Mach’s Principle and Gödel’s Rotating Universe

Oct 21, 2014

by William Brown, Resonance Science Foundation Research Scientist

The question of the nature of absolute and relative frames of space, time, and motion forms the basis of Newton’s work in The Mathematical Principles of Natural Philosophy, as the opening discussion of the book. This question is also relevant to Einstein’s theories of motion occurring at or close to the speed of light, hence the appellation relative in relativity theory. As we have seen, Newton described a fundamentally absolute frame of space and time in which an object’s motion was absolute, or independent of its orientation relative to other objects.

Newton illustrated this concept in the Principia with an example of a bucket filled with water and suspended by a rope from the ceiling. If the rope were to be turned until it was sufficiently twisted, and then released, the rapid spinning motion of the bucket would impart to the water a centrifugal force that would cause it to pull away from the center of the axis of rotation, imparting a concave curve to the surface. Newton asserted that this motion of the water would be completely independent of any other bodies apart from the bucket, the rope, and the water – it was a closed system. Ernst Mach, a 19th century physicist and philosopher, pointed out that it was not a closed system, as the Earth’s gravitational force on the water and the bucket were certainly important factors, and hence the motion of the water could never be considered independently of the other objects around it.

Indeed, Mach asserted that the water’s motion could only be defined relative to all other objects in the Universe. This has since become known as the Bucket argument and Mach’s principle. Mach’s principle [first stated by George Berkeley nearly 200 years before Mach] is the assertion that the inertial reaction forces experienced by massive objects, when they are accelerated by external forces, are generated by the action of chiefly the most distant matter in the cosmos. Since the only known force with universal coupling to mass is gravity, it is natural to assume that, if the principle is correct, the gravitational interaction is the source of inertial reaction forces. This is in fact true in general relativity theory for the class of cosmological models thought to encompass our reality. Mach's principle was in fact a major influence on the work of Einstein and even modern day physicists[1], for if we apply Mach’s principle to the effect of matter on space and time, then it will be seen that matter curves spacetime (much like the water in the bucket due to the force of spin, a fact elucidating the equivalence of gravitational forces with acceleration), and hence space and time are not independent of the matter occupying it.

However, the work of the physicist and mathematician Kurt Gödel in 1949[2], in which he introduced a new solution to Einstein’s equations[3], showed that Mach’s principle was not completely incorporated into the theory of general relativity. Kurt Gödel introduced a model in which the Universe was rotating (and contained time-like curves that loop back on themselves, essentially allowing for travel back in time). A rotating Universe does define a preferred direction of motion, and hence the motion of objects within the Universe are not entirely relative only to other objects, as there is now a kind of absolute frame of reference (although they are still relative to the observer’s motion).

Even with a rotating Universe, it appears that Newton’s conception of an absolute space must be forgone, as the latest measurements from experiments such as Gravity Probe B[4] have demonstrated the validity of a theory known as the Lense-Thirring effect (also known as frame-dragging or gravitomagnetism). Gravity Probe B demonstrated that as the Earth rotates, it ‘twists’ or drags the local vicinity of spacetime along with it, and as such the motions of objects in the vicinity of Earth will be influenced by the Earth’s spin. An amazing example of this is in the vicinity of a black hole. Black holes must have an incredible rate of spin, which produces what is known as an ergosphere around the black hole. Within the ergosphere, the Lense-Thirring effect causes such a high degree of torque that spacetime is dragged in the direction of the spin of the black hole at a velocity greater than the local speed of light relative to the rest of the Universe. This means that in order for an object within the ergosphere to appear at rest to an external observer, it would have to travel in the opposite direction of the frame-dragging at a velocity greater than the speed of light.

Faculty Article by William Brown

References

[1]  See for instance Julian Barbour, and his book entitled The End of Time.
[2]  A colleague of Einstein’s.
[3]  The Gödel solution to the Einstein field equations - http://www.math.nyu.edu/~momin/stuff/grpaper.pdf
[4]  Gravity Probe B: Testing Einstein’s Universe - http://einstein.stanford.edu/index.html