Inquire about purchasing the book | Table of Contents | Annotated Bibliography | Class on Nature of Mind

**These are excerpts and elaborations from my book "The Nature of Consciousness"**

Since
gravitation is natural motion, Einstein’s idea was to regard free falls
as natural motions, i.e. as straight lines in spacetime. The only way to
achieve this was to assume that the effect of a gravitational field is to
produce a curvature of space-time: the straight line becomes a “geodesic”, the
shortest route between two points on a warped surface (if the surface is flat,
then the geodesic is a straight line). Bodies not subject to forces other than
a gravitational field move along geodesics of space-time. The curvature of
space-time is measured by a “curvature tensor” originally introduced in 1854 by
the German mathematician Bernhardt Riemann. The Riemann geometry comprises the classical Euclidean geometry as a special
case, but it is much more general. Minkowski's four-dimensional spacetime is
characterized by a "metrics". A metrics is a 4x4 matrix, each row and
column representing one of the dimensions. The metrics for Newton's spacetime has zeros everywhere
except in the diagonal of the matrix. The diagonal has values 1,1,1 and -1.
This means that Pythagoras' theorem still
works, and time is an added dimension. The zeros in the other positions of the
matrix specify that the space is flat. When the ones and the zeros change,
their values specify a curvature for spacetime. Euclidean geometry works only
with the flat-space metrics. Riemann's geometry works with any
combination of values, i.e. with any degree and type of curvature. A specific
consequence of Riemann's geometry is
that "force" becomes an effect of the geometry of space. A
"force" is simply the manifestation of a distortion in the geometry
of space. Wherever there is a distortion, a moving object feels a "force"
affecting its motion. Riemann's geometry is based on the notion of a
"metric (or curvature) tensor", that expresses the curvature of
space. On a two-dimensional surface each point is described by three numbers.
In a four-dimensional world, it takes ten numbers at each point. This is the
metric tensor. Euclid's geometry corresponds to one of the infinite possible
metric tensors (the one that represents zero curvature). Not only space
and time are relative, but space-time is warped. With his field equations,
Einstein made the connection with
the physical world: he related the curvature of space-time caused by an object
to the energy and momentum of the object (precisely, the curvature tensor to
the “energy-momentum tensor”). Einstein therefore introduced two innovative
ideas: the first is that we should consider space and time together (three
spatial dimensions and one time dimension), not as separate; the second is that
what causes the warps in this space-time (i.e., what alters the metric from
Euclid's geometry) is mass. A mass does not voluntarily cause gravitational
effects: a mass first deforms space-time and that warping will affect the
motion of other objects that will therefore be indirectly feeling the
"gravitational force" of that mass. The mass also
has an effect on the “time” part of space-time: clocks in stronger
gravitational fields (bigger warp) slow down compared with clocks in weaker
gravitational fields (smaller warp). Summarizing: the
dynamics of matter is determined by the geometry of space-time, and that
geometry is in turn determined by the distribution of matter. Space-time acts
like an intermediary device that relays the existence of matter to other
matter. There is an
analogy with Maxwell’s theory
of electromagnetism. If one thinks of
the “metrics” as a metric field, then the metric field bends the trajectory of
bodies that have mass-energy the same way that the electromagnetic field bends
the trajectory of bodies that have electric charges. Incidentally,
this implies that mass-less things are also affected by gravitation. This
includes light itself: a light beam is bent by a gravitational field. Light
beams follow geodesics, which may be bent by a space-time warp.
Back to the beginning of the chapter "The New Physics" | Back to the index of all chapters |