Inertia... in an empty universe?

[Boerseun]

Right... another thing I don't get. Picture this:

 

Everything in the universe magically disappears. Every star, every galaxy, every gas cloud, everything - even the CMBR ... save for a small spinning top, floating aimlessly in a now empty void containing nothing but this spinning top.

 

Will the top know that it's spinning? There is nothing to measure any speed against. There are no reference points. We all know that there are limitations to this top's condition - it cannot rotate at a speed at which points along its edge will go greater than the speed of light - but greater than the speed of light compared to what? Every point on the top's edge is perfectly static in relation to everything else in this particular universe, which consist of only this top.

 

The top will experience centripetal force along the plane of rotation, but why? A plastic sphere, similarly isolated in its own personal universe will experience a flattening, and bulging at the sides, but will have no idea of rotating. So, is there a 'matrix' of some sorts, a final, static über-reference frame (very un-Einstein like) to which eveything relate?

 

Granted, this is a pure thought experiment, one which cannot be performed or even approximated, but if there is no über-reference frame, then rotational inertia is also dependent on there being multiple objects to be measured against - which is surely wrong?

 

Any thoughts?

[modest]

It's a very good thought experiment. According to Mach's principle the top will not—cannot—spin. General relativity, in a weak way, supports Mach's principle. It should not experience centrifugal forces.

 

That's my take, in any case :D

 

~modest

[Boerseun]

...so a rotating rubber ball, flattened in the axis of rotation and bulging out along the plane of rotation, will snap back into shape if you remove everything else in the universe without touching the ball and influencing its rotation?

[modest]

...so a rotating rubber ball, flattened in the axis of rotation and bulging out along the plane of rotation, will snap back into shape if you remove everything else in the universe without touching the ball and influencing its rotation?

 

That's the idea. The ball would be rotating in spacetime. Mass tells spacetime what to do and if the only thing informing spacetime were suddenly the ball then spacetime would quickly take on the exact rotation of the ball through frame dragging, so if the only thing in existence were the spinning ball then it should lose its bulging shape.

 

~modest

[Boerseun]

That's exceedingly odd.

[freeztar]

That's exceedingly odd.

 

Indeed, but so is an empty universe with a spinning top. :shrug:

[modest]

Odd indeed.

 

Newton, by the way, used the same argument from the OP. His thought experiment was with a spinning pail of water, but the idea and the conclusion that space had an absolute frame that could be found through rotation was exactly the same. Relativists were, of course, unhappy with that. So, they were very excited that general relativity, which is more Machian, replaced Newtonian mechanics. Einstein was apparently very happy with that aspect of GR too:

Einstein was so satisfied with this manifestation of Mach's principle that he wrote a letter to Mach expressing this:

it... turns out that inertia originates in a kind of interaction between bodies, quite in the sense of your considerations on Newton's pail experiment... If one rotates [a heavy shell of matter] relative to the fixed stars about an axis going through its center, a Coriolis force arises in the interior of the shell; that is, the plane of a Foucault pendulum is dragged around (with a practically unmeasurably small angular velocity).

Mach's principle - Wikipedia, the free encyclopedia

It's a very philosophically deep and interesting thought experiment, notwithstanding the results and the setup are indeed a little odd :shrug:

 

http://www.gap-system.org/~history/HistTopics/Newton_bucket.html

 

~modest

[Jay-qu]

I went to a talk by Donald Lynden-Bell from Cambridge last year and he said:

"only spaces closed by the mass of the energy they contain obey Mach's precepts"

 

That said I remember he also stated from empirical evidence that they could put an upper bound on the rotation speed of the universe as a whole - apparently if the universe is rotating it must be slower than ~1 rotations per Hubble time, for us not to have noticed it. I will go and look for a link to his work because that number may be out by a factor of 2 or 3..

[modest]

I went to a talk by Donald Lynden-Bell from Cambridge last year and he said:

"only spaces closed by the mass of the energy they contain obey Mach's precepts"

 

A closed space, like a closed universe? Interesting. I'd never heard that.

 

That said I remember he also stated from empirical evidence that they could put an upper bound on the rotation speed of the universe as a whole - apparently if the universe is rotating it must be slower than ~1 rotations per Hubble time, for us not to have noticed it. I will go and look for a link to his work because that number may be out by a factor of 2 or 3..

Yeah, wiki for Mach's principle mentions,

The Gödel rotating universe is a solution of the field equations which is designed to disobey Mach's principle in the worst possible way. In this example, the distant stars seem to be rotating faster and faster as one moves further away. This example doesn't completely settle the question, because it has closed timelike curves.

While most papers which report to support Mach's principle with GR based cosmology are based on weak field approximations, Gödel's metric is an exact solution :shrug:

 

Certainly an open question.

 

~modest