If we think of curvatures as unevenness, in much the same way as an inhomogeneous concentration of particles, then things tend to straighten out in time. In other words, motion is created by by the mean curvature of a manifold, and its evolution should tend to decrease its curvature in time. This, more or less, is the logic behind Ricci flows which have been limited to three dimensions. However, this should hold in any dimension.
We have emphasized, in previous blogs, that there is something fundamentally wrong with Einstein's field equations of general relativity. If the forces are tidal, we know that for constant curvature they are proportional to a constant mass density. Then the tt-component of Einstein's equations tell us nothing new: density=density. The other three space components give a wrong sign for the pressure, and are consequently wrong. Then what do we set mean curvature equal to?
Surfaces that tend to minimize their areas subject to constraints are well-known. Surfaces driven by surface tension that are opposed by frictional forces proportional to the velocities at which the surfaces move is a problem in brownian motion. Due to friction, the acceleration becomes proportional to the velocity and paths tend to bunch up or spread apart, much like geodesic deviation that is described by the Jacobi equation. The tendencies of the path reflect the nature of the non-euclidean geometry in which they are moving: for elliptic geometry on a sphere, the paths tends to bunch up, while, for hyperbolic geometry on a saddle, they tend to disperse.
Suppose that the motion of a surface is parameterized by time so that each point on the surface tends to move with a certain velocity which is proportional to its mean curvature. Such systems are well-known; for instance the velocity of grain boundaries are proportional to their mean curvatures. However, it is impossible to follow a single point, and, therefore we must turn to a statistical description as a measure for the set of paths.
Consider a sphere of radius R(t) at time t. The magnitude of the mean curvature is proportional to the inverse of the scale factor, which, in this case, is simply the radius, R(t). The surface tension is opposed by a viscous force, with diffusion coefficient D. Hence, the velocity at which the surface moves is
dR/dt=-D/R(t).
The velocity decreases as the radius of the circle increases. The integral is given by
R(t)=[R^2(0)-2Dt]^(1/2), (1)
showing that the radius tends to decrease in time where R(0) is the initial radius of the sphere. In time, the radius will shrink to zero; the time being t=R^2(0)/2D. But, this is none other than brownian motion where the time interval is proportional to the mean square displacement of the brownian particle. As the intervals become increasingly smaller, the paths become more and more jagged so that no instantaneous acceleration can be defined.
A surface that starts in a convex set will remain in that set for all time. So that a surface contained in the interior of a ball with radius (1), having the same center, will remain in that set for all time. As as the radius of the ball shrinks in time so too with the contained surface. This is the physical justification behind Ricci flows, and their application to the Poincare conjecture which postulates that every close simply connected 3D manifold is topologically and S^3 sphere. The proof is based on "curve shortening" described by a Ricci flow.
Ricci flow is described by the differential equation:
dg_(ij)/dt=-2R_(ij), (2)
where R_(ij) is the Ricci tensor, and g_(ij) is the metric. This is Hamilton's equation of motion for the metric. The Ricci flow equates the time variation of the metric with the spatial acceleration rate that is characterized by the Ricci tensor. It is this tensor that gives the equation of motion the form of a diffusion equation. Using this equation, the uniformization theorem can be established in which every closed surface can be attributed to one of three geometries: spherical, euclidean, and hyperbolic. That is one can deform any given geometry by means of a diffusion equation into one of these geometries.
So it would appear from Eq (2) that Ricci flow gives us something other than a energy-stress tensor to equate with a Ricci tensor. Yet, this fails on first try when we consider Einstein's condition of "emptiness" in which all eigenvalue of the Ricci tensor vanish. This would imply "no flow". But that can't be right because the vanishing of the Ricci tensor in the case of constant mass means that the sum of the tidal forces vanishes. The individual tidal forces are, however, not zero. So some flow is implied.
Consequently, instead of considering the average curvature in any given direction as the source of diffusion we are led to consider the individual sectional curvatures. As an illustration consider the Schwarzschild metric with
g_(00)=1-2M/G,
where G is the radial coordinate for the warped metric, G'=g_(00)^(1/2),
ds^2=-g_(00)dt^2+G^2(r)d phi^2,
in the azimuthal plane theta=0, where the prime stands for differentiating with respect to the radial coordinate, r. Letting F=g_(00)^(1/2), the equation of motion is:
d F^2/dt= D F"/F, (3)
where we have set the constant of proportionality equal the diffusion coefficient, D.
We can also write (3) in the form of a diffusion equation
d g_(00)/dt= D{g"_(00)/g_(00)-(1/2)g^(' 2)_(00)/g_(00)},
with the right-hand side having the form of an Onsager-Machlup potential (cf. "Nonequilibrium Statistical Thermodynamics", Eq (43.7)).
