General Relativity : the Einstein field equation
General Relativity is the theory of physics
that describes spacetime as a curved geometrical space,
extending Special
Relativity theory (only accepted as the approximative
description of small regions of spacetime), to explain
gravitation as the effect of the curvature of
spacetime.
Beyond some mere intuitive popularization comments explaining how
gravitational effects can be explained by spacetime curvature,
while the gravitational force cannot be distinguished from inertial
forces, the main formula of General Relativity that makes it
precise, is the Einstein field equation, relating spacetime
curvature, to the stressenergy tensor.
To fully, properly express it, we need the formalism of tensors.
Still it is possible to give the essential idea of the meaning
and justification of this equation without using the tensor
formalism, by taking the case of cosmology (the expansion
of the universe).
Other remark : Energy in GR can no more be
described as a precise quantity simply obtained by integrating some explicit field
along the spacelike 3D surface we consider. This does not mean that there is no
conservation of energy. But it takes a more subtle, complex form. In particular we
can roughly define the mass inside a sphere with size r, as r times the integral of
the intrinsic Riemannian curvature of space through this sphere. As this is computed
from the surface only and not from the inside, it is not possible to modify it by purely
local processes : it is necessary to do something that can affect the surface. Thus if
you take a large sphere away from the system you consider, you can only increase
your local energy by bringing it from far away.
Sorry that in this site is only an introduction to tensors,
giving some basic definitions but not yet the needed developments...
that anyway you can find in any course elsewhere about tensors.
Now the below explanations will assume the reader to have already
learned about tensorial operations not explained here (until this
gap will be filled someday), and also of course, what is the Riemann curvature tensor.
Here again, like with
other subjects in the foundations of maths and physics, it is
a pity to see that almost one century after its discovery, the
courses on General Relativity that are usually found (at least the
presentation on Wikipedia) still express this equation in its draft,
messy form, not mentioning the cleaner way that more directly
expresses its meaning.
Let us recall that expression.
There is of course the essential idea, the simple statement, saying
that we have a relation between the stressenergy tensor, and the
Riemann curvature of spacetime, that we can write in a compact form
as
G_{ij} + Λ g_{ij}
= 
8πG
c^{4}

T_{ij} 
where
 T_{ij} represents the stressenergy tensor, but
with lowered indices unlike its more canonical use as T^{ij}
in relativistic mechanics
 G_{ij} is the Einstein tensor, that is a
function of the Riemann curvature tensor.
 The constants G and c are the Newtonian
gravitational constant and the speed of light
 g_{ij} is the metric, and Λ is the cosmological
constant, that may be taken as 0 to simplify the formula by
changing the definition of T_{ij} (adding to it
a constant multiple of g_{ij}), so that,
finally, G_{ij} and T_{ij}
become simple multiples of each other.
The problem, what is messy in usual texts, is how the Einstein
tensor (with 2 symmetric indices) is expressed as a function of the
Riemann curvature tensor (with 4 indices).
Indeed they first introduce the Ricci curvature tensor, that is a
symmetric tensor with 2 indices just like the one we need, extracted
from the Riemann tensor, and that has all the necessary information
in it:
But this is still not the tensor we need. So then we need another
formula to get the Einstein tensor from it
G_{ij} = R_{ij}
− 
1
2

R g_{ij}

where R is the scalar curvature R = g^{ij}
R_{ij}.
But it begs the question: what the f**k is this formula ??? Why is
it this formula, made of 2 terms, rather than anything else (for
example, why are not G_{ij} and R_{ij }just
equal) ?
The answer, also wellknown, is that this is what we need for the
conservation of the Einstein tensor to be deduced from the Bianchi
identity.
Okay, the usual proof of this is not complicated, but here will be
another way of writing it, which I think is more elegant.
Let us introduce the symmetrizer (using the Kronecker delta δ^{i}_{j})
S^{ijk}_{lmn} = δ^{i}_{l}
δ^{j}_{m} δ^{k}_{n}
+ δ^{i}_{m} δ^{j}_{n}
δ^{k}_{l} + δ^{i}_{n}
δ^{j}_{l} δ^{k}_{m}
It has the property that for every antisymmetric tensor A_{ij},
the expression A_{ij} S^{ijk}_{lmn}
is totally antisymmetric between the indices lmn. So it
behaves as a total antisymmetrizer when applied to a tensor with 2
indices that is already antisymmetric between these 2 indices.
Okay, now let us take the Riemann tensor R^{a}_{bij}.
We already know that it is antisymmetric between indices ij.
We deduce that R^{a}_{bij} S^{ijk}_{lmn}
is totally antisymmetric between the indices lmn.
This expression can be used to write the second Bianchi identity on
the Riemann curvature, as
∇_{k} R^{a}_{bij} S^{ijk}_{lmn}
= 0
The intuitive meaning of this identity is that for every small
closed curve along which we consider a parallel transport, the
rotation produced by this transport does not depend on the choice of
surface bordered by this curve, over which we integrate the
curvature to calculate this small rotation : moving this surface
(with direction ij) towards the direction k but
keeping the same border, has no effect.
Now by uppering the index b, we get the expression R^{ab}_{ij}
S^{ijk}_{lmn}, As each of both pairs (ab) and
(lm) is antisymmetric, the contraction between these 2 pairs gives a
factor 2 of redundance, which we can factor out without fractioning
the underlying operations on coordinates. And this is how the
Einstein tensor is actually obtained :
G_{n}^{k}
= − 
1
2

