The Nature of Consciousness

Piero Scaruffi

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

Curving Spacetime

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.

 


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