Questions tagged [differential-geometry]
Mathematical discipline which studies some properties of smooth manifolds, which allow to generalize calculus to beyond $\mathbb{R}^n$. General relativity is written in this language.
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Lorentz scalar from the second derivative of the metric tensor
This might be a very simple question. Is the following second derivative of the metric tensor a Lorentz scalar: $\partial_{\mu}\partial_{\nu}g^{\mu\nu}$ ? I know that for a vector field, $\partial_{\...
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Curvature due to a Small Mass in Minkowski Spacetime
Flat spacetime is described by the Minkowski metric $\eta_{\mu \nu}$. Now let's say that we place a small test particle with a very small mass in the spacetime (we could say the particle is spherical ...
- 1,411
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Usefulness of the contact geometry structure in Non-Equilibrium Thermodynamics
As far as I know, there are quite a few questions (1, 2, 3) on Physics SE about the geometrical structure of thermodynamics.
I am really intrigued by this approach, but given that in equilibrium ...
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+50
Unimodular gravity and volume preservation: normalization with Jacobian determinant possible?
Unimodular gravity restricts the usable coordinate systems / coordinate transformations so that the unimodular condition is met. See equations (2) and (3) in https://arxiv.org/abs/2301.07641
It "...
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Physical reasons for representing Maxwell equations using TWISTED (exterior) differential forms [closed]
What are the physical reasons for representing D,H,P,M,J as twisted (odd) differential forms and E,B as untwisted (even) differential forms? Most materials I've read, even those focused on ...
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Why do we have to assume global hyperbolicity in the singularity theorems?
Why can't we just assume the strong energy conditions and the expansion $\theta_0 \leq 0$ at some point to conclude that spacetime is singular?
Given that any geodesic stops being the path of maximal ...
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How to combine Vierbein fields?
As far as I understand each Vierbein field $e^{a}_{\mu}$ and $e^{b}_{\nu}$ can be represented by a $4\times4$ matrix that can act on the Minkowski metric $\eta_{ab}$ to give the curved spacetime ...
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Derivative of geodesic $z^\mu(\lambda;x_i,x_f)$ respect to initial position $x_i$
I'm working on a problem on General Relativity where I use the geodesic $z^\mu(\lambda;x_i,x_f)$ parametrized by $\lambda$ that starts at $x_i$ and ends at $x_f$
$$ z^\alpha(\lambda_i)=x_i^\alpha\\ z^\...
- 4,647
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3
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Derivation or origin of projection tensor
Can the projection tensor that is used in General relativity be derived using formal mathematics?
In GR, the projection tensor orthogonal to a non-null vector is defined as,
$$ P_{\mu \nu} = g_{\mu \...
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2
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How does an electromagnetic wave warp spacetime?
I am trying to understand how an electromagnetic wave would curve spacetime and what effect this curvature in turn would have on the wave.
I am imagining an electromagnetic plane wave travelling in ...
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Berry curvature with Hermitian matrix interpretation
In my calculation for some effect I get terms that are proportional to
\begin{equation}
i\epsilon^{\mu \nu}\langle \partial_{\mu} \, u_n(k) | M(k) | \partial_\nu \, u_n(k) \rangle
\end{equation}
...
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Adjoint of the covariant derivative of a field?
Let's call $D$ the covariant derivative, $T$ the transposition of a field and $*$ its complex conjugate, so $T*$ is the "adjoint".
Is: $$(D_{\mu}\Phi)^{T*} (D_{\mu}\Phi)=D^{\mu}\Phi^*D_{\mu}\...
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"Large scale structure of spacetime", page 16: "the length of $V$ can be determined by $V(t)=1$"
On page 16 of the book The large scale structure of spacetime by Hawking and Ellis the authors discuss the definition of a tangent space:
so the space of all tangent vectors to $\mathcal{M}$ at $p$, ...
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Understand the Lorentz transformation in QFT
I am a beginner in QFT,so let me introduce my question by the problem in MIT8.323 (2023 spring,by Hong Liu) pset 1
.
All steps seem clear, the measure is invariant because the Jacobian is 1 and the ...
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Is it possible to use topology arguments to find analogies in thermodynamic systems?
I was contemplating whether, given the mathematical structure of thermodynamics, it might be possible to restate some of its most important propositions—or even all of them—purely in topological terms....
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PFBs and Gribov ambiguities
Recently I have been studying mathematical gauge theory and have come across some confusion regarding the connection between PFBs and physical gauge theories. I have a primarily mathematical ...
