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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Confusion on number of component of Cauchy stress tensor
The Cauchy stress tensor is often presented as a tensor having $(2,0)$ tensor having nine components in any given basis. However, I think it should actually be $6 \times 3 =18$ because a cube has six ...
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Book which deals with constraint motion using differential forms
I'm interested in books going more over the approach described in this post. Here is a quote of the total answer in case anything happens to it:
Calculating the wedge product $df_1\wedge df_2$ gives $...
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What is the difference between Dirac delta vs removing point from space approach?
Let us take for instance E&M, in it when we deal with the failure of divergence theorem to give us the right expression when evaluating the volume integral of divergence of electric field over ...
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Aristotelian vs Galilean relativity in terms of bundles
In page-385 of Roger Penrose's Road to Reality, the following is written:
In our Aristotelian scheme, it is appropriate to think of spacetime as simply the product:
$$ \mathbb{A}= \mathbb{E}^1 \times ...
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Parallel transport geometric phase derivation
In Modern Foundations of Quantum Optics by Vlatko Vedral, he derives an expression for the change in phase under parallel transport after completing a small loop as
$$\begin{align} \delta \theta &=...
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Where is the Lorentz signature enforced in general relativity?
I'm trying to understand general relativity. Where in the field equations is it enforced that the metric will take on the (+---) form in some basis at each point?
Some thoughts I've had:
It's baked ...
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What is a specific example of a Calabi-Yau manifold? are there simple ones like a 6-torus, $T^6=(S^1)^6$ or $S^3\times T^3$
What is a specific example of a 6D Calabi-Yau manifold? are there simple ones like a 6-torus, $T^6=(S^1)^6$ , $S^3\times T^3$, or similar structures with products of Spheres and Torus?
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Coordinate of a point in AdS spacetime
Is there any way to get the coordinate of a point located on a plane, where the plane is in AdS spacetime with Lorentzian signature (-,+,+,+)? I'm thinking that Lorentzian geometry will be involved to ...
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On general covariance
If newton's theory could be formulated in the language of differential geometry (symplectic manifolds), what do we really mean when we say that the theory is covariant under the Galilean group when it'...
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Pressure in stokes flow
In the stokes equation for an incompressible fluid, there is a pressure term that enforces the fluid to have $\nabla \cdot u=0$, where $u$ is the velocity field. The stokes equation reads:
$$\eta \...
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Liouville's theorem on the tangent bundle [duplicate]
One interpretation of Liouville's theorem is the determinism and reversibility of classical mechanics, i.e. the mechanical states can't converge or diverge. The theorem is often formulated on the ...
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What does Penrose mean when he talks about topology of spacetime?
Let us now set aside the question of the submicroscopic structure of space-time and concentrate, instead, on its large-scale properties. In this case, we may imagine that the smooth manifold picture ...
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Is the volume in general relativity independent or dependent on the coordinates?
The volume in curved space is calculated as:
$$V=4 \pi\int_{\Omega}r^2\sqrt{g_{rr}} d\Omega$$
Is this volume dependent or independent from the chosen coordinates? As I understand it should be ...
user336075
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Could 2D spacetime be seen as an embedded manifold?
In a chart of a 2-dimensional spacetime manifold, we can write the rule giving separation between two points as:
$$ ds^2 = - dt^2 + dx^2.$$
Could we use this to imagine how space time looks like a ...
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What is the structure group of the $SU(3)$ manifold?
It has been known for many years that the manifold of the Lie group $SU(3)$ is the non-trivial fiber bundle of the spheres $S_5 \times S_3$, in which $S_5$ is the base and $S_3$ the fibers. What ...
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Are curvilinear coordinates inertial?
At 1:46:34 of this lecture by Frederic Schuller, Inertial coordinates are defined as ones which satisfy the following equation:
I am confused by the above equation because it would imply any ...
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How to write down the metric explicitly?
I know that we can derive Schwarchild metric by imposing torsionless condition.
if the torsion vanish, we can write down spin connection explicitly by veilbein.
After variation to veilbein of Einstein-...
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Spherical polar coordinates [closed]
Please help me in deriving the details equations of motions of a particle under the influence of EM force, coriolis and centripetal force. This is for B.Sc Introduction to Astrohysics. In Some papers ...