Since the tidal force in the z-direction is F"/F=-2M/G^3, while those in the two orthogonal directions are both equal to +M/G^3, the equation of motion reduces to
dG/dt=-D/G, (4)
showing that the curvature is, indeed, inversely proportional to the scale factor, G. And although G is usually interpreted as "distance from the center of the star", neither "the travel time measurements of reflected light signals nor trigonometric distance measurement would confirm that increments in G measure radial distances." (Besse, Einstein Manifolds, p. 105.)
Hence, integrating (4) leads to:
G(t)={G^2(0)-2Dt}^(1/2), (5)
which is the hallmark of shrinkage by brownian motion. Taking into account the "natural" barrier at R_s=2M, the Schwarzschild radius, we set G^2(0)=2R_s^2 and write (5) as
G^2(t)-R_s^2=R_s^2-2Dt,
so that G(t)->R_s as t->R_s^2/2D.
If we had considered the other two tidal forces acting in orthogonal directions we would have obtained radial coordinates that continually grow in time. This has the effect of transforming a sphere into an ellipsoid. The shape of the tidal bulges on earth are caused by differences in the gravitational force acting at the center of mass and that on the surface. They compress the earth in the direction orthogonal to the line connecting the center of masses of the earth and moon and elongate along the line. The frictional frictional forces cause the earth's spinning to slow down, and to increase the distance between the earth and the moon.
In fact, Laplace assumed the contrary: that the earth moon distance is decreasing, and that the time it would take for the moon to fall to earth would be an indication of the speed of gravity. Because the speed of gravity is finite, the gravitational force would actually be an aberrated force slowed down by the ratio sin theta= v_E/v_G, the ratio of the orbital velocity, v_E, of the earth about the sun and the speed at which gravity propagates, v_G. Because theta is small, the aberrated force is
F_(theta)=-M_E*M_M/R^2 theta.
When multiplies by the orbital velocity of the speed of the moon, v_M, this becomes the rate of energy loss which must be balanced by the rate of decrease in energy of the earth moon system, GM_E*M_M/R^2 (dR/dt). Thus,
dR/dt=v_M theta=(GM_E/R)^(1/2) theta, (6)
after cancelling common terms, and introducing Kepler's law. Now, instead of a velocity proportional to D/R, we have it equal to a velocity itself. Consequently, the surface tension is inversely proportional to the square-root of the scaling factor, instead of the scaling factor itself!
Integrating (6) gives
R^(3/2)=R^{3/2)-3(GM_E)^(1/2) theta t. (7)
For normal brownian motion, the Hurst index is H=1/2. Now, for fractional brownian motion, any Hurst index H>1/2 is indicative of a long range dependence and positive autocorrelation. This occurs for H=2/3 so that the surface tension is proportional to an inverse power of the scaling factor less than one. In the opposite case where the Hurst index is less than 1/2, there is negative autocorrelation, with only short-range dependence.
Laplace calculated that for a lifetime of the moon of some 30 billion years, the ratio theta=v_G/c ~ 10^8, with c the speed of light. In Laplace's own words, "we must suppose that the gravitating fluid has a velocity which is at least a hundred million times greater than that of light." Although the model was later shown to be in error, the conclusion is not.
Newton showed that centrifugal acceleration on a rotating sphere was to make it slightly oblate, with a flattening at the poles. This was in contradistinction to Cassini who believed the opposite, as shown in old-time caricature shown below.
This is like a spinning water balloon that will start to flatten out at the poles and bulge at the equator. The tidal forces due to gravity by the pull of the moon will create further distortion. Both tidal forces and centrifugal forces squeeze a sphere into an ellipsoid. Tidal deformation and rotational deformation give rise to a surface that can be modelled by means of a Legendre polynomial of degree two implying that all of what we have said about tidal deformation can be applied to rotational deformation.
The only difference being is in the relative magnitudes of the stresses. For tidal stresses, the relative ratios were found to be -2, 1,1, while in the case of rotational stresses, they are -7/6, 1/3, 5/6. Just as in the tidal case, they sum to zero in the rotational case as well giving the impression of Einstein's "emptiness" characterized by the vanishing of the eigenvalues of the Ricci tensor. However, these systems are anything but empty! And neither Einstein's field equations, nor Ricci flow will describe such systems. What must be taken into account are the single sectional curvatures, and not their averages in any given direction for they can average out to zero giving the wrong impression that nothing is there, i.e. Einstein's condition of emptiness.
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