måndag 24 februari 2014

Physics Illusion 10: Fabric of Curved Space-Time

                Curved space coordinate system in which free fall follows coordinate lines. Believing that this coordinate system shows us the true "fabric of curved space-time" is deeply mystical, yet supposed to be a foundation of modern physics.
In general relativity the distribution of matter/energy is supposed to determine the "fabric of space and time" and make more or less "curved" according to Einstein's equation (assuming the speed of light and gravitational constant normalized to unity)
  • $G_{\mu\nu}=8\pi T_{\mu\nu}$, 
where $G_{\mu\nu}$ is Einstein's tensor measuring curvature and $T_{\mu\nu}$ is a stress-energy tensor. The folklore explanation in the words of John Wheeler reads as follows:
  • Matter tells space how to curve. Space tells matter how to move (by free fall).
In the basic case of flat Minkowski space-time and infinite speed of light (standard Euclidean space with coordinate $x$ and separate time coordinate $t$), Einstein's equations reduce to 
Newton's equations
  • $\Delta\phi =\rho$,                                            (Newton's law of gravitation)
  • $\ddot x(t) + \nabla\phi (x(t),t)=0$,                    (Newton's 2nd law)
where $\rho (x,t)$ is mass density, $\phi (x,t)$ is gravitational potential and $x(t)$ denotes the trajectory of a mass particle moving subject to gravitational forces given as $-\nabla\phi (x,t)$.  As a Wheeler paraphrase, Newton's equations could be expressed:
  • Matter determines the gravitational potential. The gradient of the gravitational potential determines how matter moves. 
Einstein's supposedly great idea was to reduce all motion to free fall in a space-time coordinate system where the equations of motion take the form $\ddot x(t)=0$ as if no gravitational forces were present. 

Einstein thus replaces the gravitational potential by space-time curvature, which makes things more complicated, and then balances by using a trivial form of Newton's 2nd law. 

The question is if there is a net gain or loss by this shift? I would vote for Newton's formulation because both Newton's laws are easy to grasp. On the other hand, Einstein's concept of "curved space-time" supposedly being realized in some "fabric of space-time" makes things much more difficult to grasp at the small gain of having only to deal with Newton's 2nd law in trivial form, thus with a net loss.

Einstein's was hooked to the idea of choosing a coordinate system in which the equations of motion reduce to the trivial form for free fall, as if the essence of physics was hidden in a coordinate system.
But real physics cannot rely on the choice of coordinate system. In principle any coordinate system can be used and a coordinate system does not need any "fabric" to be erected as it is immaterial. 

Believing that there are certain coordinate systems with higher scientific value than others, may lead to seek to identify such royal coordinate systems through some form of material existence expressed as "fabric of space-time".  But that is deeply mystical and as such can only be an illusion.

Newtonian gravitation can be combined with Maxwell's equations for electromagnetics and Schrödinger's equations for quantum mechanics into a Unified Field Theory. Einstein struggled for 40 years to combine Einstein's equations with Maxwell's and Schrödinger's equations without any success, and nobody else got further.

The effective way out of this dilemma, which has paralyzed physicists for 100 years, may well be to replace Einstein's equations by Newton's equations at no loss of anything of real value to physics. 

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