What is Christoffel symbols: Definition and 101 Discussions
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O(p, q). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.
In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection. It is common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames (called holonomic coordinates) where the torsion vanishes. For example, in Euclidean spaces, the Christoffel symbols describe how the local coordinate bases change from point to point.
At each point of the underlying n-dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denoted Γijk for i, j, k = 1, 2, ..., n. Each entry of this n × n × n array is a real number. Under linear coordinate transformations on the manifold, the Christoffel symbols transform like the components of a tensor, but under general coordinate transformations (diffeomorphisms) they do not. Most of the algebraic properties of the Christoffel symbols follow from their relationship to the affine connection; only a few follow from the fact that the structure group is the orthogonal group O(m, n) (or the Lorentz group O(3, 1) for general relativity).
Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γijk are zero.
The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900).
I would love to hear from you if you have any suggestions, feedback, or criticism. The goal is to build better and more sophisticated software that would push the boundaries of research in astrophysics!
About a month or two ago I started doing simulations of light physics around black holes and yesterday I got a fast Christoffel symbols function for the Schwarzschild metric in cartesian coordinates, but now the photon ring appears flipped. I feel as though it is wrong. But as I am still pretty...
The Hiscock coordinates read:
$$d\tau=(1+\frac{v^2(1-f)}{1-v^2(1-f)^2})dt-\frac{v(1-f)}{1-v^2(1-f)^2}dx$$
##dr=dx-vdt##
Where ##f## is a function of ##r##. Now, in terms of calculating the christoffel symbol ##\Gamma^\tau_{\tau\tau}## of the new metric, where ##g_{\tau\tau}=v^2(1-f)^2-1## and...
M. Blennow's book has problem 2.18:
Show that the contracted Christoffel symbols ##\Gamma_{ab}^b## can be written in terms of a partial derivative of the logarithm of the square root of the metric tensor $$\Gamma_{ab}^b=\partial_a\ln{\sqrt g}$$I think that means square root of the determinant of...
Hello everyone,
in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...
I am trying to find $$\Gamma^{\nu}_{\mu \nu} = \partial_{\mu} log(\sqrt{g})$$ but I cannot.
by calculations, I manage to find
$$\Gamma^{\nu}_{\mu \nu} = \frac{1}{2}g^{\nu \delta}\partial_{\mu}g_{\nu \delta}$$
and from research I have find that $$det(A) = e^{Tr(log(A))}$$ but still I cannot...
Can you approach the GR two body problem through summations of multiple Schwarzschild solutions?
Specifically, by using the Schwarzschild metric for each body of mass, then adding the Christoffel symbols together, to arrive at a new geodesic equation.
Take point C between bodies A and B...
Is a connection the same thing as a covariant derivative in differential geometry?
What Is the difference between a covariant derivative and a regular derivative?
If you wanted to explain these concepts to a layperson, what would you tell them?
I am using the code provided by Artes here, but I am missing something.
The Chrisfoffel-symbol formula is
$$\Gamma^{\mu}_{\phantom{\mu}\nu\sigma}=\frac{1}{2}g^{\mu\alpha}\left\{\frac{\partial g_{\alpha\nu}}{\partial x^{\sigma}}+\frac{\partial g_{\alpha\sigma}}{\partial x^{\nu}}-\frac{\partial...
in video "Einstein Field Equation - for Beginner!" by "DrPhysicsA" on youtube, in 01:10:56, the christoffel symbol equation is written, then i see in "Physics Videos by Eugene Khutoryansky" video with title "Einstein's Field Equations of General Relativity Explained" in minute 05:02 on how the...
I am trying to create a function to calculate the Christoffel Symbols of a given metric (in this case the Shwartzchild metric). Calculating the (non zero) Christoffel Symboles for the Shwartzchild connection, I am a double major in Physics and Computer Science so I decided to go the code rout...
I've stumbled over this article and while reading it I saw the following statement (##\xi## a vectorfield and ##d/d\tau## presumably a covariant derivative***):
$$\begin{align*}\frac{d \xi}{d \tau}&=\frac{d}{d \tau}\left(\xi^{\alpha} \mathbf{e}_{\alpha}\right)=\frac{d \xi^{\alpha}}{d \tau}...
