Posted by: Aba Cohen | April 10, 2009

Understanding what Gravity is: Time Expansion & Space Contraction

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 As I have said in the RealMagic twitter´s sentence, gravity of a given mass M comes from Time Expansions associated with transverse (to g) Space Contractions. The whole effect corresponds to tidal forces dragging, towards M, a second mass, m,   put in such a deformed spacetime tissue. We can see below two distinct approaches where gravity (= acceleration, according to Einstein´s General Theory of Relativity – GTR) is present:

You don´t have to know, a priori, anything about  the GTR to get the thing. You just need to know from the Special Theory of Relativity -STR- that space shrinks and time expands with increasing velocities, when a given phenomenon is described by an inertial observer. At first we use a MATERIAL spinning disc to explain what follows: Suppose a situation where the disc´s external border moves at a given velocity, e.g. at relativistic speed (v < ~c), but NOTE, this huge velocity is not necessary at all. An observer from outside the disc sees a time expansion, proportional to gamma, and a space contraction of the disc perimeter, proportional to 1/gamma, where gamma is the Lorentz Factor [1/ square root (1 – v2/c2)]; no change is observed in the radius as it is not material or it does not move (or, if you are not yet convinced, assume that “the radius moves” transversally to v and, from the STR, space shrinks only in the direction parallel to v – so “r gets thinner” conserving its length). As the internal concentric circumferences rotate with the same angular velocity (ω), the respective tangential velocities (v´ = ω.r´) are progressively smaller than v as you get closer to the center, where v´= 0. Being so, the internal circumferences shrink regressively less and less. The same happens to the centripetal accelerations (a = v2/r´ = ω2 r´) as they also drop with r, when described by the external observer. The consequence of progressivelly shrunk circumferences and lethargic watches along fixed radii is a bend in space-time as you can see in the figure below. An internal rotating observer lying still on the disc’s surface feels the action  of his feet (i.e. the reaction to the centripetal force, that keeps him in rotation) pulling the “floor” radially outwards –consider this floor as a step fixed on the external circumference, to avoid that observer been spit out form the disc– what keeps him in rotation. He designates this as the “force of gravity”. For the external observer, this is also gravity and is described by the GTR as a consequence of the time expansion and transverse (to g) space contraction. From the Newtonian (Classical Physics) point of view, such a “centrifugal force” is called “inertial force” and neither time-expansions nor space-contractions exist to justify its existence. In the GTR inertial-mass is synonym of gravitational-mass: what gives m that “inertial feeling” against acceleration is exactly gravity, where time “flows” slowly.

 gravity-of-a-spinning-material-disc-and-of-a-huge-mass-m-a3

In the second approach a given mass M distorts space with stronger time expansions and space contractions as you get closer to M (increase g):  Similarly, the gravity created by M comes from the time expansion and the transverse (to g) space contraction that creates tidal forces (observe the dragging gradient) pulling m towards M.

gravity-well


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