K^2 wrote:Two problems with the above. First of all, frame dragging causes acceleration of a reference frame. Agreed, you might call that semantics, but it is technically important for the next point. Suppose, the universe consists of you and a billion ton object. That object accelerates, and you feel the tug. Now suppose, the universe consists of you and a two billion ton object that accelerates at the same rate.
At the same rate as you? I.e. you and the billion ton object accelerate together, and remain comoving? (Just trying to follow your description accurately).
In GR, the frame drag will be different. According to Mach's Principle, it will be the same. No other reference objects, so you'll experience acceleration equal to acceleration of reference object regardless of later object's mass. While the two theories predict the same qualitative effect, quantitatively, they are not in agreement.
Since I'm not clear on your description above, I may be misunderstanding you, but it seems to me that if you and the billion ton object are accelerating so as to remain comoving, then in GR you would feel the pull of the moving billion ton object along one vector, and you would feel the push of your own acceleration along another vector, and the two would cancel out, leaving you feeling exactly the same forces as you did before you and the billion ton object accelerated, exactly as Mach's principle would predict (since by Mach's principle neither of you would have "really" accelerated).
Say you were inside an impossibly sealed box through which no observations can be made, completely opaque to all electromagnetic radiation, a perfect thermal and electrical insulator, chemically inert, of infinite tensile strength, etc -- a GP Hull from Known Space, if you will, but without the transparency to visible light. You are sitting in the dark in this box in freefall in open space, and then you feel a "force" like acceleration. You can infer from the vector of this force that the ship is moving in a certain fashion, and assuming that the rest of the objects in the universe are static relative to each other and your previous reference frame, you could infer in what fashion you would see the rest of the universe whizzing past you, if you could see outside the ship.
But frame dragging, caused by objects whizzing under their own power past you in a stationary ship, could cause similar measurements from inside your ship. My question is, would your inferences of what you would see outside your ship (if you could) be any different if you assumed the forces you felt were cause by frame dragging, versus being caused by your ship accelerating under it own power?
E.g. say the universe consists of a billion ton hollow sphere with your spaceship in the center of it, and the inside of the sphere is marked with degree marks. If the sphere were to spin around your ship, then due to frame dragging you would feel dizzy. If your ship were to spin inside the sphere, then you would also feel dizzy. If you could see outside your ship, you would see the degree marks passing by as you, or the sphere, spun. Given that you can't see outside your ship, and you don't know whether you're spinning or the sphere is spinning, would your prediction of how fast the degree marks would pass you by differ between the assumption that you are spinning and the assumption that the sphere is spinning?
And if your ship, or the sphere, spun up in turn to keep up with the other, wouldn't you feel nothing (and predict, accurately, that you would see no degree marks flying past) in either case?
The underlying theory might still agree with Mach's Principle. If you get effects like gravitational constant being function of universe's total energy, then the frame drag might end up being exactly the same in the above two examples. But in General Relativity the speed of light and gravitational constant are fixed. The acceleration will be a function of object's mass, and you are still left with a contradiction.
I admit I am not so well versed in the math here; can you tell me, is the gravitational constant fixed by GR because of any mathematical necessity (i.e. could we compute a priori what the gravitational constant has to be), or is it just an observed value that we plug into the equations to make the units come out right? I was under the impression that it was the latter, in which case nothing is broken by the value of that constant being a function of the universe's total energy; we would just have an explanation for why the constant is such.
It occurs to me now that your answer to the above scenario of the billion-ton sphere and the perfectly opaque ship probably depends heavily (no pun intended) on the mass of the sphere; a lighter or heavier sphere would give different degrees of discrepancy from what you'd expect from Mach's principle, given our current value for the gravitational constant
, which will be used to calculate how much the spinning sphere drags your frame. In that case, then we should be able to infer from Mach's principle and the observed value of the gravitational constant what the mass of the universe is. That gives us an empirical test for Mach's principle: what is the mass M which such a hollow sphere would have to have, for the frame-dragging effects of it spinning (as calculated with the observed value of gravitational constant) to equal the acceleration experienced by a ship spinning at the same speed in open space (as calculated with the same gravitational constant value)? Mach's principle predicts that that the mass of the universe should then be M. What is the mass of the universe as observed by other means, and how well does it match M?
It would make sense that we should be able to measure the mass of the universe from some local phenomenon, anyway, given that every mass in the universe affects every other mass. We should be able to pick a local mass, and measure how much it is being affected by the mass of the rest of the universe, and from that calculate what the mass of the rest of the universe is, no?
Addendum: Really going out on a limb here now, but follow this. Given Mach's principle, the gravitational constant is proportional to the mass of the observable universe, as above. The rate of the universe's expansion is (uncontroversially) inversely proportional to the gravitational constant, i.e. higher constant would mean slower expansion, lower constant would mean faster expansion. In an expanding universe, more and more distant mass is constantly leaving the cosmic event horizon, i.e. becoming unobservably distant, as more distant objects appear to be moving away from us proportionally faster and so sufficiently distant objects appear to recede faster than light and vanish from observability. Since the effects of gravity also propagate at c, the gravitational effects of such objects beyond the cosmic event horizon should also be lost... decreasing the mass of the observable universe, which per Mach's principle should decrease the gravitational constant, in turn accelerating the expansion of the universe.
Mach's principle explains the accelerating expansion of the universe!
EDIT: Expansion. (No pun intended).