Tensors in Space-Time Curvature
by Aditya Mittal – April 14, 2006 to April
26, 2006
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General Relativity Overview
The essence of General
Relativity is that there exists a 4 dimensional space-time consisting of 3
dimensions of space and 1 of time that is shaped by matter. Often, space-time is compared to a
trampoline. A heavy object on a
trampoline causes it to sag. Then if we
add other objects in the trampoline, the objects will fall inwards. This picture is nice; however, in reality
space-time is different from the trampoline in that the trampoline is not only
below the object but throughout it and all around it. More precisely in 2 dimensions for simplicity
we can imagine a vector field, namely force field, spreading the Euclidean
Plane. We can then place objects in this
force field that have a certain mass and cause the vectors in their region of
the plane to change. Then, we can define
the motion of any such object as being given by
Einstein’s model goes beyond
this however, since it deals not in Euclidean space but in Minkowski space,
including time as the fourth dimension.
This fourth dimension comes from Special Relativity since time just like distances undergoes Lorentz transformations and is relativistic to
an observer. Of course, we define the
concept of time based on the properties of light and therefore, light always
travels at a constant speed since if light has traveled more distance than more
time must have passed. This is the case
with all electromagnetic radiation.
Since electromagnetism led to the conclusion that mass and energy are
related by speed of light squared, we ended up with the result that energy can
be converted to mass and vice versa. So,
we collide particles and release light energy and vice
versa. From this it is shown that
photons must also lose energy when they fall in a gravitational field or else
conservation laws are violated.
Therefore, if we add light in our vector field it too will experience
the force vectors and its energy will affect the vector field.
Such a bent up space can be
said to be a manifold as long as it looks Euclidean locally. A manifold is an abstract mathematical
space which, in a close-up view, resembles the spaces described by Euclidean geometry but which may have a more
complicated structure when viewed as a whole.
In the case of Einstein’s space-time this is the case and therefore, it
is a manifold. A similar term is orbifold. See my
Superstrings page for an explanation of that: http://www.scientificchess.com/articles/Superstrings.htm
Tensors
A tensor is an abstraction of
scalars, vectors, matrices, and linear operators and is used in describing
things like fluid mechanics, heat transfer, and in this case space-time
curvature. A scalar is a 0th order
tensor, whereas a vector is a first order tensor and a matrix is a second order
tensor. There are three types of
tensors: Contravariant,
covariant, and mixed composed of both the former.
They are written using the Einstein summation convention which allows a
concise algebraic presentation of vector and tensor equations. The Einstein notation for example takes, A1
+ A2 + A3 +…An and
simply writes it as Ai where i goes from 1
to n and the summation is implied. Both contravariant and
covariant types have specific
mathematical definitions. Let us,
however, concentrate on the Riemann curvature tensor which is a mixed tensor of
rank 4 defined by Rαβμν. The superscripts are for the contravariant indices and subscripts are for covariant
indices. A more formal mathematical
discussion of contravariant and covariant is
available at http://en.wikipedia.org/wiki/Contravariant. One can then work through topics like
conjugate tensor, associated tensors, christoffel
symbols and transformations of christoffel symbols.
Riemann Tensor
“As Eddington
nicely explained, Einstein's field equations can be constructed from components
of Riemann curvature tensor. In a nutshell, the tensor G Eddington mentioned (sometimes called "Einstein
tensor") is a sort of average of the Riemann curvature over all
directions. Thus, Riemann curvature is the basic notion for expressing
gravitational fields; and although the expression of Riemann curvature tensor
is different depending on our choice of a coordinate system, this curvature is
an invariant quantity. A sphere has a definite (positive) curvature, and it is
the same whatever coordinate system you may choose, and likewise a Euclidean
plane is flat (zero curvature), independent of any coordinate system. Thus
although metric is different in different coordinate systems, the curvature
characterized in terms of metric is an invariant quantity.”
http://www.bun.kyoto-u.ac.jp/~suchii/Einstein/riemann.curv.html
Friday, April 14, 2006
Riemannian Space
The distance ds between two adjacent points where rectangular Cartesian
coordinates are (x,y,z) and
(x+dx, y+dy, z+dz) is given by ds2 = dx2 + dy2
+ dz2. Riemann extended the
concept of distance to a space of n dimensions and defined the distance ds between two adjacent points xi and xi
+ dxi by the relation written as ds2
= aijdxidxj using
the Einstein summation convention discussed above. The coefficients aij
are functions of the coordinates xi. This quadratic form ds2 is called
the Riemannian metric and we have referred to it often in class. Any space in which the distance is given by
such a metric is called Riemannian space.
