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. The first variation of energy is defined in local coordinates by. The christoffel symbols calculations can be quite complicated, for example for dimension 2 which is the number of symbols that has a surface, there are 2 x 2 x 2 8 symbols and using the symmetry would be 6. If anyone can help me derive this geodesic equation, it would be great. Christoffel symbols and geodesic equation ucsb physics. Christoffel symbols and geodesic equation this is a mathematica program to compute the christoffel and the geodesic equations, starting from a given metric gab. Often an easier way is to exploit the relation between the christoffel symbols and the geodesic equation. Another is that the indices on unprimed and primed christoffel symbol in the last equation have moved around in a very odd way. If you like this content, you can help maintaining this website with a small tip on my tipeee page. Polymerforschung, ackermannweg 10, 55128 mainz, germany these notes are an attempt to summarize some of the key mathe. Lots of calculations in general relativity susan larsen tuesday, february 03, 2015 page 2. Geodesic equation from christoffel symbols mathoverflow. Mar 31, 2020 this is known as the geodesic equation. Geodesics on a torus are shown to split into two distinct classes.
Dynamical systems approach is used to investigate these two classes. I would like to compute the christoffel symbols of the second kind using the geodesic equation. The connection form and parallel transport on the torus are investigated in an appendix. Most of the algebraic properties of the christoffel symbols follow from their relationship to the affine connection. The geodesic equation more generally, suppose that the metric coe. Derivation of the geodesic equation and defining the christoffel. Derivation of the geodesic equation and defining the christoffel symbols. In other words, the name christoffel symbols is reserved only for coordinate i. Lagrangian method for christoffel symbols and geodesics equations calculations basic concepts and principles the christoffel symbols calculations can be quite complicated, for example for dimension 2 which is the number of symbols that has a surface, there are 2 x 2 x 2 8 symbols and using the symmetry would be 6. Christoffel symbols for schwarzschild metric pingback. However, the connection coefficients can also be defined in an arbitrary i.
This equation can be useful if the metric is diagonal in the coordinate system being used, as then the. Calculating the christoffel symbols, and then the geodesic equation can be a really tough and time consuming job, especially when the metrics begin to get more and more complicated. I know one can get to an expression for the christoffel symbols of the second kind by looking at the lagrange equation of motion for a free particle on a curved surface. General relativity phy5genrel u01429 16 lectures alan heavens, school of physics, university of edinburgh. Chapter 5 schwarzschild solution university of minnesota. Lagrangian method christoffel symbols calculations. Another is that the indices on unprimed and primed christoffel symbol. Geodesics on a sphere and the christoffel symbols physics. Geodesic equation in terms of christoffel symbols pingback.
Schwarzschild solution 69 this is in full agreement with schwarzschild metric 5. Sphere geodesic and christoffel symbols physics forums. A straight line which lies on a surface is automatically a geodesic. The geodesics on a round sphere are the great circles.
It is worth noting that our method is different from the conventional methods of directly solving the geodesic equation i. This notation is a simple way in which to condense many terms of a summation. We have already shown how to derive the geodesic equation directly from the equivalence principle in in our article geodesic equation and christoffel symbols here our aim is to focus on the second definition of the geodesic path of longer proper time to derive the geodesic equation from a. Jun 02, 2015 in most spacetime geometries general exact solutions of the geodesic equation are difficult to obtain, and when they are available they are often expressed in terms of nonelementary. We first have to find the derivative of the metric tensor in the primed coordinate system. The geodesic equation and christoffel symbols part 3 youtube. Again, the point is not to be able to understand the details with extreme rigor, but to grasp the. Christoffel symbols and geodesic equations example ps, example pdf, the shape of orbits in the schwarzschild geometry. Apr 07, 2014 the geodesic equation and christoffel symbols part 3 duration. There is a factor of two that is a common gotcha when applying this equation. Then the affinely parameterized geodesic equation in an.
