# Can you explain tides in our water bodies with respect to Theory of General Relativity and spacetime curvature?

We read everywhere that Moon's "gravitational pull" causes tidal waves. Is there any explanation of this "gravitational pull" in terms of General Relativity and curvature of spacetime? And how does this work? Or Newtonian gravity is the only way to explain this phenomenon?

• Newtonian gravity can be derived as a special case of general relativity. So anything understood from Newtonian gravity follows from that. Are you looking for a specific derivation? Lots of questions and answers on that. Dec 27, 2020 at 18:43
• To be exact, the Moon's gravitational pull causes tides. Tidal waves or "tsunamis" are caused by undersea landslides, earthquakes, or volcanic eruptions. Dec 27, 2020 at 19:05

Either set of tools can be used to get an accurate picture of tidal dynamics. Here is the key point in either case:

If you trace out the gravitational field lines emanating from a compact, massive object, you will find that the field lines diverge as you move away from the object and converge as you get closer to it.

Now imagine that you are distant from that object, and you spread your arms out and drop a golf ball from each hand towards the massive object. Each ball will follow its own field line as it falls, and since those lines converge on the center of mass of the massive object, the golf balls will move toward each other as they get closer to the object.

Imagine further that you now drop another pair of golf balls from your hands, but this time you turn sideways so the two balls will follow the same field line, but one starts closer to the object and the other starts farther from it. Both balls will fall toward the object, but the one that started closer to it will experience a stronger pull and the one behind it a lesser pull. This means that as they approach the object, the distance between them will increase.

If you had an arrangement by which you could drop all four of these balls at the same instant, you would then observe the side-by-side balls moving towards each other and the fore-and-aft balls moving away from each other, as they got progressively closer to the object.

The shift in the positions of the falling balls is what is meant by tidal effects and to an observer, it looks as if there is a force stretching the fore-and-aft balls apart and squeezing the side-by-side balls together. And if you yourself were falling instead of the balls, you would feel that stretch and squeeze.

Since both general relativity and newtonian gravity feature field lines that possess divergence in three-dimensional space, both will accurately predict tidal effects as long as the field strengths are not large and the speeds with which the objects fall are not really big. In the cases of huge masses and high speeds, newtonian gravity will yield the wrong answers and general relativity will give the right ones.

Tidal forces find a very natural expression in the language of spacetime curvature, via the concept of geodesic deviation. Here's how:

First, we need the concept of a geodesic. This is the equivalent of a "straight line" on a plane; effectively, it is a path in your space (or your spacetime) that is "locally straight". An example would be a great-circle path on a sphere. One of the fundamental postulates of general relativity is that freely falling objects follow geodesics in a curved spacetime.

Now, an important property of curved spaces (or curved spacetimes) is that nearby geodesics may converge or diverge. Suppose I'm on a flat plane with a friend, who is standing the COVID-required 2 meters away from me. At a given time, we both start off walking in the same direction, and we walk at the same speed along a geodesic (a straight line) thereafter. As we walk, we will continue to be exactly 2 meters apart at all times.

But now let's suppose that we go to the equator of the Earth, each get in identical planes that are 2 km apart, and we each take off heading due North. Even if we start off going in the same direction, and even if we travel at the same speed, the distance between us will change with time. We're both travelling along lines of longitude, but all lines of longitude meet at the North Pole. So as we travel north, the distance between us will decrease from 2 km to (eventually) 0. The fact that we both travelled along straight lines and yet didn't remain the same distance apart is due to the fact that the Earth is curved.

The analogue of this phenomenon in curved spacetime is tidal effects. Suppose my friend and I are now in space, freely falling towards the Earth. We both follow our own geodesics through the curved spacetime around the Earth. But because the spacetime is curved, the distance between our "nearby" geodesics will change. Depending on where my friend is relative to me and the Earth, it will look like there is a force pushing them towards me or pulling them away from me as we both freely fall. In other words, I see my friend undergo a "tidal acceleration" relative to me. If we replace my friend & me with the Earth and its oceans, and think of both of them as freely falling towards the Moon (or the Sun), we would say that the oceans are experiencing tidal forces that pull them away from the Earth at some points and push them towards the Earth at other points.

Of course, my friend isn't really experiencing any forces, any more than they were actively steering their plane towards me back when we did that experiment. It is the curvature of the spacetime that causes the distance between us to change as we both freely fall.

It's non-Relativistic, but I suggest you read about Amphidromic points. https://en.wikipedia.org/wiki/Amphidromic_point . It deals with the water-sloshing-around stuff, not gravitational gradient stuff.

As someone who lives on the shore a few miles away from truly impressive tidal events https://www.youtube.com/watch?v=QEfUblSDzww , its hard to believe tides were caused by the Moon. OK. not true. I believe it's caused by the Moon.

The High school physics explanation of diurnal tides is so simplistic, it is useless.