Euclidean Space
In particular, if the
quadratic form of the coordinate system Xi reduces to ds2
= (dX1)2 + (dX2)2 + (dX3)2
+ … (dXn)2,
Then it is called a Euclidean
metric and the corresponding space is called Euclidean Space. Geometry based in this space is called Euclidean
geometry and the geometry based in Riemannian space is known as Riemannian
geometry. Compare this way of defining
Euclidean geometry with the definition on http://www.scientificchess.com/articles/Geometry.htm. In one case we are looking at the geometry in
terms of the transformations and in the other case we are looking at the
geometry in terms of the space in which it arises. This seems analogous to the idea of defining
motion in the way
“The
equivalence between accelerated motion and gravity suggests a relationship
between spacetime coordinates and gravity… Geometry tells matter how to move,
and matter tells geometry how to curve.”
Thus, Riemann’s formulation
of distances in higher dimensions is clear and from it we can get the metric
for different coordinate systems in different dimensions. Distances and curvature characterized in
terms of the metric will always be invariant regardless of the coordinate
system used.
Spacetime Curvature from GR
In GR we will use the 4
dimensional space-time coordinate systems and thus get the metric
. Also, we will call ‘ds’ the spactime interval. This quantity is invariant under the Lorentz
transformations which we studies at the very beginning of the relativity and
cosmology course although we only worked in a single space dimension then for
the purpose of being able to draw. Clearly,
we are in a very special subset of the Riemannian space and the gi correspond to the aij in the Reimann
concept of distance. The Euclidean case
on the other hand would only arise if gi =
1 for i = 0,1,2,3.
General relativity gives a
relationship between curvature and the density of mass and energy in space:
Curvature of space = 
(mass-energy density) (Equation
15.8 Krane)
In this we can look at the
classical limit as speed of light goes to infinity the curvature seems to be 0
and thus everything acts as if it were in a Euclidean space. The other case where spacetime curvature
becomes 0 is when there are no significant gravitational fields (the
weak-fields limit).
Ricci Tensor
We briefly defined a manifold
in the GR overview at the beginning of the page. The Ricci curvature is “the mathematical
object which controls the growth rate of the volume of metric balls in a
manifold.” http://mathworld.wolfram.com/RicciCurvatureTensor.html
Essentially, the Ricci tensor
is also a special case of the Riemann tensor where the contravariant
index is the same as one of the covariant indices.
(Definition
of Ricci Tensor).
See
John Baez page on Ricci and Weyl tensors for an
interesting conceptualization using coffee grounds http://math.ucr.edu/home/baez/gr/ricci.weyl.html
Specifically, let us take a look
at how this Ricci tensor is used in GR and what role it plays in the Einstein
Field Equations. The Einstein field equation (EFE)
are usually written in the form
(Einstein Field Equation)
with Rab as the Ricci
tensor, R as the Ricci scalar, gab is the metric tensor, and Tab
is the stress-energy tensor we talked about in class. We also said in class how each of these 4x4
tensors representing have 10 independent components. The 10 independent components of the Ricci
tensor and the 10 independent components of the Weyl
tensor put together give the complete curvature of a point in the 4-dimensional
spacetime. Usually, the EFE is solved
only for special cases such as the Vacuum field or the Linearized EFE. These special cases are of interest in cosmology
when studying objects like blackholes.
Vacuum Field Equations
The vacuum field equation is the special case when the stress energy tensor
goes to 0. Making Einstein field
equation become
The
solutions to the vacuum field equations are called vacuum solutions. Minkowski space (flat space) is the simplest example of a
vacuum solution. Nontrivial examples include the Schwarzschild solution and the
Kerr solution.
Additional Links:
http://www.jb.man.ac.uk/~jpl/cosmo/friedman.html
http://mathworld.wolfram.com/RiemannTensor.html ; mathematical
description
http://www.bun.kyoto-u.ac.jp/~suchii/Einstein/riemann.curv.html ; simple curvature
description
http://en.wikipedia.org/wiki/Contravariant
http://www.mth.uct.ac.za/omei/gr/chap6/node2.html#255
http://www.mth.uct.ac.za/omei/gr/chap6/frame6.html
http://math.ucr.edu/home/baez/gr/outline2.html
http://math.ucr.edu/home/baez/gr/ricci.weyl.html
http://en.wikipedia.org/wiki/Einstein_field_equation
http://en.wikipedia.org/wiki/Exact_solutions_of_Einstein%27s_field_equations