Hi All
Given that the Riemann Curvature Tensor may be derived from the parallel Transport of a Vector around a closed loop, and if that vector is a covariant vector
Having contravariant basis
The calculation gives the result
Now:
Given that the Christoffel Symbols represent the...
In Landau Book 2 (Classical Field Theory & Relativity), he mentions that the transformation rules of the christoffel symbols can be gotten by "comparing the laws of transformation of the two sides of the equation governing the covariant derivative"
I would believe that by the equations...
I've noticed that for both the surface of a sphere and a paraboloid, one arrives at the same Christoffel symbols whether using
\Gamma^i_{kl} = \frac {1}{2} g^{im} ( \frac {\partial g_{mk} }{\partial x^l} + \frac {\partial g_{ml}}{\partial x^k} - \frac {\partial g_{kl}} {\partial x^m} )...
What is the general difference or importance between using christoffel symbols of the first kind and those of the second kind in terms of geometry and their application. The christoffel symbols of the second are identical to those of the first except with the inverse metric tensor in front...
So I'm trying to get sort of an intuitive, geometrical grip on the covariant derivative, and am seeking any input that someone with more experience might have. When I see ##\frac {\partial v^{\alpha}}{\partial x^{\beta}} + v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}##, I pretty easily see a...
I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are:
\Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y}
knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...
Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element:
ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy
The result I have obtained is that the only non-zero component of the Christoffel symbols is:
\Gamma^x_{xx}=\frac{1}{x}
Is this correct?
MY PROCEDURE HAS BEEN:
the...
Let the metric be defined as ##ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2##
Through some calculations, we then see that our connection one forms are ##\omega_{12} = -d \theta## and ##\omega_{21}= d\theta##, ##\omega_{13} = -sin\theta d \phi## and ##\omega_{31} = sin\theta d\phi##...
Hello everyone,
I'm sure a lot of you know that the Christoffel symbols are not tensors by themselves but, their variation is a tensor.
I want to revive a post that was made in 2016 about this: The Variation of Christoffel Symbol and ask again "How is that you can calculate ∇ρδgμν if δ{gμν} is...
Let us assume a "toy-metric" of the form
$$ g=-c^2 \mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2-\frac{4GJ}{c^3 r^3} (c \mathrm{d}t) \left( \frac{x\mathrm{d}y-y\mathrm{d}x}{r} \right)$$
where ##J## is the angular-momentum vector of the source.
Consider the curve
$$ \gamma(\tau)=(x^\mu...
I have a couple of questions about how Christoffel symbols work. Why can they just be moved inside the partial derivative, as shown just beneath the first blue box here: https://einsteinrelativelyeasy.com/index.php/general-relativity/61-the-riemann-curvature-tensor
And if you had the partial...
Homework Statement
My textbook states:
Since the number of particles of dust is conserved we also have the conservation equation
$$\nabla_\mu (\rho u^\mu)=0$$
Where ##\rho=nm=N/(\mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z) m## is the mass per infinitesimal volume and ## (u^\mu) ## is...
Hi, I really wonder how these second derivatives can be written in terms of christofflel symbols. I have made so many search but could not find on internet What is the derivation of equations related to second derivatives in attachment?
Homework Statement
Hi, We are trying to calculate the Coriolis acceleration from the Cristoffel symbols in spherical coordinates for the flat space. I think this problem is interesting because, maybe it's a good way if we want to do the calculations with a computer.
We start whit the...
Consider a force-free particle moving on a geodesic with four-velocity v^\nu.
The formula for the four-acceleration in any coordinate system is
\frac{dx^\mu}{d\tau} = - \Gamma^\mu_{\nu\lambda} v^\nu v^\lambda
Since the four-acceleration on the left side is orthogonal to the four-velocity, this...
