General theory of relativityCoordinate transformation | Field equation | Gravitational waves | Differential Equation
Coordinates are the identity of points in spacetime. The rules of assigning coordinates are neither arbitrary nor specific, which depends on observer. If you are living on a curved spacetime you must have suitable non-Cartesian coordinate system that will represent events having unique coordinates of that particular system , whereas in flat space you can have different coordinate representations for the same region, which includes curvilinear coordinates as well. The metric tensor of flat spacetime have the values like that of Minkowski metric. Coordinates transformation allows us to switch between different coordinate systems. Coordinates themselves do not have any physical significance , which is clearly evident in the representation of vectors and tensors. They represent certain relationship as vectors are some kind of analytic expression. But relation is not anything physical. Coordinates are terms between which the relations hold. They are merely convenient symbols to fabricate(form) mathematical equations. That is why they are still important entities in the field of physics.
The mapping between coordinate systems is a function of coordinates. For example we can map coordinates of Cartesian plane to the coordinates of Polar coordinates. The later
will be the functions of the former. So we can differentiate this function with respect to the coordinates that are mapped with. When we do that we get
a Jacobian matrix whose determinant can be calculated. It can be used to represent the relationship between two coordinate systems. Total differential is the function of
coordinate variable's differential, which is given as a summation of differential changes along each coordinate. Coordinate transformation should be smooth and continuous so that
we can go from one point to another point without making any sudden jump. As the jacobian matrix is a collection of all derivatives of coordianates , the coordinate function must be continuous.
We can make a coordinate transformation to get rid of gravity , which is the same thing as saying that we go to another reference frame. The origin (0,0,0,0) can be transferred to new coordinate system and corresponding coordinate transformation can be made like this :
The (`) marked symbols represents coordinates in the new coordinate system and Christoffel symbols are represented by the letters inside the second bracket. The value of Christoffel symbol is evaluated at the origin. The value of the metric g will be something as given in the equation (36.45). So the second derivative of this is the coefficient of the second term (x`.x`) in the original equation(36.4).
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The Christiffel symbol transform like the way shown in equation (31.3). After substituting the value of the second derivative in this equation we see that the christoffel symbol in the new coordinates vanishes. The christoffel symbol is the derivative of metric tensor. So our proof is complete.
One typical, yet important coordinate transformation is that between Cartesian to spherical polar coordinate system. There are three different equations for three variable x, y and z, which relates to coordinates in the surface of a sphere at constant distance from the center. The spherical polar coordinate system is a curvilinear coordinate system so is cylindrical coordinate system. Its coordinates lines are curved, which lie on the surface of sphere. The surface of a sphere is curved. If we can find out the metric tensor on this coordinate system we can calculate distance on the surface on the sphere between two points. The notion of usual Euclidean distance is not applicable here. Geodesics on such a surface are the great circles which are paths of shortest distance. The planes containing them passes through the centre of the sphere. using the language of calculus we can say that geodesics are such that distances between two points along them are stationary. It is Spherical coordinate system possesses some symmetry. Using this symmetry calculations can be made simplified.
Later we will use this coordinate system to solve Einstein field equation.
Cartesian to Polar coordinate transformationCartesian coordinates represent flat space. Polar coordinates can be used too. There is certain transformation rules from Cartesian to Polar coordinate system as given below:
Some useful formulasWhen an infinitesimal displacement is made in coordinates x(u), metric tensor changes correspondingly. Accurate change in the metric involves tylor series approximation :
Some specific kind of transformation matrix has specific names :
Various kinds of coordinate systems along with their unit vectors can be listed in a chart :
Coordinates are needed to fix an event in spacetime. Previously in Newtonian framework three coordinates were needed to fix a position in space but in theory of relativity time is itself a coordinate. So we need four instead of three to describe any physical occurance.
In lots of hollywood movie we see that word " coordinates" are intensively used , usually in war movies. Army personal and individual ask for coordinates. They are seen to ask for three space coordinates but this is not sufficient from the relativistic viewpoint. We always need four coordinates to correctly specify something happening in spacetime. You can provide your comment and response below: