Question about rotating frames

[dand5]

In flat spacetime, if you are at rest in a rotating frame, how would you be able to detect that rotation? For instance, on a small planet surrounded by nothing, how can a reference frame that rotates at the angular velocity of the planet be declared noninertial?

Thanks for any replies.

 

[pervect]

Probably the easiest thing to do is look for coriolis forces. Using the Earth's rotation as a reference, an object moving straight up at a velocity v will experience an acceleration east-west of 2 w v, where w is the angular velocity of rotation of the Earth.

The same thing will happen on your planet - an object moving upward will experinence no transverse acceleration if and only if the planet is not rotating.

 

[daniel_i_l]

For one thing you would feel centrafugal force because of the spin.

 

[dand5]

Thanks. My real confusion was over how a rotating frame cannot be declared inertial even in the absense of any other objects to compare the motion to. But, it's pretty clear just from looking at a spacetime diagram in flat space that you are accelerating.

 

[Garth]

Thanks. My real confusion was over how a rotating frame cannot be declared inertial even in the absense of any other objects to compare the motion to. But, it's pretty clear just from looking at a spacetime diagram in flat space that you are accelerating.

It is only a thought experiment - but consider your tiny rotating planet in an otherwise empty universe. The question of whether there is an absolute inertial frame of reference by which it can be deemed rotating, with the corresponding corriolis and centripetal forces, is an open one. For if the planet is rotating what is it rotating with respect to?

The problem with comparing with the flat Minkoski space-time is that after GR we know that space-time must be completely empty of matter and energy. If it were not then there would be gravitation and some space-time curvature. This is the question around Mach's Prinicple that has not been resolved yet although it might be in a year or so when the results of the Gravity Probe B satellite experiment are published.

In GR the whole universe may be rotating (although observational constrains that possibility to less than a very small angular rate) However a Machian response would be to ask, "What is it rotating with respect to?"

Garth

 

[pervect]

The only solution in GR that I'm aware of for a rotating universe is the Godel universe. This has a number of undesirable features, it's not globally hyperbolic for one thing (i.e. it has closed time-like curves, or time-travel).

I don't think it's possible to have a rotating universe within GR without these undesirable and unphysical behaviors, though I could be wrong.

 

[dand5]

For if the planet is rotating what is it rotating with respect to?

That is what confuses me. But even in a curved space (for instance, the schwarszchild geometry), coriolis forces don't appear unless you boost
to a reference frame that is rotating (of course that also changes the geometry, but I didn't consider it yet because the math gets messy). So the
fact that they are observable suggests that some aspect of the rest of the universe creates them or that the rotation is absolute. Something seems wrong with the above reasoning, but I'm not quite sure what it is.
Also,
do shear stresses in the velocity along the radial direction, which can
be detected in a lone planet contribute at all to spacetime curvature in a way that makes it impossible to eliminate the rotation of the outer planet shell?
Thanks

 

[masudr]

Easy to know if your planet is rotating. Release 2 particles of different inertial mass and observe what happens.