Homework Statement
Show that g(d \sigma ^k, \sigma _p \wedge \sigma _q) = \Gamma _{ipq} - \Gamma _{iqp}Homework Equations
Given $$\omega_{ij}=\hat e_i \cdot d \hat e _j = \Gamma_{ijk} \sigma^k$$, we can also say that $$d \hat e_j = \omega^i_j \hat e_i$$. Where $$\sigma^k, \sigma_p, \sigma_q$$...
As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative.
The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain...
So the Schwarzschild metric is given by
ds2= -(1-2M/r)dt2 + (1-2M/r)-1dr2+r2dθ2+r2sin2θ dφ2
and the Lagragian is ##{\frac{d}{dσ}}[{\frac{1}{L}}{\frac{dx^α}{dσ}}] + {\frac{∂L}{∂x^α}}=0##
with L = dτ/dσ. So for each α=0,1,2,3 we have
##{\frac{d^2 x^1}{dτ^2}}=0## for Minkowski spacetime
also...
The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as:
$$
R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n}
$$
My question is that it seems that...
If you want to define a covariant derivative which transforms correctly, you need to define it as ##\nabla_i f_j = \partial_i f_j - f_k \Gamma^k_{ij}##, where ##\Gamma^k_{ij}## has the transformation property
##\bar{\Gamma}^k_{ij} = \frac{\partial \bar{x}_k}{\partial x_c}\frac{\partial...
In Carroll's GR book (pg. 96), the transformation law for Christoffel symbols is derived from the requirement that the covariant derivative be tensorial. I think I understand that, and the derivation Carroll carries out, up until this step (I have a very simple question here, I believe-...
Here's what I'm watching:
At about 1:35:00 he leaves it to us to look at a parallel transport issue. Explicitly to caclculate
##D_s D_r T_m - D_r D_s T_m##
And on the last term I'm having some difficulties, the second christoffel symbol.
So we have
##D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]##...
Hi, I recently tried to derive the equations for a geodesic path on a sphere of radius 1 (which are supposed to come out to be a great circle) using the formula \dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0 for the geodesic equation, with the metric...
Hi friends,
I'm learning Riemannian geometry. I'm in trouble with understanding the meaning of ##g^{jk}\Gamma^{i}{}_{jk}##. I know it is a contracting relation on the Christoffel symbols and one can show that ##g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})## using the...
I've attempted to derive an expression for the Christoffel symbols (of the 2nd kind) solely in terms of the covariant and contravariant forms of the metric by only using the definition of the Christoffel symbols. I would like to know if my approach is correct or not.
The Christoffel symbols are...
Hi,
Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.
In Sean Carroll's Lecture Notes on General Relativity (Chapter 3, Page 60), in the chapter on Curvature, he derives the definition of the Christoffels Symbols by assuming the connection is metric compatible and torsion free. He then takes the covariant derivative of the metric and cycles through...
Homework Statement
Write down the geodesic equation. For ##x^0 = c\tau## and ##x^i = constant##, find the condition on the christoffel symbols ##\Gamma^\mu~_{\alpha \beta}##. Show these conditions always work when the metric is of the form ##ds^2 = -c^2dt^2 +g_{ij}dx^idx^j##.Homework...
Homework Statement
[/B]
(a) Find christoffel symbols and ricci tensor
(b) Find the transformation to the usual flat space form ## g_{\mu v} = diag (-1,1,1,1)##.
Homework EquationsThe Attempt at a Solution
Part(a)
[/B]
I have found the metric to be ## g_{tt} = g^{tt} = -1, g_{xt} = g_{tx} =...
Homework Statement
I'm having trouble figuring out how to use Christoffel symbols. Apart from the first three terms here, I can't understand what's going on between line 3 and 4. What formulas/definitions are being used? How do you find the product of two chirstoffel symbols? Where are all the...
So, it is defined that:
Γλμυ = Γλμυ + δΓλμυ
This makes obvious to see that the variation of the connection, which is defined as a difference of 2 connections, is indeed a tensor.
Therefore we can express it as a sum of covariant derivatives.
δΓλμυ = ½gλν(-∇λδgμν + ∇μδgλν + ∇νδgλμ)
However...