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We all know from General Relativity theory that movement of objects around, for example, a star actually straight in spacetime continuum. It just time distortion causes the movement to curve.

Considering this, I have the following questions:

  1. Do planets experience coriolis effect in terms of rotation around a star?
  2. And do I correctly understand that space distortion happens only in time dimension?
Andrew Shaban
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    It is not just time distortion that causes the object to move in a curve. It is certainly true that is is mostly the curvature associated with the time coordinate because the time component of the 4 velocity is usually the largest, but all the curvature terms (the Christoffel symbols) affect the motion. So your question is based on a false premise. For more on this see Why does mass bend the temporal dimension more than the spatial dimensions of spacetime?. – John Rennie Mar 22 '23 at 15:42
  • @JohnRennie Your comment addresses a different issue - but it made me think - is there a tensor-based derivation of the Coriolis pseudforce, or is it as simple as substracting the frame motion into the centrifugal term? – AtmosphericPrisonEscape Mar 22 '23 at 16:21
  • There is a possible way to think about this, which may help: if the orbit is near circular, you can imagine the planet being attached to a rotating disc which just happens to rotate at the same speed. Clearly, there are Coriolis effects on this disc. Now, since our planet is supposedly not a point mass but has some area along the plane of the orbit, the Coriolis effect of our entire imaginary disc will extend to it as well. – Amit Mar 22 '23 at 17:50

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Whether the force is Coriolis or centrifugal, or if the planet accelerates radially is frame dependent. For mass $m$ planet revolving at speed $v$ in a radius of $R$:

In an inertial frame, there are no fictitious forces, and the centripetal force caused by gravity is:

$$ F_g = \frac{mv^2} R $$

which keeps the planet in orbit

In a rotating frame ($\omega=v/R$) in which the planet is fixed, there is a centrifugal force:

$$ F_{cent} = {m\omega^2}R = {m(v/R)^2}R = \frac{mv^2}R = F_g$$

Since the planet is stationary, there is no Coriolis force. The total fictitious force balances gravity, and the planet remains fixed.

In a frame moving at $\omega'=\omega/2$, there is a centrifugal force:

$$ F'_{cent} = \frac{m\omega'^2} R = \frac{mv^2}{4R} $$

In this frame, the planet's velocity is:

$$ v' = v - R\omega' = v-v/2 = v/2 $$

The Coriolis force is:

$$ F'_{cor} = 2m\omega' v' = 2m(v'/R)v' = \frac{mv^2}{2R}$$

which is in the anti-radial direction (see Eötvös effect, in while the motion is manifest as a change local gravity).

The total fictitious force is:

$$ F'_{fact} = F'_{cor} + F'_{cent} = \frac 3 4 F_g $$

Hence,the planet experiences a radial force:

$$ \frac 1 4 F_g = \frac{mv'^2} {R} $$

which keeps it on its orbit.

The latter analysis holds for any $\omega'$.

JEB
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