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Questions tagged [differential-geometry]

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

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How to calculate the derivative of scale factor as a function of conformal time from the solution of Friedmann equation

For the flat geometry of lamda CDM model, the solution for Friedmann equation is $$ a(t) = \left\{ \frac{Ω_{m,0}}{Ω_{Λ,0}} \sinh^2 \left[\frac{3}{2} \sqrt{Ω_{Λ,0}} H_0(t - t_0)\right] \right\}^{1/3},...
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Acceleration as the logarithmic derivative of the redshift factor $U^\sigma \nabla_\sigma U^\mu = g^{\nu\mu}\nabla_\nu \ln V$

Let $K^\mu = V(x)U^\mu$ be vector proportional to the four-velocity $U^\mu$. (e.g. $K^\mu$ is a normalized time-like Killing vector for an observer at infinity). Then, $V(x) = \sqrt{-K_\nu K^\nu}$ is ...
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1answer
48 views

Why is $\chi_{[\mu}\nabla_\nu \chi_{\sigma ]} = 0$ at the Killing horizon?

Let $\chi$ be a Killing vector field that is null along a Killing horizon $\Sigma$ Why is $\chi_{[\mu}\nabla_\nu \chi_{\sigma ]} = 0$ at $\Sigma$?
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2answers
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Curl and circulation of a vector field that is ill-defined at the origin: any interesting physical effects?

In the cylindrical polar $(\rho,\phi,z)$ coordinate, suppose the velocity field in a liquid is given by $$\vec{v}=\frac{K}{\rho}\hat{e}_{\phi}, \qquad K=\text{constant}.\tag{1}$$ It can be easily ...
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0answers
38 views

General relativity: congruence and integral curves

I've been doing some research on GR for a assignment on the Raychaudhuri equation. To do so, I've had to pick up some math I had not learnt earlier. My university offers a really poor choice of ...
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19 views

Torsion Tensor covariant and contravariant

I want to know the relation between covariant and contravariant torsion tensors. Also please tell me that can we change the order of indices in torsion tensor?
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1answer
55 views

Covariant derivative in field theory

I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-half - spin-1 field: $$ ...
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40 views

Fields transforming under an exceptional Lie group

We may think of tensors as sections of an associated vector bundle to a principal $\mathrm{GL}(n,\mathbb R)$ bundle, with a fibre chosen to be $\mathbb R^m \times (\mathbb R^*)^n$ - these play a role ...
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Doubt about energy conditions: the Time-like Convergence Condition

First of all, consider a congruence of smooth time-like geodesics parametrized by proper time $\tau$. So, a tangent vector to a time-like geodesic is indeed a four-velocity up to a factor constant; ...
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1answer
48 views

How does a tensor from cotangent and tangent spaces transform?

In Sean Carroll's Spacetime and Geometry An Introduction to General Relativity Chapter 2, there is an example of tensor transformation from $x,y$ coordinates to primed ones using $$(x',y') = (\frac{2x}...
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I could not proof that curl of gradient is zero. How can I do this by using indiciant notation? [migrated]

I could not find a way to equal this to zero. $\vec{\nabla}\times(\vec{\nabla}\phi)= \epsilon_{ijk}\partial_{j}(\partial\phi)_{k}= \epsilon_{ijk}(\phi(\partial_{j}\partial_{k}) + \partial_{k}(\...
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Positive frequency definition in general spacetime for general fields

In Quantum Field Theory the positive frequency solutions to the classical field equations are quite important since they are the basis of the definition of particles. In Minkowski spacetime we have a ...
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0answers
29 views

Connectedness on Special Kaehler manifolds

I just wanted to make a short/concise question which is quite mathematical but the aim is physical so I would like to ask it here. Anyone knows if there is a general statement about connectedness on ...
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1answer
27 views

Gravitational force between large square plate and spherical object touching the plate [closed]

How do I calculate the gravitational force between a very large square plate object and a small sphere touching that plate? I am assuming it is some kind of integration exercise but my mathematics is ...
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1answer
49 views

Derivation of Komar formulae from ADM formalism?

Is there a way to derive the Komar formulae from the ADM formulae? Both the formulae give the same answer for mass and angular momentum, so I was wondering if one can be derived from another. On an ...
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1answer
50 views

How are the differential forms for Maxwell's Equations used?

I am currently reading up on Maxwell's Equations (specifically Ampere's Circuital Law- with Maxwell's Addition) for a presentation on differential equations. I chose the topic ignorant of how the ...
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Connected components of conformal group $ {\rm Conf}(p,q)$ containing $P$, $T$ and conformal inversion are same or different?