Computing the christoffel symbols with the geodesic equation. I hope this small tutorial was useful to you in understanding how to extract the lagrangian from the proper interval, and then use it to compute the geodesic equation and the corresponding christoffel symbols. The christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. This is a mathematica program to compute the christoffel and the geodesic equations, starting from a given metric. We find the general form of the geodesic equation and discuss the closed form relation to find christoffel symbols. The geodesic equation is where a dot above a symbol means the derivative with respect to. Consider for example the schwarzschild metric, given by. So christoffel symbols are like the metric they do tell us about curvature. Then for we obtain the equation, where is a solution. The line element and metric our model of a torus has major radius c and minor radius a. If the metric is diagonal in the coordinate system, then the. So, we combine these three equations and solve for the christoffel symbols. The geodesic equation and christoffel symbols part 3 duration.
Grss 031 derivation of christoffel symbol part one. The geodesic equation and christoffel symbols part 1 youtube. For instance, the above equation could be written as 16 terms ds2. Every geodesic on a surface is travelled at constant speed. Geodesic equation in terms of christoffel symbols physics pages. A theoretical motivation for general relativity, including the motivation for the geodesic equation and the einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. Here our aim is to focus on the second definition of the geodesic path of longer proper time 1 to derive the geodesic equation from a variationnal approach, using the principle of. As a qualitative example, consider the airplane trajectory shown in. Christoffel symbols k ij are already known to be intrinsic. Grss 031 derivation of christoffel symbol part one duration. To do that, it is convenient to transform the second order equation to a system of two rst order equations by. Techniques of the classical calculus of variations can be applied to examine the energy functional e.
Note that we can also write this equation as d 0 or 0. In most spacetime geometries general exact solutions of the geodesic equation are difficult to obtain, and when they are available they are often expressed in terms of nonelementary. Lagrangian method christoffel symbols calculations mathstools. A smooth curve on a surface is a geodesic if and only if its acceleration vector is normal to the surface. Then it is clear that if the metric does not depend on we obtain the equation, hus. Pdf geodesic equations and algebrogeometric methods. Christoffel symbols and the geodesic equation the easy way. For dimension 4 the number of symbols is 64, and using symmetry this number is only reduced to 40. Christoffel symbols 1 the metric and coordinate basis. Einstein relatively easy geodesic equation from the. Mar 04, 2011 the geodesic equation is given by, where. Actually, we will see in christoffel symbols in terms of the metric tensor how the christoffel symbol at the heart of the gravitational force can be calculated from the space time metric. But he hasnt given any explanation how he obtained this.
Christoffel symbols and the geodesic equation the easy. Then the righthandside of the geodesic equation 20 vanishes, which implies that gkc x. The usual way of deriving the geodesic paths in an ndimensional manifold from the metric line element is by the calculus of variations, but its interesting to note that the geodesic equations can actually be found simply by differentiating the. Solving the initial value problem of discrete geodesics. We see that the gravitational term in the geodesic equation depends on the gradients of g. In mathematics and physics, the christoffel symbols are an array of numbers describing a metric connection. Schwarzschild solution quick recap to begin, lets recap what we learned from the previous lecture. We have already shown how to derive the geodesic equation directly from the equivalence principle in in our article geodesic equation and christoffel symbols. Another way to write equation 1 is in the form ds2. There were a lot of abstract concepts and sophisticated mathematics displayed, so now would be a good time to summarize the main ideas. Aug 02, 2014 calculating the christoffel symbols, and then the geodesic equation can be a really tough and time consuming job, especially when the metrics begin to get more and more complicated. In differential geometry, an affine connection can be defined without reference to a metric, and many additional. This trajectory is the shortest one between these two points.
In general, this is the way to proceed, but if the problem has some symmetry to it, then a variational approach is easier see chapter 6. General relativity and geometry 230 9 lie derivative, symmetries and killing vectors 231 9. So here, i present a well known method of calculating the geodesic equation just from a knowledge of the lagrangian, and then simply reading off the christoffel. However, if we assume that they actually do define a connection, you just have to check the definition. Once this geodesic equation is obtained, my claimed equation is obvious. To do that, it is convenient to transform the second order equation to a system of two rst order equations by going into the tangent bundle tm.
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