R^{ab}_{ij} S^{ijk}_{abn}

Remark: another notation for this expression R^{ab}_{ij}
S^{ijk}_{abn} would be by using the [ ] for
antisymmetrization:
3 R^{ab}_{[ab} δ^{k}_{n]}
Indeed when replacing S by its definition, this expression
develops as :
−2 G_{n}^{k} = R^{ab}_{ab}
δ^{k}_{n} + R^{kb}_{bn}
+ R^{ak}_{na}
We see here the Ricci tensor R_{n}^{k}
= − R^{kb}_{bn} = − R^{ak}_{na}
and the scalar curvature R = g^{ij}
R_{ij} = R^{ab}_{ab}.
This gives the (slightly rewritten) previous formula of the Einstein
tensor
G_{n}^{k}
= R_{n}^{k} − 
1
2

R δ^{k}_{n}

The conservation of the Einstein tensor is directly deduced from the
above Bianchi identity:
2∇_{k}G_{n}^{k} = ∇_{k}
R^{ab}_{ij}S^{ijk}_{abn}
= 0
Now what is the interest of writing the Einstein tensor using S
rather than the usual expression :
It presents this operation as a contraction between the Hodge duals
of the antisymmetric pairs in the Riemann tensor (this contraction
has order 1 = n−3, where n = 4 is the dimension of
spacetime, and 3 is the number of indices on which S
operates), instead of between these pairs themselves.
Namely, it can be written using the LeviCivita symbol ε as
G^{nk}
= 
1
4

