Graphically the above matrix can be rerpresented by
the following figure:
Calculus
To properly understand the mathematics behind general
relativity one most understand the foundations of calculus.
As the solutions to GR are given in the form of partial
dirivative equations. First we begin with the definitons
of an ordinary derivative equation:
the change of f by the parmeter k gives f' (f prime)
or the derative (also known as the rate of change [a
very imporatant calculus concept!]) with respect to
the function f(x).
Below is an example of an ordinary derivative with a
logerithmech function using d notation:
Partial Derivatives
The key to understanding general relativity (at least
in the mathematical sense) comes from partial dirivative
equations. What makes a partial dirivative differnt
from an ordinary one is that the function has two or
more indpendent variable, and notation wise the d symbol
is replaced by a curl, below are two simple represenatations
of a partial derivative:
,
The uniniated may be curios as to why there are two
subscripts 1 and 2 used for f prime, there meaning
can be extrapolated from the demoninators on the right
hand side of the equations. To simplify the meanings
1 can only solve for x and 2 only for y, hence the
name partial derivative.
Example Partial Derivatives used in General Relativity:
Here it will be introduced an arbitary tensor T, usaly
such a notation reprsents a stress-energy tensor, but
here we are just interested in the symbolics of the
defintions:
The specific tensor above is considered a contrivariant
tensor because of the superscript indices (which in
this case can be taken as vectors as this is not a
true stress-energy tensor). If we assume T + some
indice to reprsent some vector we can deduce an objects
locations due to covariance (the mathematical interpretation
of relativity) of some other indicie and from this
we can calculate the partial derivate of a particle
"world line" in spacetime. Recalling the
before mentioned difintion of a partial derivative,
to solve for the above tensor equation for the x coordinates
one has a solution of order:
This gives a partial derivative of some contrivariant
four vector T in respect to the row cooridnate 1,0
or x0
, thus to calulate the total vector of T, one need to
perform three more calcuations (four if relativitic
velocities are reached), or do what most physicst do
simpyify the tensor through some symmetry type, e.g.
spherical (it may not be to realsitic but it gives
you a quick answer for general questions).
Also note if there is only one field component it is
an ordianry matrix however when there are two compontent
such as an electromagnetic field it becomes a tensor.
But even if that seemed easy enough to grasp, there
are yet other kinds of tensors in general relativity,
called mixed tensors which get real messy real quick,
below is such an example:
Coceptually the physical foundations of General Relativity
are given from what is known as the Einstein Tensor
G (for simplicity one can think of it as the gravitational
tensor). It is the addtion of all the components which
tell spacetime how to bend, which is done by means
of the stress-energy tensor T, or:
Where 8 pi represents the four-dimensions of spacetime
and k is the speed of light devided by the speed of
light. The above equation is a covariant represenation
which is fine mathematically, but for us physicits
it is the contrivariant soultions that tell us the
physical properties of the space. The next question
to answer is how do we know how spacetime bends, for
this we need a form of nonlinear geometry. The form
of nolinear geometry used in GR is known as Riemannian
Geometry, however this goes beyond the GR introduction
that this work is meant to be. However rather than
living you hanging I will simply spit out the Einstein
Field Equation, which the reader should be able to
grasp at some level:
As can be seen Riemannian Geometry is highly complicated
consiting of a number tensors, metrics, and mixed tensors
to get this final result. In short Riemannian Geometry
describes a very specific type of spatial curvature,
and mathematical this is how one can derive the statment
that gravitation is a curvature in spacetime.
Keys to Understanding and Exploring Tensors within General
Relativity
produced and converted to HTML in Mathcad
Edward Halerewicz, Jr.
hal_warp@hotmail.com
About: This document is meant to be a very brief introduction
into some of the mathematics used within general relativity,
it is in no way meant to be a complete treatise on
the subject. These very breif notes were produced
within in my spare time, and have not been spell checked
grammar cheked etc, it is only meant to help out curious
people who may be intersetd in knowing the mathematics
requried to understand the underlying principles behind
General Relativity. This document focuses on the concept
of tensors and there applications to General Relativity,
if time permits I hope to add more to this docuement
in the future.
Example Matrices:
Let us start by constructing simple surface with arbritrary
coordinates, it is natural to do so using Eulcidean
geomtery the number of dimensions for a given structure
is defined through n*(n+1)/2. Since a surface is a
two dimensional object in order to calcuate the properites
of such an object in Eulcidean geometry one must describe
such in object in three-dimensions as seen from 2*(2+1)/2=3.
From that we can contruct an imaginary grid for our
surface in two dimensions by making the following defintions
# columus =
# rows=
Thus we can now set up an arbritray matrix M:
We see that M has 9 component this is because the matrix
resides in Euclidean 3 spaces which is defined by by
the aribrary dimensions 3 x 3. One can seperate the
components of the surface through the following procedure:
Matrices are subdivied into two catoagoreis vectors
and scalars, we can see this by writing out the cooridnates
of of matrix M:
Matrices are further subdivied through scalar and vectors
where a=b are vectors and
are scalars. The scalars are count once such that
the 9 componets reduce to 6, which is seen through
3*(3+1)/2=6.
Now we plot an antisymmetric or non orthogral cooridnates,
first we give the defitions:
angles
Since the mteric is nonothogal one can only write one
coordinate at time, here k is chosen for an indice.
are scalars. The scalars are count once such that
the 9 componets reduce to 6, which is seen through
3*(3+1)/2=6.