As we known (see this post), the global conformal group for $\mathbb{R}^{p,q}$ is $$ {\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}$$ The global conformal group ${\rm Conf}(p,q)$ has 4 ...
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Where can I find 'General relativity without coordinates ' by T.Regge? [closed]

Or any other such article, I have searched everywhere but haven't found any other coordinate free treatment of GR.
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1answer
93 views

Implication of the Jacobian map for the structure of the Euclidean space-time

I'm listening to Alain Connes "On the Fine-Structure of Space-Time" around minute 23 saying that it was disappoing that the solution Y to the equation $$ <Y[D,Y]^{2m} >= \gamma $$ with $D$ a ...
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2answers
142 views

Intuition behind Manifold

As the majority of concepts in dynamical systems are based on Manifolds. How can one think/imagine about the concept of a manifolds intuitively? (A Lucid explanation is highly encouraged!!!)
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Intuitive meaning of Maxwell action in terms of geometry (differential form formulation)

For the electromagnetic field strength, $F^{(2)}$, which is an exact $2$-form, i.e. $F^{(2)}=dA^{(1)}$ for a $1$-form $A^{(1)}$, we can define its Hodge dual, $*F^{(2)}$ and then define the action $$\...
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1answer
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Are the horizon generators radial null geodesics also?

What I am going to ask is probably a result of unrigorous treatment of the submanifold in question. Radial Null Geodesics of Schwarzschild So start with Schwarzschild spacetime. The metric tensor is ...
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What is Lie-dragging? What is the relationship between Lie-dragging and Lie-derivative? [duplicate]

I found lie-dragging at following picture, i can not understand why L and N have property of (12)
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1answer
70 views

MTW Exercise 4.4: Rotation free 1-forms [closed]

MTW in Exercise 4.4 calls a 1-form $A_\alpha$ a rotation free 1-form if $$\textbf{A}\wedge\textbf{dA}=0.$$ And claims that all such 1-forms may be written as $$\textbf{A}=\phi\,\textbf{d}\psi$$ for ...
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1answer
76 views

Schroedinger equation in Differential geometric language

I have reading about manifolds and tangents spaces and lie derivatives. I have been wondering is there is a way to write Schrödinger equation in this formalism?
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1answer
42 views

Determinant of ADM metric

I am studying inflation and for the calculation of the bispectrum we are using the ADM formalism where the metric is the following form: $$g_{\mu\nu}=\begin{bmatrix}-N^2+N^iN_i&N_i\\N_i&h_{ij}...
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0answers
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Question about Lie derivative of connection [closed]

This is from Hughston and Tod, Ex 8.5. Please give me an idea how to start with this. If $\mathcal{L}_V g_{ij} = 2\phi g_{ij}$ then prove that $$ \mathcal{L}_V\Gamma^{i}_{jk} = \phi_{,j}\delta^{...
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0answers
39 views

Identically null Einstein equations in Schwarzschild spacetime

In deriving Schwarzschild solution one assumes many constraints on the metric, in particular parity invariance (invariance of $g _{\mu \nu}$ under $t \rightarrow-t, \phi \rightarrow-\phi, \theta \...
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1answer
30 views

Question about explicit notation of averaged energy conditions integrals

Beyond the basics of general relativity, we rapid encounter the so called Averaged energy conditions. The mathematics of these quantities are related to line and volume integrals. As given by [1], ...
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0answers
20 views

Einstein field equations in terms of invariants [duplicate]

Is it possible to express Einstein field equations of general relativity in terms of invariants of Riemann tensor and Stress-energy tensor? I suppose field equations should lead to an algebraic ...
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How to determine whether a set of coordinates are independent and sufficient to determine the system completely?

In Analytical mechanics, when we formulate our principles, in general, it is assumed that we start with a cartesian coordinate system, and then find some generalised coordinates $q_j$s they are all ...
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2answers
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Geometric interpretation of the second Bianchi identity?