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Requesting reliable references for the Physical Meaning of Differential Geometry objects in General Relativity
I am seeking to understand the physical significance of certain differential geometry objects commonly used in general relativity, such as:
The Ricci tensor
The Ricci scalar (curvature scalar)
The ...
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1
answer
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When is a classical field theory on curved spacetime supersymmetric?
Our setup is a classical field theory which is defined on any (generally curved) spacetime $(M,g)$. For Minkowski spacetime $\mathbb{R}^{1,n-1}$, the theory is invariant under the action of a ...
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Affine parameterization
Considering a curve $x^\mu$ parameterized by parameter $u$, the tangent to the curve satisfies
\begin{equation}
\eta^\mu \nabla_\mu u =1
\label{affine}
\end{equation}
where $\eta^\mu = \frac{dx^\mu(u)}...
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Definition of global supersymmetry on curved spacetimes and use of constant spinor fields
According to "Quantum Fields and Strings: A Course for Mathematicians" (Deligne et al.), we call theories supersymmetric if their action functional is invariant under the action of a super ...
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Non-metric gravity calculations
According to "Gravity and Strings" by T. Ortin (2015), the non-metricity tensor is calculated as
$$
Q_{\rho\mu\nu}\equiv\nabla_\rho g_{\mu\nu}=\partial_\rho g_{\mu\nu}-\Gamma^\beta_{\rho\mu}...
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Showing that a generator exponentiates to a $\mathbb{R}$ group
I have a generator $G$ that acts on the phase space of Schwarzschild and maps geodesics into each other.
In order to discuss the corresponding symmetry group, I need to exponentiate this generator and ...
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What is the difference between "maximal Cauchy development" and "maximal global hyperbolic development" within General Relativity?
Following the theorem of Choquet-Bruhat and Geroch, we can define the maximal Cauchy development of initial data (see Theorem 10.2.2 of Wald's 1984 GR book, herewith attached). However, in the ...
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On the Background Independence condition
In General Relativity, one has that the equations of motion for any matter distribution are given by extremizing the following action:
\begin{equation}
S[g] = \int\left[\frac{1}{8\pi}(R - 2\Lambda) + ...
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EM junction conditions from Faraday tensor
In classical electrodynamics, we have the Maxwell's junction conditions:
$$
\mathbf{n}\cdot[\mathbf{B}]=0
$$
$$
\mathbf{n}\cdot[\mathbf{D}]=\Sigma
$$
$$
\mathbf{n}\times[\mathbf{H}]=\mathbf{K}
$$
$$
\...
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What is the difference between the Penrose Conjecture and the Penrose Inequality?
The Penrose Inequality applies to specific initial data sets in general relativity and has been proven under certain conditions, particularly in asymptotically flat spacetimes with non-negative scalar ...
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In a Spatially One-Dimensional Universe, is a Minkowski Space-Time Diagram accurately graphable if we include the effects of "gravity"?
I've been working on studying Special Relativity and General Relativity for the past few years. As I think we all know, GR gets a lot more complicated than SR, and my knowledge is limited. I am very ...
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Covariant derivative with torsion
The covariant derivative is defined (on contravariant vectors) as:
$$\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu \rho} V^\rho \tag{1}$$
The purpose of the covariant derivative is to ...
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Why general relativity agrees with Newtonian theory in the limit?
I'm currently reading "The Large Scale Structure of Spacetime" by Hawking and Ellis. While the mathematical computations are clear to me, I find myself puzzled by the physics, particularly ...
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Is a spin connection with torsion possible whereas the affine connection is only Levi-Civita (torsion-free) in Supergravity?
In the paper "Simple Supergravity" from G. Dall'Agata & M. Zagermann (arXiv:2212.10044v2 15 Feb. 2023) on page 8 when it comes to the antisymmetric part of the covariant derivative of ...
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Proof of Bianchi identity with the use of Riemann normal coordinate
In many GR books the Bianchi identity is proved by taking covariant derivative of the Riemann tensor in the Riemann normal coordinate $$R_{\rho\sigma\mu\nu}(p) = \frac{1}{2}(\partial_\mu \partial_\...
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Hamilton's characteristic function, wave-particle duality and constant-action surfaces
So, I'm currently doing some research about the way in which classical physics connects to quantum physics, and I came across the Hamilton-Jacobi equation, and the implications of Hamilton's ...
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Can one encode topology into the position operator in quantum mechanics?
We work with the concrete example of electromagnetism, but we intend to ask a question in broader scope at the end.
In classical field theory, the electromagnetic potential $\mathcal{A}$ lives on a ...
