)}$ as an analytic function and expanded it. ...
0
votes
0answers
36 views

Tensor components multiplication vs. matrix multiplication [duplicate]

I'm teaching myself general relativity at the moment and I'm not sure I understand the difference between the product of tensor components and matrix multiplication. First of all, if $A$ is a $(2,0)$ ...
1
vote
0answers
74 views

Will a black hole cause scattering of a gravity wave?

In my GR textbook, it states that gravity waves can undergo interference but not scattering. I am just starting the chapter on linearised gravity concepts (weak field approximation) and my apologies ...
2
votes
0answers
30 views

When can a $k$-cycle wrap around a manifold?

According to the paper ``Heterotic and Type I String Dynamics from Eleven Dimensions'' (page 7): Even when the topology is wrong -- for instance on $\mathbb{R}^{11}$ where there is no two-cycle ...
1
vote
1answer
58 views

What is a “local” lorentz transformation of vielbein? How does it transform?

I'm struggling with Anthony Zee's chapter on differential forms in Einstein Gravity in a Nutshell, page 600. He asks us to prove that $$\omega= \Lambda \omega' \Lambda^{-1} - (d\Lambda) \Lambda^{-1}$$ ...
3
votes
2answers
111 views

Calculation of Einstein tensor for weak gravitational field

I am studying A First Course in General Relativity (2nd Ed.) by Bernard Schutz. I have some difficulty in deriving Eq.(8.32) on P.193, the form of Einstein tensor for weak gravitational field, which ...
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0answers
50 views

Nabla or semicolon notation for covariant derivative? [closed]

$$A_{\,;\alpha}^{\mu}=\nabla_{\alpha}A^{\mu}$$ Are there any pros and cons regarding these two notations for denoting the covariant derivative?
1
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1answer
63 views

Difference of connections in the Killing vector equation

For the Killing vector equation, I sometimes see it written in terms of spin connection $\omega$ and other times in terms of the affine connection $\Gamma$. More clearly ...
4
votes
0answers
159 views

What exactly is the relationship between the symplectic 2-form and the frequency of leaves of integrable systems in classical mechanics?

In classical mechanics we equip a differential manifold with a closed symplectic 2-form $\omega$. The symplectic leaves of integrable systems also have a unique frequency, in literature denoted ...
4
votes
3answers
125 views

Is there a way to see that $ \nabla_\mu g_{\nu \rho} = 0 $ without explicit computation, where $\nabla_\mu$ refers to the covariant derivative?

In books, it is usually said that this is a consequence of the fact that parallel transport preserves dot product. How ?
2
votes
0answers
28 views

How does one show that asymptotically $AdS_3$ spacetimes are locally $AdS_3$?

Time and again I keep reading that any asymptotic $AdS_3$ spacetime is locally isomorphic to $AdS_3$. I tried to find proof of this by analyzing the Riemann tensor $R_{\rho\sigma\mu\nu} $ in Ricci ...
33
votes
5answers
2k views

Why do we need coordinate-free descriptions?

I was reading a book on differential geometry in which it said that a problem early physicists such as Einstein faced was coordinates and they realized that physics does not obey man's coordinate ...
1
vote
1answer
41 views

Interpretation for negative energy of curves

Let $(M,g)$ be a Lorentz manifold and $\gamma :[a,b] \to M$ a differentiable curve. I understand we define the energy of $\gamma $ as: $$E[\gamma] = \frac{1}{2} \int_a^b ...
0
votes
2answers
81 views

What experience tells us that gravitational acceleration cannot vanish everywhere?

In attempt to describe the consequences of the Equivalence Principle, Papapetrou in his book, said: When there are gravitational accelerations present, as for example in the gravitational field of ...
2
votes
2answers
107 views

What is a Null Geodesic? [duplicate]

What is a Null Geodesic? My textbook only explains it as the Minkowski metric which equals to zero, but I'd appreciate a more detailed explanation.
2
votes
2answers
73 views

Different signatures

I was working out the christoffel symbols, once where the metric that I am using has (+---) signature and another time where it has (-+++) signature because two books had different signatures and I ...
3
votes
0answers
59 views

Projector and delta function on a cycle $\Sigma$ of a manifold $\mathcal{M}_6$

In the paper ``Hierarchies from Fluxes in String Compactifications'' by Giddings, Kachru and Polchinski, the following example is considered for a localized source that may have negative tension (my ...
0
votes
1answer
49 views

Spherical Symmetric Metrics

In the case where all books try to illustrate a spherical metric, the procedure goes this way: First they impose isotropy in terms of polar coordinates so that one can write: $$ds^2=-A(r)dt^2 + ...
-1
votes
1answer
52 views

Is a planets orbit really a straight line through curved spacetime? [duplicate]

My understanding is that general relativity concludes that gravity isn't real because it does not exist in all frames of reference. Also that mass and energy warp spacetime into a curved geometry. ...
0
votes
2answers
80 views

How to show flat FRW metric has a time-like conformal Killing vector?

I would like to derive the fact that the flat FRW metric has a time-like conformal Killing vector. Is there an easy way to do this? @ValterMoretti showed how one can do this for metrics with a ...
20
votes
5answers
4k views

What exactly is a dimension?

How do you exactly define what is and isn't a dimension? I heard somewhere that it is "anything you can move through" but if that is right, why wasn't time and space considered a dimension before ...
5
votes
3answers
334 views

Is there a natural (suitable) definition for functional derivative in Curved space time

If $$\delta S = \int \sqrt g F[\phi] \delta \phi\tag{1}$$ Then is it natural to define the functional derivative as follows, $$\frac{\delta S}{\delta \phi} = F[\phi].\tag{2}$$ In particular does ...
1
vote
0answers
46 views

Conformal time-like Killing vector near null geodesics in all spacetimes?

Is it true that in all spacetimes there is some conformal time-like Killing vector $\tau^a$ in the vicinity of null geodesics? If the above statement is true then can one argue that, for all ...
2
votes
2answers
104 views

How are FRW metric and Minkowski metric physically different?

According to GR, matrices are coordinate invariant. Does this mean we can transform FRW metric to Minkowski metric with a coordinate transformation like $$dx'=dx\cdot a(t), dy' = dy\cdot a(t), dz' = ...
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0answers
8 views

How does impulse lose energy in rotation, forward and striking earth?

I have this kind of empirical signal in Electronics. where the electronics simulate events signal in rotation signal going forward and signal hitting the earth. The area of the signal is ...
1
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1answer
91 views

Fiber bundle understanding of the wavefunction

Usually people say that given a wavefunction $\Psi$ although $|\Psi(\cdot, t)|^2$ is the probability density for the position random variable at time $t$, the wavefunction $\Psi$ itself has no ...
1
vote
1answer
87 views

Covariant derivative ordering

I was working on a problem involving Bianchi identities, in a particular case I have to take the covariant derivative of the following, which indeed is the Ricci tensor in linearised limit ...
2
votes
2answers
60 views

Topological strings: Why is the complex structure for $T^2$ denoted as $\tau$ in string theory?

In these notes by Vafa on topological string theory he says in page 7 that the moduli of the 2-torus can be repackaged into two quantities: $$A=iR_1/R_2 \,\,\,\,\,\,\,\,\, \tau=iR_2/R_1$$ where $A$ ...