Questions tagged [differential-geometry]
Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
2,637
questions
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Curvature from change of coordinates
This is from Zee's book on General Relativity. Using
$$g_{\mu\nu}=\delta_{\mu\nu}+B_{\mu\nu,\lambda\sigma}x^\lambda x^\sigma+...$$
$$x^\mu=x'^\mu+M^\mu_{\nu\lambda\sigma}x'^\nu x'^\lambda x'^\sigma+.....
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3answers
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Definition of spacetime in GR
In all the references/textbooks that I have looked at, the precise definition of spacetime is never really clear. By gathering the hypothesis that we need to make, I get the following definition: $$\...
1
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1answer
58 views
Locally Flat Understanding
I wanted to make sure that I was definitely understanding the proof of locally flat correctly. I can't see to find a similar proof to the one in the book, so I'm not super sure if my understanding/...
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0answers
38 views
Doubt on Newman-Janis algorithm for a traversable Wormhole
Recently in a paper $[1]$ the researchers presented a rotating traversable wormhole solution using the famous Newman-Janis Algorithm $[2]$. But something is anoying me. In $[1]$ they presented the ...
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1answer
63 views
What are the local Lorentz transformations in general relativity?
What is the exact form of local Lorentz transformations (from the point of view of the metric) in a curved spacetime background like in general relativity? It should deviate substantially from ...
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3answers
89 views
An identity between the d'Alembertian and the covariant derivative
Suppose $f$ is function which depends on $\phi$, $f = f(\phi)$; and $\phi$ is a scalar field. We define
$$\square \equiv g^{\mu\nu} \nabla_\mu \nabla_\nu$$ and $$\nabla_\mu f(\phi) \equiv f_{;\nu}$$
...
2
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1answer
59 views
Covariant derivatives in a rank 2 tensor
I was trying to prove that for any second order tensor:
$$A^{\mu\nu}_{;\mu\nu}=A^{\mu\nu}_{;\nu\mu}$$
considering the torsion free property and locally flat coordinates. Considering the point where ...
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1answer
46 views
Conditions on one forms [closed]
I am trying to solve exercise 8.3 from Lightman's problem book, but I don't know where to start to get a sufficient and necessary condition on a field of one forms $\tilde\sigma$ for there to exist a ...
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2answers
48 views
Is there any way to prove that contrarly to a flat 3D space, a curved 3D space can only be constructed in a 4D manifold?
This question is a result of me trying to understand how this universe can be possibly infinite if it isn't infinitely old. So to compare with an area that is flat it can be constructed both in 2D and ...
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1answer
37 views
Commutators and coordinate-induced basis (General Relativity)
There is an exercise in MTW that asks to prove that, given two vectors u and v, there exists a coordinate system for which
$$
\textbf{u}=\frac{\partial}{\partial x^1} \mbox{ and }\textbf{v}=\frac{\...
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2answers
55 views
Proof of the differential Bianchi identity
I was trying to prove the differential Bianchi identity by applying the covariant derivatives to each of the Riemann tensor terms
$R^{\lambda}_{\sigma\mu\nu;\rho}+R^{\lambda}_{\sigma\nu\rho;\mu}+R^{\...
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Weyl transformation vs diffeomorphism; conformal invariant vs general in/covariant
Background info:
My understanding:
1.
Weyl transformation is a local rescaling of the metric tensor
$$
g_{ab}\rightarrow e^{-2\omega(x)}g_{ab}
$$
A theory invariant under this Weyl transformation is ...
2
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1answer
54 views
Complex Lie algebra vs Real Lie algebra in Physics [closed]
A Lie algebra is a vector space $\mathfrak{g}$ over some field $F$ together with a binary operation $$\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$$ called the Lie bracket satisfying the following ...
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0answers
33 views
Sectional curvature to Ricci tensor
We know that to measure the if the n-dimensional manifold is curved or not, we look at the Riemann Tensor. If it is 0, then it is flat otherwise it is curved positively or negatively (which each of ...
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0answers
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Deforming a nematic line defect to a uniform configuration
In Nakahara section 4.9, "Defects in nematic liquid crystals", it is discussed that the order parameter for a nematic should be the real projective plane $\mathbb{R}P^2$, which has ...
2
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1answer
63 views
Weyl connection
Considering the Weyl connection $\Gamma^{\lambda}_{\mu\nu}$ in a torsion-free space, defined by the covariant derivative of the metric tensor and a vector field $A_{\lambda}$:
$D_{\lambda}g_{\mu\nu}=...
4
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0answers
58 views
Writing the EL equations in the language of differential geometry
I want to explore generalised Noether currents obtained from $q$-form symmetries in an action.
The regular way we obtain Noether currents is fairly straightforward: We have a 0-form symmetry $\phi \to ...
