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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Does General Relativity satisfy the Homotopy (or “h”) Principle?
By this I mean in the standard second order (whether metric or tetrad/verbein-based) form of General Relativity. I've been reading about the homotopy principle of late (see Eliashberg's introduction ...
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Is there a physical interpretation of why Christoffel symbols do not transform like a tensor? [duplicate]
I understand mathematically why they don’t, but I was hoping someone could provide a physical interpretation to this. Is there a physical consequence of this fact?
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Duality and corrections to second-order gravity without and with torsion terms
Recently, there appeared a paper by Giacomo Pollari, A Nieh-Yan-like topological invariant in General Relativity, where the action for gravity looks like:
$$S_g=S_{EHP}+S_{HO}+S_{PO}+S_{GB}+S_{NY}+S_{...
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Upper index covariant derivative $\nabla^\mu$
In the book Cosmology by Daniel Baumann, the author states that $\nabla^\mu g_{\mu\nu}=0$, where $g_{\mu\nu}$ is the metric tensor considered (usually the one associated to the Minkowski metric or to ...
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Why no cosmological constant in momentum constraint?
In the ADM formalism of general relativity, one decomposes the Einstein equations in (3+1) dimensions. More explicitely, if the Einstein equations are given by
$$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\...
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Parallel transport in curved spacetime
This is sort of a very introductory question and I am not finding any reference regarding this. And let me know whether my answer is correct or not.
For example, we are parallelly transporting a ...
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Generalized Stokes theorem in superspace
Do the generalized Stokes theorem apply in superspace? Any issues or uncommon behaviour of the gradient, divergence and rotational in superspace?
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What does it mean to have a zero-dimensional induced metric?
I have an integral on the form
\begin{equation}
S=\int d^dx g_{\mu \nu} h^{ab}.
\end{equation}
In this example, $g_{ab}$ is a $d$-dimensional metric, $h_{ab}$ is an co-dim 2 induced metric. I wanted ...
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How does this derivation of the proper time derivative of a covariant vector work?
Define the operator $\frac{D}{D\tau}$ by its action on an arbitrary contravariant vector $A^\lambda$:
$$\frac{DA^\lambda}{D\tau} = \frac{dA^\lambda}{d\tau} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{d\...
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Lie derivative of the partial derivative of the metric
I am currently studying this metric 'object'
$$g_{\mu \nu}\partial_\rho g^{\rho \mu} \partial_\sigma g^{\sigma \nu}$$
which is clearly not a tensor. I want to compute the Lie derivative.
My problem is ...
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Positive and negative kinetic terms for metric degrees of freedom in GR
I am currently studying the paper "Path Integrals and the indefiniteness of the Gravitational Action" by Gibbons & Hawking and Perry. I'm trying to prove (to myself) that for any ...
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What is natural about the Berry connection?
So I asked a similar question here and even though I still believe it's a valid question, the formulation may have been a bit too complicated to pique people's interest, so let me try to break it up ...
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Asymptotic symmetry and Conserved charges in presence of radiation
Question:
For an asymptotically flat space-time, is the asymptotic symmetry group in presence of radiation the same as standard BMS group? If, instead, it is the larger symmetry group (see definition ...
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Why is the acceleration vector the spatial gradient of the lapse function?
If we have a Lorentzian manifold $(M, g)$ with a foliation by spacelike surfaces $\Sigma_t$ with unit-normal vector field $n$, we can define the lapse function $N$ by
$$
\partial_t = N n + X
$$
where $...
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Concise formulation of Berry phase as holonomy of "natural" connection
I've been trying to understand the Berry phase (abelian/non-abelian) as the holonomy of some "natural connection". I almost have all the pieces together, but there are a few parts that are a ...
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Differential geometric perspectives on Lorentz invariance
Consider a general spacetime $(M,g)$. In a differential geometric language, Lorentz transforms (other than being a part of the isometry group of Minkowski space) can be recovered as the ...
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Names of the proposals for higher $p$-form non-abelian gauge symmetries
Beyond 1-form non-abelian YM theories with
$$F=dA+A^2$$
What are the alternative names and proposals to generalize YM to non-abelian higher $p$-forms? What is the current state-of-art of those?
Remark:...
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How do the Christoffel symbols appear in the equations of motion for the Nambu-Goto action in a general targetspace?
I'm trying to derive the equations of motion from the Nambu-Goto action for a general target space metric, and I know that the final result should have a term containing a Christoffel symbol. I couldn'...
