Shape of Earth

[PatrickPowers]

I have read that at sea level gravity is the same everywhere on the surface of the Earth. I have also read that motionless clocks at sea level will stay synchronized.

It seems to me that this should be a consequence of general relativity. If one instead uses Newtonian gravitation then gravity would be less at the equator because both the rotation of the Earth and the shape of the Earth as an oblate spheroid would decrease gravity.

Is there any easy explanation of the shape of the Earth due to GR?

 

[Q-reeus]

...Is there any easy explanation of the shape of the Earth due to GR?

Not about the shape of the Earth due to GR, but how the shape of the Earth does according to GR give almost exact clock synch:
http://www.mathpages.com/home/kmath182/kmath182.htm - starts near half-way down the page.

 

[PeterDonis]

I have read that at sea level gravity is the same everywhere on the surface of the Earth. I have also read that motionless clocks at sea level will stay synchronized.

The second sentence is true (at least to a very good approximation; see below for some further comments). The first is not, at least not if by "gravity is the same" you mean the "acceleration due to gravity" is the same. The acceleration due to gravity is slightly smaller at the Earth's equator (on the surface) than at the poles.

The difference is due to the fact that, speaking somewhat loosely, the clock rate depends on the potential, and the acceleration depends on the spatial gradient of the potential. The Earth's surface is an equipotential surface--again, at least to a very good approximation. If the Earth were a perfect fluid, its surface would be a perfect equipotential surface. But the spatial gradient of the potential is not constant over its surface, because of rotation. See below.

http://arxiv.org/abs/gr-qc/0411060It [Broken] seems to me that this should be a consequence of general relativity. If one instead uses Newtonian gravitation then gravity would be less at the equator because both the rotation of the Earth and the shape of the Earth as an oblate spheroid would decrease gravity.[/QUOTE]

This is true in GR as well, as the page Q-reeus linked to makes clear. As I noted above, the acceleration due to gravity is slightly smaller at the equator than at the poles, basically for the reasons you give. That page also gives Newton's reasoning for why the surface of the Earth is an equipotential surface, at least to a very good approximation: if the Earth's surface were not an equipotential surface, then the oceans would slosh around to fill up the lower places (i.e., lower potential) from the higher places (i.e., higher potential) to *make* the surface equipotential. The fact that the Earth's surface is not entirely ocean means that it can't adjust itself completely in this way; but over long enough time scales, even the solid crust of the Earth is not completely rigid, so it too will adjust itself to make the surface equipotential to a very good approximation.

This reasoning is still valid in GR; but GR gives us an additional tool, the spacetime metric, with which to specify more precisely exactly what the shape of the equipotential surface will look like. The problem is, as the page Q-reeus linked to notes, we don't have an exact solution for the metric of a rotating oblate spheroid. However, we can make a very good approximation as follows:

(1) The Earth's radius is much, much larger than its mass in geometric units (about a million times larger). (The "mass in geometric units" is GM / c^2, where M is the mass in conventional units--i.e., it is half the Schwarzschild radius of a black hole with the same mass.) This implies that the metric at the Earth's surface is very close to flat; i.e., the metric is the Minkowski metric plus small correction terms.

(2) Also, the Earth is rotating very slowly (i.e., the speed of a point on its surface is much much less than the speed of light). This implies that the metric is very close to the Schwarzschild metric, with small correction terms for the rotational velocity.

(3) The key metric coefficient we're interested in is $g_{tt}$, since that's the one that governs the "rate of time flow"; also, to a very good approximation, its spatial gradient is what governs the "acceleration due to gravity". (In other words, $g_{tt}$ is basically the "potential" I referred to above.)

Given all the above, we can write:

$$\frac{d \tau}{dt} = \sqrt{g_{tt}} = \sqrt{1 - \frac{2M}{r} - v^{2}}$$

where I have used units where G = c = 1. M is the mass of the Earth, r is the radius at a given point on the surface, and v is the velocity of that point due to the Earth's rotation.