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How to make sense of the length of a rod in Special Relativity, using the mathematical framework of General Relativity? [duplicate]
This question asks how exactly Special Relativity (SR) emerges mathematically as a special case of General Relativity (GR).
In GR, spacetime is modeled as a pseudo-Riemannian manifold with generally ...
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Normal to the hypersurface
I have a given metric and I want to choose a spacelike hypersurface and find the normal to that hypersurface. I know that if the hypersurface is spacelike, then the normal is timelike.
The given ...
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Identity $\nabla_{j}\partial_{i} = \Gamma ^{k}_{ij} \partial_{k}$ involving covariant derivative
I am trying to understand the identity i have came across, but i am not being able to:
$$\nabla_{j}\partial_{i} = \Gamma ^{k}_{ij} \partial_{k}$$
I thought that such equality would become obviously if ...
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Could Spacetime have a more General Geometry than just a Metric Field? [closed]
In standard general relativity the fundamental structure of spacetime is given through the six tuple $({M},\mathcal{O},\mathcal{A},{g},\nabla, {W}[\Phi,{g}])$, where ${M}$ is a smooth four-dimensional ...
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Isometry between Minkowski space and Tangent space
In this notes Geometric Wave Equations by Stefan Waldmann at page 70 they have
Having a fixed Lorentz metric $g$ on a spacetime manifold $M$ we can
now transfer the notions of special relativity, ...
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How do we figure out what is the right geometry of space?
In page-319 of Visual Differential Geometry, the following is written:
When we speak of a solution to Einstein's equation, we mean a geometry of space time (defined by it's metric) that satisfies the ...
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A question for expert in geometrical method and Riemannian metrics
I'm a physical oceanographer with great interest in Theoretical Geophysical Fluid Dynamics. I have some ideas on the possibility to derive the so-called: geostrophic equilibrium (i.e. on a rotating ...
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Vanishing expansion of a geodesic congruence
I am considering a timelike geodesic in the outer region of the Schwarzschild spacetime, whose tangent vector field is denoted by $X$.
I know that we can construct geodesics moving on a plane $\{\...
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How to simplify the process of calculating spacetime geodesics?
I want to study the movement of a particle along geodesics in an expanding universe with metric (FRW metric)
$$
ds^2 = -dt^2 + a^2(t) \left( \dfrac{1}{1-kr}dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\...
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Does physicists have a pre-conceived notion of continuity?
In many physics lecture on GR/ mathematical physics, one of the first things discussed is topology. I have seen many times that the reason for topology being discussed is that it's the weakest ...
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Smooth vs analytic spacetimes
Recently in more technical settings (I was learning algebraic QFT), I encountered the term "real analytic" manifolds (Lorentzian manifolds, to be precise). This is in contrast to smooth ...
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Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?
Newton's second law is a coordinate agnostic statement, we can use it to calculate the forces in a coordinate system, and hence, the motion of the body in that coordinate system. However, depending on ...
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About the equation $\frac {d^2} {dt^2}\vec x(t) = \nabla \times \vec F(x(t))$. Motion in a curl vector field
I was wondering if there is a physical interpretation of ODEs of the form
$$\frac d{dt}\vec x(t)=\vec y(t)$$
$$ \frac d{dt} \vec y(t) = \nabla \times \vec F(x(t))$$
(or equivalently $\frac {d^2} {dt^2}...
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Change of Metric Under Coordinate Transformation
Under a local change of coordinates $x\to x'=x+\delta x$, the metric transforms as
$$g_{\mu \nu}^{\prime}\left(x^{\prime}\right)=g_{\lambda \rho}(x) \frac{\partial x^{\lambda}}{\partial x^{\prime \mu}}...
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Generalizing Fermi-Walker Derivative/Transport to General Vector Bundles
The usual definition of the Fermi derivative (eg as given in Hawking and Ellis) is to consider a Lorentzian manifold $(M,g)$ and a unit timelike curve $\gamma$, a smooth vector field $X$ along $\gamma$...
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Are tensors constructed such that one forms "act" on some complex vector field?
I have some confusion understanding the motivation in constructing tensors (or tensor fields). On a differentiable manifold $\mathcal{M}$ consider a vector field $X$. At any point $p\in \mathcal{M}$, ...
