Hello all,



I happened to stumble upon a tutorial on general relativity authored
by John Baez (http://math.ucr.edu/home/baez/gr/ricci.weyl.html). In
the section called "The Ricci and Weyl Tensors" the author tries to
give a geometric interpretation of these tensors. However, I am not at
all clear about the precise meaning of what he writes:


" [...] We consider a bunch of initially comoving coffee grounds near
a point P in spacetime, with the coffee ground that actually goes
through P having velocity v at that instant. (Hence the term "instant
coffee".) Working in the local rest frame of the coffee ground that
goes through P, we consider a small round ball of comoving coffee
grounds centered at P, and see what happens as time passes. Each
coffee ground moves along a geodesic, but since spacetime is curved,
the ball may shrink, expand, rotate, and/or be deformed into an
ellipsoid.

The Ricci tensor Rab only keeps track of the change of volume of this
ball. Namely, the second time derivative of the volume of the ball is
-Rabva vb times the ball's original volume."

(Actually, Lee C. Loveridge may also be quoted, writing something
along the same lines: "The Ricci tensor governs the changing size of a
small volume propagating through a curved space.")


It is the last paragraph quoted from Baez that I am very unsure about.
There are several naturally defined balls associated with a manifold,
so I am unclear about which exact definition is being referred to.
Further, some vector field on the manifold or some family of geodesics
needs to be chosen for the "ball" to be uniquely determined.


My questions are as below:

1. How would the claim made by Baez be expressed precisely, as a
theorem of differential geometry? I should like very much to see a
proof also. A reference to some bibliography would be nice.

2. Is there some direct geometric interpretation of the Ricci tensor,
apart from it being defined as "the" contraction of the Riemann
tensor? In terms of sectional curvature, perhaps?

3. Is there some way of interpreting the Weyl tensor, physical as well
as geometric? (In Baez's tutorial, the Weyl tensor is related to
gravitational waves, but in my opinion the argument given there
doesn't seem to pinpoint waves as the only possible interpretation.)



Thanks in advance!

Kasper J. Larsen

Kasper J. Larsen wrote:
I happened to stumble upon a tutorial on general relativity authored
by John Baez (http://math.ucr.edu/home/baez/gr/ricci.weyl.html). In
the section called "The Ricci and Weyl Tensors" the author tries to
give a geometric interpretation of these tensors. However, I am not at
all clear about the precise meaning of what he writes:

" [...] We consider a bunch of initially comoving coffee grounds near
a point P in spacetime, with the coffee ground that actually goes
through P having velocity v at that instant. (Hence the term "instant
coffee".) Working in the local rest frame of the coffee ground that
goes through P, we consider a small round ball of comoving coffee
grounds centered at P, and see what happens as time passes. Each
coffee ground moves along a geodesic, but since spacetime is curved,
the ball may shrink, expand, rotate, and/or be deformed into an
ellipsoid.

The Ricci tensor Rab only keeps track of the change of volume of this
ball. Namely, the second time derivative of the volume of the ball is
-Rabva vb times the ball's original volume."

(Actually, Lee C. Loveridge may also be quoted, writing something
along the same lines: "The Ricci tensor governs the changing size of a
small volume propagating through a curved space.")

It is the last paragraph quoted from Baez that I am very unsure about.
There are several naturally defined balls associated with a manifold,
so I am unclear about which exact definition is being referred to.

His "ball" is most definitely not an open set or any other topological
property of the manifold. Instead it is a REAL set of coffee grounds (or
any other suitable set of small particles, such as grains of salt). He
is imagining that one begins with a bunch of such dustlike particles
carefully arranged in a small spherical ball, and then observes how the
particles of that initial ball evolve over time (treating their totality
as a volume).

Note, please, that the manifold is a model of the world.
The coffee grounds are in the world being modeled, and
are represented in the model as pointlike test particles with
negligible mass. This intermixing of world and model is quite
common, but can be confusing to the novice -- especially ones
with a mathematical background. Strictly speaking it is
wrong to intermix these VERY different concepts, but is such
a common shortcut that this mode of speaking is normal.


Further, some vector field on the manifold or some family of geodesics
needs to be chosen for the "ball" to be uniquely determined.

No additional vector field or family is required. The ball is defined by
a set of worldlines of the coffee grounds through spacetime (which are
contained in a spherical volume initially[#]); as these are test
particles with negligible mass, each individual coffee ground follows a
geodesic. So this does define a family (congruence) of geodesics (in a
suitable limit with # of grounds = infinity and the mass of each = 0).

[#] Any "spherical volume" will do here -- just pick any
foliation of spacetime into space and time, and define
the sphere in one hyperplane of constant time at the
initial time you select. His discussion holds only for
a "short" time duration, and the way you select the
foliation will affect how "short" is short enough.


1. How would the claim made by Baez be expressed precisely, as a
theorem of differential geometry? I should like very much to see a
proof also. A reference to some bibliography would be nice.

I believe this is the contracted Bianchi identity (but I'm not 100%
certain).


2. Is there some direct geometric interpretation of the Ricci tensor,
apart from it being defined as "the" contraction of the Riemann
tensor? In terms of sectional curvature, perhaps?

Baez just gave one -- the time evolution of the volume of a small ball
of coffee grounds.

I believe in a Riemannian manifold there is an interpretation of Ricci
in terms of radii of curvature in all possible directions. But it's not
clear to me how that might work in a Lorentzian manifold of GR.

Lorentzian manifold = a semi-Riemannian 4-d manifold with
signature 2. If you care, Hausdorff and para-compact are
also required, and physicists are only concerned with
connected spacetime manifolds.


3. Is there some way of interpreting the Weyl tensor, physical as well
as geometric? (In Baez's tutorial, the Weyl tensor is related to
gravitational waves, but in my opinion the argument given there
doesn't seem to pinpoint waves as the only possible interpretation.)

No, gravitational waves are not the only interpretation of Weyl.

Remember from the Einstein field equation one can show that any region
of the manifold with T=0 has Ricci=0. So in a vacuum region that is not
flat, Weyl must be nonzero -- that is what causes gravitational effects
at distant locations. So the falling of an apple is due to nonzero Weyl.

Of course this can also be expressed in terms of the connection
rather than the curvature tensors.

"vacuum region" = region in which T=0 throughout. T is the
energy-momentum tensor.


Tom Roberts