makc wrote:
...what exactly do you mean when you write "the time and space are coupled", mathematically?
JimJast wrote:... identity (d^2)T/dtdr+1/R=0, where d means "partial", T is proper time at point in deep space, t is coordinate time (of observer), r is radial coordinate (coordinate distance from observer to observed point in deep space), and R is radius of curvatue of space under consideration ...
... it implies Hubble type time dilation in deep space
(d/dr)(dT/dt) proportional to
1/R. And so the Hubble constant of Einstein's universe of radius
R (a.k.a. "Einstein's radius") at observer is
H=c/R, where
c is speed of light.
The above identity "couples time to space" making the intrinsic curvature of spacetime vanish. Feynman:
"As you know from special theory of relativity, measurements of space and measurements of time are interrelated. And it would be kind of crazy to have something happening to space, without the time being involved in the same thing." It is involved
in the same thing making the space looking like expanding to the "curved spacetime" (BB) folks.
You may of course follow the BB line of thinking saying that space and time are independent as it used to be in Newtonian math that would require to have the curved spacetime (
"cosmic time" being flat and
space curved). Mathematically it produces the same Hubble constant, however without the acceleration of expansion which has to be patched separately with
"dark energy" and besides, another small detail, the BB requires constant creation of energy (parhaps through the divine intervention as page 1218 of MTW
"Gravitation" and Leibniz quote at it suggests:
"One suffices to create Everything of nothing!" while "intrisincally flat specetime" requires neither expansion nor creation of energy and that's why, not beliving in miracles, I vote for "Einstein's universe". You don't have to of course especially if you believe in divine intervantion suggeted by MTW's about $100
"Bible" (possibly the real reason for BB) and George LeMaitre.
If you need more math it is in Narlikar's papers and so you may further discuss the matter with him and reject his rather exotic physics of increasing mass of particles (as you may see the same math may fit many different theories), but at least you will have the math the same as Einstein's since Narlikar, being a math professor, noticed already also that the spacetime must be intrinsically flat not to
"be kind of crazy" (according to Feynman). Narlikar and Arp call
"intrinsic redshift" what I call
"general time dilation" but the physical reason for it is the same: Inability of nature to create energy (that would be
"be kind of crazy" in Feynman words). So, as you may see there is some kind of logic in our crazy ideas. And as I said you don't need to share them especialy if you believe (as BB folks do) that creation of energy is possible.
I'll try to make "more effort" to explain gravitational force in Einstein's gravitation later (if I won't be banned from here till then) but you could look up Landau's texts in his "Theory of fields" where he explains relativistic spacetime. Just differentiate his equation of page 285 and get
"gravitational force" since this equation (possibly unknown to Landau himself since I don't see him differentiating it anywhere) expresses the
"gravitational energy" of a particle.