- 2,762
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Spacetime coordinates 4-dimensional vs curvature of space time? [closed]
what is the difference between spacetime coordinates 4-dimensional and curvature of space time I dont know the difference between both of them
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Inverse problem for geodesic
If I know the expressions for geodesic distance between any points $x$ and $y$:
$$L=L(x^\mu,y^\nu) \ .$$
How do I find the metric of the corresponding space?
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When dealing with a space like $AdS_5$, why can we take the universal cover?
In the context of the AdS/CFT correspondance, when considering $AdS_5$ we can pick the coordinates
$$x_0=R\:cosh(\rho)\: cos(\tau)
\\ x_5=R\:cosh(\rho)\:sin(\tau)
\\ x_i=R\:sinh(\rho)\:\hat{x}_i,\;\;\;...
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What's the relevance of geometric rigidity/flexibility to physics?
I'm currently working on a mathematics research problem in differential geometry that deals with the rigidity of closed manifolds described by non-trivial induced metrics. I'm curious what the ...
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1
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Confusion on the definition of the Christoffel symbols
A common way to introduce the Christoffel symbols is by defining them as determining how the basis vectors change when you translate along your coordinates:
$$\frac{\partial\hat{e}_{(\gamma)}}{\...
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Classical Yang-Mills Versus Quantum Yang-Mills
What insights do the study of instantons and minimizers of the Yang-Mills classical functional give for physics?
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"A coordinate transformation changes components, but not one-forms themselves" [closed]
I'm very puzzled by a statement made by Sean Carroll in his Spacetime and Geometry: An Introduction to General Relativity textbook.
Discussing tensor densities at page 89, he says:
"But the ...
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Calculating Ricci tensor given the metric - undergraduate GR [closed]
I’ve been trying to solve the following problem:
Consider an $(N + n + 1)$-dimensional spacetime with coordinates $\{t, x^I, y^i\}$, where $I$ goes from 1 to $N$ and $i$ goes from 1 to $n$. Let the ...
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1
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Conformal compactification of Minkowski spacetimes
Let's say we compactify the Minkowski space-time through the following coordinate transformation:
$$u=t-r, \\
\Omega=\frac{1}{r},\\
\theta=\theta,\\
\phi=\phi.$$
The conformally rescaled unphysical ...
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Why is $*F^{\mu \nu} *F_{\mu \nu}$ not considered in Lagrangian of EM?
In Lagrangian of electromagnetism $F_{\mu \nu} F^{\mu \nu}$ and $*F_{\mu \nu} F^{\mu \nu}$ are used, where the $*$ denotes Hodge dual. But $*F^{\mu \nu} *F_{\mu \nu}$ is not used, why?
I know an ...
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Global Hyperbolicity of Spacetimes implying Connectedness
I am currently working on a problem and right now I want to show that the global hyperbolicity of a spacetime M implies, that M is connected. Therefore, I assumed the following:
We could write $M$ as ...
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Killing vectors on fixed point surfaces
Given a fixed point surface $S$ for the Killing vector $\xi$ in a $4$-dimensional spacetime. If one has an orthonormal basis for this spacetime with two basis vectors tangend to $S$ and two normal to $...
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Hawking and Ellis Lemma 4.3.1 Proof
I have a few questions about Hawking and Ellis' proof of this lemma (pages 92-93):
Write the $(2, 0)$ stress-energy tensor in coordinates as
$\mathbf{T} = T^{ab} \partial_a \otimes \partial_b$ and ...
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Metric under conformal transformation
I have a question regarding the conformal factor $\Omega(x)$ when dealing with a conformal transformation. We know that under a change of coordinates $x\rightarrow x^{'}=x^{'}(x)$ our metric changes ...
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How does the Ricci tensor describe the changing separation of two airplanes flying from the equator? Conceptually understanding the Ricci tensor
I'm trying to understand the concept of the Ricci tensor and its physical implications using a concrete example involving two airplanes. Suppose two airplanes start at the equator, separated by a ...
- 2,170
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Notation for vector density in Lagrangian density
Consider a manifold $M$ and a Lagrangian density $\mathcal{L} \equiv \mathcal{L}(\phi, \nabla \phi)$. By varying the action, one obtains the equation
$$\int_M \, dV \; \Big( \frac{\partial \mathcal{L}}...
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0
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Index theorem of Callias operator in physics
In the article "On the index type of Callias-type operator" (https://doi.org/10.1007/BF01896237) Anghel study the index of a Callias type operator over an odd dimensional complete ...
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