The transformation of the sun into a red giant is approximately 5 billion years from now according to our current understanding of the sun and how it is changing. As for the moon, the theoretical prediction for reaching an orbit that matches the rotation of the Earth would be in about 50 billion years. As for the Earth's rotation, it should indeed be a decaying change in rotation over time due to the moon moving farther away (i.e. the Earth slowed down more when the moon was closer). As far as gravity is concerned, it is inversely proportional to the square of the distance between two bodies. The math for changing tidal forces is more complicated though.

As for the leap second, that has more to do with keeping the atomic clocks in sync with the background stars due to the fact that the Earth doesn't rotate in exactly 24 hours. What this really means is that inserting leap seconds has to do with the current rotational speed of the Earth as opposed to the slowing of the rotation of the Earth, even though that is the technical cause of the leap seconds being inserted. The actual change in Earth's rotation is only about 2 microseconds since 1820. A good explaination can be found here: http://novan.com/earth.htm

But getting back to the original question, what would the rotation rates of the Earth be for the last 4 billion years? I'd guess that the slow-down rate is somewhat proportional to the tidal forces, but that the tidal forces are not linearly related to the change in the Earth-moon distance. Move the moon twice as far away from the Earth and the tidal forces should drop by a factor greater than two. But what's the relationship? 1/radius^2 ? 1/radius^3 ? I suppose the shape of the curve as a function of time is pretty much a decaying exponential, more or less, and that would mean the change in rotation rate was quicker in the past and will asymptotically approach a final value in the future.

24 seconds in 36 years seems like a fast rate of slowing. Are there any estimates for rotation rates of Earth as a function of time? Oh, say maybe over the last billion years or so? I assume the rate is not linear. Also what would cause the slowdown? More dams holding water at higher elevations on the planet (a small effect no doubt)? Energy dissipated by tidal action in the oceans?
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Reply to Cynthia:

Technically speaking, anything that is accelerated by coming near the earth is slowing down Earth's rotation – which means that all of those missions that come back past Earth to do a slingshot to increase their speed are slowing down the planet. On the other side, this effect is negligible based on the masses involved. In theory, there are probably a large number of things that may affect Earth's rotation but again it comes down to a matter of the masses involved and most of them are negligible.

If you wanted to point a finger at the largest culprit however, it would be the moon which is slowly being pushed by the Earth away from us so in a way you are close with the concept of the tidal action in the oceans – dry land also experiences a very minor flexing as well as the oceans but since it is solid it is hard to notice today (according to the collision moon formation theory it would have been more noticeable billions of years ago when the moon was much closer to the Earth). At any rate, the moon's gravity causes a minor bulge in the Earth, which moves slightly ahead of the moon due to the Earth's rotation and this causes a gradual acceleration of the moon. Theoretically, in billions of years, the moon should make its way out of Earth orbit.

Earth's (or any other planet's) gravity has nothing to do with its rotation, only its mass. (and density matters for the force of gravity at its *surface,* but not at a distance).

However, centrifugal force (yes, I know that strictly speaking it's not a 'force') does mean that you'd be very slightly lighter standing at the equator, compared to the poles, and as the planet gradually spins down over tens of billions of years (through tidal interactions with the Moon, that also cause its orbit to enlarge) that difference will fall toward zero…

Jupiter however, its outer parts being made mostly of its low density atmosphere, and turning once in about 11 hours, is stretched into a visibly oblate spheroid.

"What we call mass would seem to be nothing but an appearance, and all inertia to be of electromagnetic origin." — Henri Poincaré, physicist, 1908

"Gravitation is an electromagnetic phenomenon." — Immanuel Velikovsky, cosmologist, 1946

Velikovsky, I., Cosmos Without Gravitation, 1946

For more information about the leap second and the retardation of the rotation of the Earth due to the expansion of the Earth you can see below:

McCarthy, D.D., Precision Time and the Rotation of the Earth, Transits of Venus: New Views of the Solar System and Galaxy, Proceedings IAU Colloquium, Number 196, 2004

McCarthy, D.D., et al., The Physical Basis of the Leap Second, The Astronomical Journal, Issue 5, Nov 2008

Yes, the Earth is slowing down, so according to general relativity, the cesium clock muss be affected as well, Is this taken in account?

Relativistic effects on time are infinitesimally small at the speeds involved. Earth orbits at 0.01% of the speed of light which isn't slow, but the change in speed that results in the a one second adjustment once every six months or so is much much smaller than that.

So, while I am sure that relativity does make a difference, I suspect that for that difference to add up to a second or more, we'd have to wait millions of years.

If the Earth stopped spinning the gravitational effect of the Earth would increase not decrease.

Like putting a cork on a spinning record turntable (which I remember doing when I was a child) makes the cork fly outwards away from it's centre point of rotation.

But seriously, the time problem is nothing compared to the month and years problem.
The current setup is just bad, there are plenty of good alternatives out there.
But of course, changing everything over in this world we live in now would be near impossible…

Oops… just wasted it…

"Then what are you doing on here???

No wonder employee production is down"
coffee break, which actually increases my productivity, and it's my only chance to get the job done…. btw, I'm self employed….

I reckon it's either taken into account, or irrelevant.
Aren't all measurements of "stuff" involving time made RELATIVE to the cesium clock?

Then what are you doing on here???

No wonder employee production is down.

Thanks alot Universe Today!