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Trouble with the Ricci tensor formula giving alternate answer than the answer given by the computer
Suppose we have a metric $g_{\mu\nu}$ and want to find the Ricci tensor components. The formula for the Ricci tensor is $$R_{\mu\nu}=\partial_{\alpha}\Gamma^{\alpha}_{\mu\nu}-\partial_{\nu}\Gamma^{\...
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Where did Henry Poincare state that we should not sum linear movements with curvilinear movements?
Years ago, I was arguing with a P.h.d. in Relativity, and he showed me a print screen that Henri Poincare stated that we should not sum curvilinear movements with linear ones.
But, I lost that print ...
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Physical significance of metric compatibility
When we try to construct a covariant derivative, we impose several conditions on it so that the resulting derivative is unique. However, I can't make sense of the condition of metric compatibility. I ...
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Geodesic deviation and Lie dragging
Suppose that $𝑥_\mu(\lambda,𝑠)$ represents a family of curves. Let $𝑣_𝜇$ represents the the tangent vector to a curve $𝑥_𝜇(\lambda,𝑠_0)$ with $𝑠_0$fixed that is $𝑣_𝜇=∂𝑥_𝜇/∂\lambda$ and ...
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Kerr Solution metric ansatz for EFEs
The Schwarzschild metric ansatz is given by $$ds^2=-A(r)dt^2+B(r)dr^2+r^2d \Omega^2$$ where upon applying the Einstein Field Equations $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we obtain the normal ...
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What is a signature of pp-wave metric?
pp-wave spacetime metric is defined in Brinkmann coordinates as $$ds^2 = H(u,x,y)du^2 + 2 du dv + dx^2 + dy^2.$$ Since it's lorentzian (https://en.wikipedia.org/wiki/Pp-wave_spacetime), I wonder what ...
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How to find a complete understanding the 2nd law of Thermodynamics in terms of forms?
I have two straightforward question, and below I introduce more context to interpret them:
What is, or is there, an order relation for forms that one can use to make sense of the 2nd law of ...
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What is the idea behind 2-spinor calculus?
In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
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A particular contraction of Levi-Civita symbols and tetrads
Consider a four-dimensional spacetime. Consider the following contraction between Levi-Civita symbols and tetrads
$$\epsilon_{\alpha \beta i j}\,{\epsilon^{ij}}_k\, e^\alpha\!\wedge e^\beta\!\wedge e^...
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Programs/Scripts for plotting penrose diagrams
What programs do people generally use to plot penrose diagrams? I need to plot some simple ones for the dS and AdS and Schwarzschild metric and my hand drawn ones dont look very good.
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Transformation of vielbein and spin connection
I'm trying to derive the transformation of vielbein and spin connection under local Poincare transformation:
$$
\delta{e_{a}}^{\mu} = {e'_{a}}^{\mu} - {e_{a}}^{\mu} = (\Lambda^{\nu}\partial_{\nu}{e_{a}...
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Geodesic equations with varying mass and the variational principle
Consider the action,
$$
S
=
\int d\lambda\
\phi(x)
\left(
-g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{dx^\nu}{d\lambda}
\right)^{1/2}.
$$
Using the variation principle we obtain,
$$
\delta S
=
\int d\...
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Basis Vectors as Partial Derivatives Issues
I have been introduced a number of times to people defining vectors as derivatives of a curve, with basis vectors as partial derivatives, but I have several issues with this that make this formalism ...
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Components of the fully contravariant Kronecker Delta in Schwarzschild Metric
I thought the kronecker delta $\delta^{\mu\nu}$ should always be of value $1$ if both indices are equal and $0$ if they are different. However it seems that the delta has components different from $1$ ...
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Can the metric tensor be treated as a linear transformation?
In general relativity, the metric tensor $g$ is a covariant, second rank, symmetric tensor that can be written down as a 4x4 matrix. The metric tensor generalizes the notion of distance between points ...
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Ricci Identity with Torsion Proof
In the notes I'm using for General Relativity, the author begins their proof of the Ricci identity with torsion by writing
$$\nabla_{[\mu}\nabla_{\nu]}Z^{\sigma}=\partial_{[\mu}(\nabla_{\nu]}Z^{\sigma}...