On the equator, we can write $v = \omega r$, where $\omega$ is the rotational frequency of the Earth, and since the metric is very close to flat (all the terms under the square root other than 1 are << 1), we can expand out the square root to obtain, to first order, the equation near the end of the mathpages page Q-reeus linked to:

$$\frac{d \tau}{dt} = 1 - \frac{M}{r} - \frac{1}{2} \omega^{2} r^{2}$$

Taking the derivative of this with respect to r, as the mathpages page notes, gives the net acceleration due to gravity, and you can use these formulas to run the numbers and see that (a) the potential $d \tau / dt$ is the same, to a very good approximation, at the equator and at the poles, but (b) the acceleration due to gravity is slightly smaller at the equator than at the poles.

 

[D H]

I have read that at sea level gravity is the same everywhere on the surface of the Earth. I have also read that motionless clocks at sea level will stay synchronized.

The second sentence is true (at least to a very good approximation; see below for some further comments). The first is not, at least not if by "gravity is the same" you mean the "acceleration due to gravity" is the same. The acceleration due to gravity is slightly smaller at the Earth's equator (on the surface) than at the poles.

Patrick's first statement is incorrect, as you correctly noted. Patrick's second statement is exactly correct.

It seems to me that this should be a consequence of general relativity. If one instead uses Newtonian gravitation then gravity would be less at the equator because both the rotation of the Earth and the shape of the Earth as an oblate spheroid would decrease gravity.

The gravitational self-attraction of a planet tends to make the planet's surface follow that of an equipotential of gravitation plus centrifugal force. Newton himself derived this, so general relativity is not needed to explain the shape of the Earth. The observed gravitational acceleration (which includes gravity plus centrifugal forces) is the gradient of this potential.

Is there any easy explanation of the shape of the Earth due to GR?

Not really. Look at Peter's derivation. He used a spherical Earth, which is only approximately valid even in Newtonian mechanics. Going beyond a spherical mass distribution is a bit tough in GR. Going to a full-blown spherical harmonics model, which is what is needed to explain the real shape of the Earth (the geoid) is tougher still with general relativity. This is one of those case where it is better to use Newtonian gravity.

Where general relativity does come into play is in explaining why all clocks at sea level tick at the same rate.

 

[PeterDonis]

Patrick's first statement is incorrect, as you correctly noted. Patrick's second statement is exactly correct.

Do you say "exactly" because "sea level" is *defined* as being the equipotential surface? I.e., "sea level" means "geoid" in the sense given here:

http://en.wikipedia.org/wiki/Geoid

The reason I qualified my statement was that the geoid is not exactly the same as the actual "sea level" we observe.

Look at Peter's derivation. He used a spherical Earth, which is only approximately valid even in Newtonian mechanics. Going beyond a spherical mass distribution is a bit tough in GR. Going to a full-blown spherical harmonics model, which is what is needed to explain the real shape of the Earth (the geoid) is tougher still with general relativity. This is one of those case where it is better to use Newtonian gravity.

Actually, I didn't mean to imply that I was assuming a spherical Earth, only that I was approximating the actual metric using the spherically symmetric Schwarzschild metric plus an extra correction for velocity. The Earth's surface in the coordinates I was using has a different r at the equator than at the poles, but the change this induces in the 2M/r term in g_tt, to the level of approximation I was using, is exactly compensated for by the change in the v^2 term (v = 0 at the poles but v > 0 everywhere else, by just the right amount to compensate for the increase in r). For this to work, r *has* to change as v changes, so the Earth has to be oblate. But I agree that I haven't derived all this rigorously from a solution to the EFE (as far as I know no such exact solution is known); I've simply handwaved an approximation that works reasonably well.

 

[D H]

Do you say "exactly" because "sea level" is *defined* as being the equipotential surface? I.e., "sea level" means "geoid" in the sense given here:

http://en.wikipedia.org/wiki/Geoid

The reason I qualified my statement was that the geoid is not exactly the same as the actual "sea level" we observe.

To be pedantically correct, I should have said "All clocks on the geoid tick at the same rate". You are right in that mean sea level is not quite the same as the geoid. It's very close, and it would be the same as the geoid if there weren't any continents in the way, if there weren't any ocean currents and tides, etc. The Atlantic is on average about 0.4 meters lower than the Pacific, and sea level at the Bahamas is about a meter higher than it is in New York. (The Gulf Stream, just like any other river, flows downhill.)

 

[YummyFur]

The GOCE project may have some relevance.