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Finding a closed timelike curve for a specific metric [closed]
I am tasked with finding a timelike curve in a specific metric that supports it. How can I approach this problem? Metric is $$ds^2=-dt^2+a(r) dr^2+ b(r) d\varphi^2 + c(r) dtd\varphi,$$ where $a(r)=(1+...
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Proving statements about Killing vector fields
I'm proving some identities on Killing vectors and the like and I've stumbled on this one which I can't seem to figure out.
Suppose $A^\mu = K^\mu$ is a Killing vector and we have the following field $...
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Metric tensor determinant under coordinate transformation
I've been studying GR through Wald's and Carroll's books, and I've been trying to derive one expression.
$$g(x^{\mu^\prime}) = \left|\dfrac{\partial x^{\mu^\prime}}{\partial x^{\mu}}\right|^{-2} g(x^\...
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Metric compatibility and torsion-free condition of GR
In an introduction to general relativity, we see the unique connection of a manifold is described by both the conditions, matric compatibility and torsion-free condition. The metric free condition can ...
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Christoffel symbols and metric compatibility
In some coordinate system ,we can calculate the Christoffel symbols using the following procedure
Basis vectors $\Gamma^k_{ij}\vec{e_k}=\frac{\partial \vec{e_i}}{\partial x^j}$
multiply $\vec{e_l}$ ...
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Path integral quantization for scalar, spinor, and Yang-Mills gauge fields on a general differentiable manifold?
Most of the recommendations on path integrals on differentiable manifolds I've found here (like Kleinert) focus only on formulating quantum mechanics via path integrals on differentiable manifolds. ...
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Homogeneous Maxwell equations from the Bianchi identity
It is easily proven that:
$$\partial_{\lambda} F_{\mu \nu} + \partial_{\mu} F_{\nu \lambda}+ \partial_{\nu} F_{\lambda \mu} = 0$$
Lots of sources say this equation implies the Homogeneous Maxwell ...
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Tortoise coordinate transformation
The differential form $dr$ can be written $\left(1-\dfrac{2GM}{r}\right)dr^*$ where $r^*$ is the tortoise coordinate. Writing the Schwartzchild metric then gives
$ds^2$ = $\left(1-\dfrac{2GM}{r}\right)...
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What would the Raychaudhuri Equation be for accelerated geodesics?
What would the Raychaudhuri Equation be for accelerated geodesics? Suppose we are not able to assume the geodesic equation but rather have to assume for some tangent vector $u^\alpha$
$$u^\alpha\...
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Vector field coordinate transformation
On Carroll's spacetime and geometry book, page 67, the book gives the component form of vector field commutator
$$[X,Y]^\mu=X^\lambda\partial_\lambda Y^\mu- Y^\lambda\partial_\lambda x^\mu \tag{2.23}$$...
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What does the Einstein-Hilbert action look like in terms of Riemannian metric of positive signature?
For a 4-manifold to admit a Lorentzian metric is equivalent to that manifold having vanishing Euler characteristic. Any spacetime that admits a Lorentzian metric $g^{\mathcal{L}}$ can have that metric ...
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How to derive the free rigid body equations from Euler-Lagrange? [closed]
I'm trying to retrieve the equations of motion for a free rigid body:
$$
I(t)\dot{\omega}(t)+\omega(t)^T \times (I(t)w(t)) = 0
$$
where $$ I(t)=R(t)I_{0}R(t)^T $$
I know that Euler-Lagrange equations ...
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The definition of "total curvature" for a scalar field
In Modern Electrodynamics, Zangwill remarks that the total curvature vanishes at every point where $\nabla^2 \varphi = 0$.
Now my question(s): how is "total curvature" defined for a scalar ...
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A conformally-flat metric is also Ricci-flat?
If I have a conformally-flat metric like this one:
$g_{\mu\nu}=e^{2\phi(x)}\eta_{\mu\nu}$
Where $\phi (x)$ is a scalar function of the coordinates.
Shouldn't this metric be Ricci-flat? This means that:...
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Killing field and vector field type
The Killing field condition can be defined as;
$$\mathcal{L}_{X}g=0$$
for a metric $g$ and vector field $X$. In this case does it matter the type ($X^{\alpha}$ or $X_{\alpha}$) of the $X$ used in this ...