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0answers
32 views
Taking Chart-Coordinate Derivatives of Geodesic Tangents
I've been working to understand cuvature in relation to geodesic deviation. Reading Wald, he uses a family of gedesics parameterized by two coordinates $s,\tau\in\mathbb{R}$ called $\gamma(s,\tau)$, ...
3
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0answers
83 views
How did the Lagrangian and Hamiltonian theories of motion inspire the idea that forces should be treated as one-forms instead of vectors?
On page-5 of this paper1 by E. Minguzzi titled "A geometrical introduction to screw theory", he writes:
Who adopts this point of view argues that it should also be adopted for forces in ...
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0answers
33 views
Spinor covariant derivative conventions
The covariant derivative of a spinor $\psi$ is given by
$$ \nabla_\mu \psi = \partial_\mu \psi + \Omega_\mu \psi $$
where $\Omega_\mu$ is the spin connection. In equation (7.227) of Geometry, Topology ...
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3answers
107 views
Why isn't $g^{\mu \nu}\partial_{\nu}$ equal to $\partial^{\mu}$ in General Relativity?
The question is pretty short. I have been told that unlike for flat spacetime, in curved spacetime we cannot write
$g^{\mu \nu}\partial_{\nu}f=\partial^{\mu}f$
With $f$ an scalar function, but I don't ...
4
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1answer
119 views
Could spacetimes have singular manifolds?
Let's take a spacetime as a pair $(M,g)$ where $M$ is the manifold and $g$ the metric.
I've seen that there exist a generalization of manifolds. This generalization consist in accept singularities in ...
3
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2answers
93 views
How can a 2-sphere exist in Euclidean 3-space?
I don't know if this is a simple question to answer however, I have trouble understanding how a spherical object (such as a planet) with positive curvature can exist in Euclidean 3-space with no ...
2
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2answers
87 views
How momentum is dual to the velocity vector at a point on a differentiable manifold?
The tangent space $T_pM$ which is a real vector space on a point $p$ of a differentiable manifold $M$, has a cotangent bundle $T_p^*M$ at $p \in M$, such that for any $v \in T_pM$ and for any $w \in ...
3
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1answer
66 views
Weyl Tensor Theorems
Is there any quick way of proving that for a conformal metric of the form:
$g^{}_{\mu\nu}=\Omega^2\eta^{}_{\mu\nu}$
where the $\eta^{}_{\mu\nu}$ is the usual Minkowski metric, the Weyl tensor vanishes ...
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0answers
26 views
Extrinsic Description of spacetime curvature, is it possible? How many dimensions would we need? [duplicate]
Imagine we had an extra spatial dimension $f$ and so an overall spacetime interval:
$ds^2=-dt^2+dx^2+dy^2+dz^2+df^2\tag{1}$
Now let's say that we're restricted to a 4-dimensional "plane" in ...
2
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0answers
68 views
Christoffel Symbols Different Representation?
I was given the $\mathbb{R^2}$ metric in polar coordinates, as follows:
$$
ds^2=dr^2+r^2d\theta^2
$$
In this context we denote $e_1=\partial_r=(\cos(\theta), \sin(\theta))$, $e_2=\partial_{\theta}=(-r\...
2
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0answers
42 views
Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives
For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives.
I have tried ...
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0answers
35 views
The role of diffeomorphism in physics
Suppose space time is the manifold $M $ isomorphic $ \mathbb{R^4}$ with the metric $-\eta_{00}=\eta_{11}=\eta_{22}=\eta_{33}=1$ in the Cartesian coordinates $\Psi(p)=(x^0,x^1,x^2,x^3)$ for $p \in M $ ....
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0answers
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Generalisation of Equilateral Triangle in curved space [migrated]
It is known that in flat Euclidean Space the internal angles of a triangle sum to $180^{\circ}$ but if we draw a similar triangle in curved space then the sum of interior angles will be more or less ...
2
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1answer
81 views
Variation of d'Alembertian operator
I am working in a higher derivative quantum gravity theory, and I'm having trouble with the variation of d'Alembertian operator.
Suppose we have the following action:
\begin{equation}
\mathcal S[g]=\...
2
votes
1answer
59 views
Do separation vectors exist in general relativity?
Deriving the equation of geodesic deviation one looks at two test masses on positions $x^\mu$ und $\tilde{x}^\mu$ and defines the separation vector $\boldsymbol{\chi}$ as
$$\tilde{x}^\mu=x^\mu+\chi^\...
3
votes
1answer
62 views
Quantum Geometric Tensor and Berry Connection [closed]
The Quantum Geometric Tensor is given by
$$
\begin{split}
Q_{\mu\nu}=\langle\partial_{\mu}\psi|\partial_{\nu}\psi\rangle-\langle\partial_{\mu}\psi|\psi\rangle\langle\psi|\partial_{\nu}\psi\rangle
\end{...
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0answers
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Horizontal vector subsepace of electromagnetic connection in Minkowskian spacetime
For Minkowskian space-time $M$, the principal bundle for electromagnetism is $(M \times U(1), M, proj_{1}, U(1))$. I imagine there is a global gauge potential (since I can choose a global section, ...
