13/03/2022
Newton's Laws and General Relativity
We explain the inadequacies of the Newton’s laws in the case of curved space-time. The general relativity asserts that the laws of physics take the same form in all reference frames. Also, we cannot construct an inertial frame in the Special Relativity sense in the presence of a (non-uniform) gravitational field. The reason being MS, which is backbone of STR, is squashed up but not curved even due the presence of mass/energy or gravity. This, however, no longer is a relevant issue. It is implicit in the Equivalence Principle that there is no preferred coordinate system, for we no longer distinguish between gravitational forces and fictitious forces, such as the centrifugal force of Newtonian mechanics. The latter may be introduced by coordinate transformations. The objective of Newton’s first law, to establish a frame with no fictitious forces, in which to formulate the rest of the theory, is made obsolete by general relativity. We may replace the first law with the following law governing freely falling motion, that is, the motion of a body acted on by no non-gravitational forces: A freely falling body travels along a space-time geodesic emphasizing the importance of the given space-time template. As a special case, in flat Euclidean space-time, this implies the first law. Also, the homogeneous and isotropic propagation of light. Now it is turn of the second law: general relativity is a nonlinear theory, though the nonlinear effects only appear at very high field strengths, as is the case with electromagnetism. Perhaps the most dramatic demonstration of the failure of the second law is the black hole. In the inadequate Newtonian picture, a body that has fallen beyond an event horizon can only be described as experiencing an “infinite” force, since no force can prevent it from falling further in and colliding with the so-called singularity. The action of the black hole on the in-falling body cannot be described as the sum of the actions of the individual particles making up the mass of the hole. The “event horizon” (which is a geometric feature) mathematically distinguishes the world outside of the black hole from the inside. This horizon has nothing to do, however, with the Newtonian idea of an escape velocity. While the former is where we live and it contains our universe, inside the event horizon space-time behave in very abnormal way. Once inside, matter and light cannot get back out into the rest of the universe. These two regions have very different geometric properties, due to the vastly different behaviour of space-time. Therefore, the second law is totally inadequate tool to investigate the black-hole physics. The third law implies global conserved momentum in a flat Euclidean spacetime. In a general space-time manifold, there is local but no global conserved momentum. So, the law direly needs modification. In the particular case of our universe, which is expanding, photons get red-shifted as the universe expands. Consequently, they lose both energy and momentum ( Tμν ) , which simply disappears into the void. Having stated this, we note that energy and momentum are not arbitrarily created or destroyed; they change according to the law of parallel transport. In other words, the curvature of space-time causes them to evolve in a predictable way. This is captured by the modified third law: Tμν;ν=0. In flat space and time this reduces to the (local) conservation of energy and momentum. In conclusion, gravity is not a force acting on an object in space causing the path of the object to curve; this is synonymous with the curvature of space-time itself! To draw an analogy, gravity is the animate dance floor where all particles strut their funky stuff. However, the dance floor itself bends and warps, under the influence of all possible interactions between actor particles, re-choreographing their dance. The loop quantum gravity, the string theory, etc. are brilliant formulations to integrate all possible permutation and combination of the dance floor antics with those of the particles.