R^{a}_{bij} ε_{a}^{bn}_{l}
ε^{ijkl} 
(with sign changed because of the odd signature of spacetime)
Note that the lower position of ij in R^{a}_{bij}
justifies to have k in upper position (to use ε^{ijkl}
independently of the metric), while the good position (up or down)
of n, is more ambiguous, as it depends on the more arbitrary
choice of position of indices a and b.
This explicitly shows that the curvature in a given pair of
dimensions contributes to the Einstein tensor only in the orthogonal
dimensions. Which is more elegant than to first write the
contribution as if it was in these dimensions (in the Ricci tensor),
then make a global substraction in all dimensions by (1/2) R g_{ij},
which just happens to cancel the contribution in these dimensions
and leave an opposite contribution in the rest of dimensions.
Basic examples
The uniform unidirectional curvature
Take the 4dimensional Riemannian manifold (=curved 4dimensional
space that is approximately Euclidean at small scales) defined as
the cartesian product M=E×F where E is a (2dimensional) sphere, and
F is an Euclidean plane.
It has the remarkable combination of properties of being very
simple, with only one nonzero component of its Riemann curvature
tensor at each point (the given by the Gaussian curvature of E), and
still it turns out to be general enough so that any possible value
of the Riemann curvature at a point of any 4dimensional Riemannian
manifold, equals some superposition of a number of rotated images of
this one (this superposition is defined in first approximation near
the point, by adding up the geometric distortions observed on a
given figure when it is embedded in such rotated images of this
manifold).
So, it suffices for us in a first time to describe the Ricci tensor
and the Einstein tensor in this manifold, as those in any other
manifold will be deduced from this case by superposition of its
rotated images.
The Ricci tensor at every point of this space M, can be imagined as
a symmetric bilinear form on vectors from this point, with only 2
nonzero components, that are in the 2 dimensions of E. Like any
symmetric bilinear form it can equivalently be represented by a
quadratic form. That is, a field whose values are a quadratic
function of the position. The sets of points where it takes any
given value, are quadrics centered on the chosen point ; and for
every (small) value taken as reference, the more intense the field,
the smaller the quadric (more precisely, shrinking the size by 2,
corresponds to multiplying the field by 4). Here in particular,
these quadrics are just the cylinders, cartesian product of a small
circle C in E, with F.
We can modify the intensity of the curvature, just by changing the
radius of E. Dividing its radius by 2, results in multiplying the
curvature by 4, so that the cylinder also shrinks by 2. So, the
circle C, that is a small circle around a point of the sphere E,
keeps the same proportion to E when the size of E changes: The
proportion of C's radius in E being a small angle, its square is the
small dimensionless number that is the value the quadratic form
defined by the Ricci tensor, takes on C×F.
Okay, now let us describe the Einstein tensor there.
If we look at its definition as G_{ij} = R_{ij}
− 1/2 R g_{ij}, it would be represented by another
quadratic form, that cancels in the direction of E and only varies
as a function of the projection in F (because g_{ij}
just gives the dot square function of vectors, and 1/2 R g_{ij}
coincides with R_{ij} in the direction of E).
However, its other expression that we gave above, suggests to
represent it as twice contravariant (i.e. as a combination of tensor
squares of vectors, or a quadratic form on the space of covectors)
rather than twice covariant (combination of tensor squares of
covectors, or quadratic form on the space of vectors), which better
fits with its identification with the stressenergy tensor (that is
twice contravariant).
But the visual representation of a twice contravariant symmetric
tensor, differs from that of a twice covariant one. It is still
represented as an ellipsoid (if it is positive definite), but its
size is bigger when the tensor is multiplied by a scalar quantity a
(it is dilated by the square root of a) instead of being
smaller (shrinked by the square root of a). And when, like
here for our manifold M=E×F, it is not positive definite (namely the
matrix is diagonal but some diagonal coefficients cancel), then this
ellipsoid shrinks onto a disk, instead of extending as a cylinder.
Namely here, this disk extends in the (flat) direction of the plane
F, with no extension in the (curved) direction of E.
(The circularity of this disk comes from the implicit use of the
metric when we lowered the index l to form the expression R^{a}_{bij}
ε_{a}^{bn}_{l} ε^{ijkl})
The isolated (dense or singular) axis of curvature
Let us modify the above example by replacing the sphere E by a
surface that is only curved in one region, and flat outside. For
example we can represent E by a sheet of paper forming a cone in the
3dimensional Euclidean space, and whose vertex would have been
smoothed, replaced by a spherical cap. So here the curved region is
shaped like a disk, but it does not really matter which shape it
has: it may be a triangle or anything. What matters is that it is a
small region that is curved, and extended by a flat surface.
The "total curvature" (or surface integral of the curvature) of this
region, is measured by parallel transport around it, that is, how
the flat surface glues itself. When we imagine it as made of a piece
of paper that was cut to be glued as an extension of this region
like a cone around a spherical cap, this piece of paper is then
glued back to itself, and the total curvature of this the angle by
which the paper is glued to itself.
We can consider the limit case of a singular curvature, when the
curved region shrinks to a point while its total curvature remains
constant. Then the surface is just a cone, flat except at one point
of singularity (the vertex), whose "total curvature" is the angle of
difference between the full turn around it, and 2π radians.
Now let us still take the cartesian product of this surface E with
the simple Euclidean plane F.
And describe the Einstein tensor of the result:
 Inside the curved region, it is just the same as the previous
case: the Einstein tensor is shaped like a disk in the direction
of F
 Outside, since the space is flat, the Einstein tensor cancels
too.
Natural, isn't it ? The forces are just flowing in the direction
where the curvature extends and must be conserved. They cannot flow
in any other direction.
Let us now describe this more concretely in the usual spacetime
language.
 If the time dimension is one of the dimensions of F, we see a
3dimensional space with a fixed curvature on an axis. In the
case of a positive curvature (an angle of full turn around this
axis smaller than 2π radians), this axis is endowed with a
positive linear density of energy, together with an attractive
force along this axis.
 If, instead, the time dimension is one of the dimensions of E,
we have an instantaneous "flash" of presence of a spacetime
curvature along a plane F'={0}×F where 0 is the singularity of
curvature in E. Only at a precise time on F', something happens,
while no effect is noticed elsewhere. What happens is that if
there were 2 objects initially at rest relatively to each other,
and then at time of the flash, they happen to separated by F',
then this makes them turn out to be moving relatively to each
other after the flash, with a direction of speed orthogonal to
F'.
 The case of "repulsive acceleration" between sides of the
space separated by F' (the acceleration is orthogonal to F')
corresponds to the case of presence of an attractive force
along F'. This is just like the surface tension that is found
at the surface of a liquid, except that it is only at one time
(while the surface of a liquid persists in time), and cannot
come together with any energy density (because the energy,
that needs to be preserved, has nowhere to go here).
 The case of "attractive acceleration" between sides of F',
corresponds to the presence of a repulsive force along F'.
(To be continued)
See also
Amplitudes of Effects of
General Relativity
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