Would bending of spacetime make tides an invisible effect? [closed]

Question

Similar to this question: How does general relativity explain tides?

But I’m specifically interested in if General Theory of Relativity would predict that bending spacetime means the water and land on earth bend together as the below authors suggest.

“According to the General Theory of Relativity, mass do not interact gravitational [sic] with each other by the gravity force, but because they bend the space. If, according to the General Theory of Relativity, gravity manifests itself as a curvature of space, then it is difficult to explain tides and outflows. At tides and outflows, the differences in water levels reach several meters. It follows that the changes in curvature of space, postulated by GTR, caused by variable gravitation of the Moon are very large. The problem is why tides and outflows are very clear, while the deformations of mainland are invisible. If the mass of Moon bends the space, it is the same when that space is filled with water and when it is filled with the mainland. Therefore, sea water should be deformed in the same way as the mainland. The banks should rise in the same way as the water. Then, of course, tides and outflows would be invisible. However, because they are visible, it follows that the mass of Moon does not distort space, but rather the matter. The water is malleable, and therefore it deforms more than stiff rocks. This proves that the assumption of GTR, that gravity is a curvature of space, is incorrect.”

(PDF) Gravitational Waves in Newton’s Gravitation and Criticism of Gravitational Waves Resulting from the General Theory of Relativity (LIGO). Available from: https://www.researchgate.net/publication/326034030_Gravitational_Waves_in_Newton's_Gravitation_and_Criticism_of_Gravitational_Waves_Resulting_from_the_General_Theory_of_Relativity_LIGO [accessed Feb 20 2023].

What, if anything, is wrong with this assessment? Would the bending of spacetime make the tides invisible?

Answer 1

If the mass of Moon bends the space, it is the same when that space is filled with water and when it is filled with the mainland. Therefore, sea water should be deformed in the same way as the mainland. The banks should rise in the same way as the water. Then, of course, tides and outflows would be invisible.

The "therefore" is incorrect. In the language of Newtonian gravity, tides happen because of a differential gravitational pull between different places on Earth and its center of mass acting as a forcing term for the system of all the bodies of water, whose response function is rather complex. See this excellent SE answer or Theory of tides on Wikipedia.

The different response accounts for the fact that bodies of water move by much more compared to land.

The thing that differs between Newtonian gravity and general relativity is the way the forcing term is handled; in this low-velocity, low-density regime their predictions are almost identical, although the language is different.

In this regime, the Newtonian language is to consider any given particle (molecule of water, say) to have an acceleration due to gravity:

$$ \frac{\text{d}^2 x^i}{\text{d}t^2} = F^i_{\text{grav}} + F^i_{\text{other}} $$

where the $F_{\text{other}}$ term accounts for all non-gravitational forces.

In the language of GR, the main equation is the geodesic one:

$$ \frac{\text{d}^2 x^\mu}{\text{d}s^2} + \Gamma^\mu_{\nu \rho} \frac{\text{d} x^\nu}{\text{d}s} \frac{\text{d} x^\rho}{\text{d}s} = F^\mu_{\text{other}} $$

which in this case reduces to

$$ \frac{\text{d}^2 x^i}{\text{d}t^2} \approx - \Gamma^i_{00} + F^i_{\text{other}} $$

and in particular, again in this approximation, $- \Gamma^i_{00}$ is quite close to the Newtonian expression for the gravitational force $F^i_{\text{grav}}$.

Answer 2

The mistake in this analysis is that water can flow, while rock cannot on a time scale of hours. Thus, the tide is different for water and land. The oceans don't simply deform: they also flow. This makes the ocean tide larger than the body tide, and also shifts it in phase, making it even more apparent.