Assuming a torsion free Christoffel symbol, the covariant derivative can be shown to satisfy the second (differential) Bianchi identity: \begin{equation} [[\nabla_a,\nabla_b],\nabla_c]+[[\nabla_c,\...
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3answers
78 views

Geodesics - Reparameterization

I am reading Wald's textbook Chapter 3. I am struggling with Section 3.3 and problem 5. It states that any curve that satisfies the weaker condition $T^{a}\nabla_{b}T^{b} = \alpha T^{a}$ is $eq.(3.3....
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2answers
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Question about vector field on a manifold [closed]

Arnold defined a vector field on a manifold M is a map from M to the tangent space of M (which has all derivations, roughly). In his ODE book, he talks about $\dot{x}(t) = v(x(t))$ for a vector field ...
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4answers
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Question about holonomic constraints

Goldstein says that when a system of $N$ particles is subject to $k$ holonomic constraints, the positions $\mathbf{r}_1, \dots, \mathbf{r}_N$ can be parameterized by $3N - k$ independent coordinates $...
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2answers
77 views

Meaning of $n+1$ dimensions

Is there anything mathematically significant about studying a theory in $n$ dimensions or $m+1$ dimensions if $n=m+1$? For instance in the context of general relativity I hear people talk about the ...
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72 views

Solving the biharmonic equation $\square^2f(\vec{x}) =0$ on a curved space

How to solve the biharmonic equation $\square^2f(\vec{x}) =0$ on a curved space (e.g. $d$ dimensional sphere or hyperboloid)? I was thinking in the following line : I know how to solve $\square f(\...
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1answer
27 views

Why is a single function sufficient to specify a canonical transformation?

Spivak argues at page 577 in his book Physics for Mathematicians: What are the $2n$ relations he is talking about?
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3answers
64 views

Unique null geodesic between two points

Given two points in Lorentzian spacetime $p,q\in M$, is it true that there is only a unique null geodesic (up to affine reparametrization) that connects that the two points? On the one hand, it seems ...
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1answer
59 views

What's the variation of the Christoffel symbols with respect to the metric?

By the Leibniz rule, I expected it to be $$\delta \Gamma^\sigma_{\mu\nu} = \frac 12 (\delta g)^{\sigma\lambda}(g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu}-g_{\mu\nu,\lambda}) + \frac 12 g^{\sigma\lambda}(\...
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0answers
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Tangent vector to a curve [migrated]

I am trying to relate things simply. If a curve is on a flat 2D space represented by the parameter $\lambda$. In polar coordinate system $(r,\theta)$ at any lambda the tangent vector components are $$...
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0answers
20 views

What is the definition of coordinates of an observer? [duplicate]

I am confused about coordinates of an observer. Suppose we have an observer $A$ whose world line $\gamma$ is a geodesic and another observer whose world-line $\gamma$ is not. Since coordinates is just ...
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2answers
58 views

Transform mixed vielbein expressions to $\sqrt{-g}$ times traces in massive gravity?

In the Massive Gravity review by Claudia de Rham the massive gravity action is given by with mass potential in vielbein formulation. Equivalently, the same action can then be described by with I ...
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Question about the geometric structure of Newtonian mechanics

My point here is about the mathematical structure of Classical pre-relativistic physics and general relativity (GR). It became more clearly, after GR, about the fact that pseudo-riemannian are a nice ...
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0answers
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Trouble in proving geodesic action invariance under diffeomorphism

In this post Diffeomorphism invariance and geodesic action it is said: You found (by computing in local coordinates) that this is invariant under a diffeomorphism $\phi: M \to M$. This statement ...
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28 views

Symplectic and Euclidean structure invariance

Consider a $2n$ real symplectic space - the usual $\mathbb R^{2n}$. Suppose that the same space could be endowed too with an Euclidean structure, by which the vectors of the symplectic basis are ...
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1answer
101 views

How to calculate initial conditions to integrate a null geodesic

Suppose, this is the line element of a FLRW metric, $$ ds^2 = -[1 + 2ψ(t,x_i)]dt^2 + a^2(t) [1 - 2ϕ(t,x_i)]dx_i^2 $$ and the geodesic equation is, $$ \frac{d^2x^α}{dλ^2} = - Γ_{βγ}^α \frac{dx^β}{dλ} \...
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0answers
29 views

Understanding the variational / functional derivative (in relation to the Euler-Poincaré equation)

I'm trying to understand the Euler-Poincare equations, which reduce the Euler-Lagrange equations for certain Lagrangians on a Lie group. I'm reading Darryl Holm's "Geometric mechanics and symmetry", ...
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1answer
53 views

How can the exponential generator apply to all Lie groups (not just rotation)?

How can it be shown that any element of a Lie group can be represented as $A=e^{ig_A V^A}$? I think this results from the exponential map. In the case of $SO(3)$ it can be shown through the Taylor ...
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1answer
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Active Diffeomorphism

I am a little confused about active diffeomorphism $f:M\to M$. Let us focus on translation. When we say that we are doing an active translations does that means that all the particles $\gamma$,and ...