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1answer
38 views
How Would Projectile Motion Be Described in Non-Euclidean Space?
If I and a friend found ourselves in a world described by spherical geometry (as simulated in this linked video https://youtu.be/yY9GAyJtuJ0 ) how would the kinematics equations need to be augmented ...
3
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1answer
105 views
Heat kernel: DeWitt iterative procedure
The DeWitt ansatz for the heat kernel is given by
$$K(t ; x, y ; D)=(4 \pi t)^{-n / 2} \Delta_{V V M}^{1 / 2}(x, y) \exp \left(-\frac{\sigma(x, y)}{2 t}\right) \Xi(t ; x, y ; D)$$
where $\sigma$ is ...
0
votes
1answer
105 views
Prove $\displaystyle{\not}{\nabla}^2=\nabla_{\mu} \nabla^{\mu}-R/4$ [closed]
I am trying to show $\displaystyle{\not}{\nabla}^2=\nabla_{\mu} \nabla^{\mu}+R/4$ where $R$ is Ricci scalar. $\nabla_{\mu}$ is covariant derivative for spinor:
\begin{equation}
\nabla_{\mu}=\partial_{...
2
votes
2answers
50 views
Definition of parallel transport
The definition of parallel transport is $t^iD_i u^j=0$, where $\vec{t}$ is the tangent vector to the curve and $\vec{u}$ is the vector being parallel transported along the curve. In flat space, using ...
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0answers
26 views
Transverse metric being 2-dimensional in null case
In Wald section 9.2 page 221 he says that
We turn our attention; now , to null geodesic congruences. Again, we
parameterize the geodesics by an affine parameter $\lambda$, but ,
unlike the timelike ...
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0answers
32 views
Cauchy Surfaces inside Schwarzschild Black Hole
Context: Working in a globally hyperbolic spacetime we know by definition that we can foliated our spacetime $\mathcal M$ with disjoint cauchy surfaces $\Sigma_\alpha$ $\left(\therefore \bigcup \...
3
votes
1answer
104 views
Does spacetime topology have importance in physics?
Generally in textbooks they represent spacetime as $(M,\nabla,g,t)$
where $M$ is a Lorentzian manifold,$\nabla$ a torsion-free connection,$g$ a metric and $t$ a time orientation.
But they do not talk ...
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0answers
41 views
What does a universe with a boundary look like?
Physically, what would it look like if we lived in a universe with a boundary at finite distance?
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1answer
62 views
Alternative formula for the affine connection in a new coordinate basis
In Hobsons's General Relativity: An Introduction for Physicists, pg. 64,
he gave two different expressions for the affine connection $\Gamma'^a_{bc}$ in a transformed coordinate basis $x'^a$ (the ...
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2answers
40 views
Isn't the following addition wrong on manifold as done in Frankel book?
In ch-$4$ when calculating expression of Lie derivative using Hadamard's Lemma before $(4.4)$ Frankel's do following manipulation:
$$\lim_{t\rightarrow0}\frac{\textbf{Y}_{\phi_tx}(f)-\textbf{Y}_x(f)}{...
1
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1answer
39 views
Conformal vector fields in $m$-dimensional Euclidean manifold
A vector field $X=X^\mu\partial_\mu\in\mathfrak{X}(M)$, where $M$ is a (pseudo-)riemannian manifold with a generic metric tensor $g_{\mu\nu}$, is a conformal Killing vector field if the conformal ...
4
votes
1answer
70 views
What motivates defining vectors as first order differential operators?
I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
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18answers
5k views
Does it make sense to take an infinitesimal volume of shape other than a cube?
The question clearer: Is the infinitesimal cube the absolute smallest infinitesimal volume?
(Sorry if people thought that it meant: "Is it possible and is it done in daily life to use anything ...
3
votes
1answer
69 views
Meaning and application of the connection coefficients (Christoffel symbols)
I know that in polar coordinates, it is
$\frac{\partial \,{{\mathbf{e}}_{r}}}{\partial \theta }={{\mathbf{e}}_{\theta }}$ and $\frac{\partial \,{{\mathbf{e}}_{\theta }}}{\partial \theta }=-{{\mathbf{e}...
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0answers
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Conformal Transformation and angle preserving [duplicate]
For general co-ordinate transformation $V^\mu$ transforms as
$V^\mu \rightarrow V'^\mu= \large\frac{\partial x'^\mu}{\partial x^\alpha}\ V^\alpha $
therefore the inner product between two vectors ...
1
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0answers
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From Reissner-Nordstrom to Dilatonic Gravity and Einstein Equation
I am doing some calculations of this paper: https://arxiv.org/abs/1711.08482 and in particular I am having trouble with 2 parts:
Dimensional Reduction (16):
I have managed to get equation (16) from ...