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QFT on curved spacetime, uniqueness of spacelike hypersurface
Consider the Lagrangian of a real, scalar field coupled to gravity via the metric $g_{\mu\nu}$ and covariant derivative $\nabla_\mu$
$$\mathcal{L} = \sqrt{-g} (-\frac{1}{2} g^{\mu\nu} \nabla_\mu \phi \...
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D'Alembertian operator in linearized gravity
In linearized gravity, where we take a general background metric $g$ with perturbation $h$, the linearized Einstein equations become
$$-\square h_{\alpha\beta}+\nabla^{\delta}\nabla_{\alpha}h_{\beta\...
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Is Einstein's GR field equations covariant under local diffeomorphisms?
Given that a local diffeomorphism is not necessarily a [global] diffeomorphism, I wonder if Einstein's GR in covariant under local diffeomorphisms of a Lorenzian manifold?
If so, why do we interpret ...
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Identity with Riemann tensor
Is there a fast way to derive the identity
$$(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha})T_{\gamma\delta}={R_{\alpha\beta\gamma}}^{\lambda}T_{\lambda\delta}+{R_{\alpha\beta\delta}}^{\...
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Killing vector definition [duplicate]
How is Killing vector defined? Could you please suggest a book with basic physical examples for Killing vectors application?
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$\sqrt{-g}$ under orientation/time reversal
I want to check whether $\sqrt{-g}$, $g = $ det$g_{\mu\nu}$ picks up a minus sign under time-reversal $t \rightarrow -t$, which reverses the orientation of the manifold. I've been reading a lot of se ...
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Is a perturbation of a tensor field a tensor field?
Let say I take some $2$-tensor field $T_{\mu\nu}$ on some pseudo-Riemannian manifold. Now, often, we are interested in its linearization, which means that we take a family of tensor fields $T_{\mu\nu}(...
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Christoffel Symbols for a Perturbed Metric
If a metric $g$ is given by the sum of a background metric $g_B$ and a perturbation $h$ ie. $g_{ij} = g_{Bij} + h_{ij}$, then the difference of the Christoffel symbols for the background metric and ...
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How is the full Riemann curvature tensor determined if the Einstein field equations only present the Ricci curvature tensor? [duplicate]
I'm currently learning general relativity following Leonard Susskind's lectures, and I was very surprised by the fact that the components of the full Riemann curvature tensor are relevant even though ...
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Closure of Lie brackets associated to Brown-Henneaux boundary conditions
When we impose Brown-Henneaux boundary conditions to the metric field on AdS$_3$,
\begin{align}
\begin{split}
g_{tt}&=-\frac{r^2}{\ell^2}+\mathcal{O}(r^0)\,,\\
g_{t\phi}&=\mathcal{O}\...
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What is a line always pointing at 45° on a sphere like?
I can easily imagine a line pointing dead vertically or horizontally on a sphere.
Say I want to draw a line which is always pointing to some degree (eg 45°) from an origin. What is this line like?
In ...
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How is this term an angle?
This question is regarding the $\Theta_{\mu\nu}$ term given in equation (3.8) of the paper (https://arxiv.org/abs/2206.07725). The term ( I have typed it below )is defined right below in (3.9) and ...
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How to prove formula for contraction of a vector with a Multivector?
I am currently reading "Space-Time Algebra" by David Hestenes and the following proof is given for the formula of contraction of a vector $a$ and multivector $b_1 \wedge b_2 \wedge b_3 \...
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Integration constants in geodesic equation for wave equation
I am stuck at a following "hello world problem". Let us consider the most common (d'Alembertian) wave equation:
$$
\frac{\partial^2 \psi}{\partial^2 x} - \frac{1}{c_0^2}\frac{\partial^2 \psi}...
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Finding the type of underlying geometry from the metric
I am having a confusion regarding the type of geometry of a coordinate system, whether it is Euclidean or not. Intuitively if we consider the spherical polar coordinates, it is a Euclidean geometry ...
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Covariant divergence and derivatives of coordinate basis vectors
I've been trying to derive the covariant divergence of a vector field and I've ran into 2 problems.
Basically I have $\nabla_aA^a=\partial _{a}A^a+\Gamma^{a}{}_{ab}A^b$, and then I found $\Gamma^{\mu}{...
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Is the EM gauge potential on a trivial bundle?
I am studying mathematical gauge theory and its relations to physics (specifically field theory). The reading I’m doing (Gauge Theory and Variational Principles by Bleecker) explains that the EM field ...
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