Mistake in column about relativity and aging?

Cecil's Columns/Staff Reports
1

I was just reading a classic column on the website:

http://www.straightdope.com/classics/a4_183.html

The question is whether people at the equator, being whirled about at 1000 miles an hour, age more quickly than those at the pole, who are relatively stationary. Cecil’s response was that the general relativity effect of being higher up in a gravitational field would precisely cancel the special relativity effect of the circular motion.

This is pretty clearly wrong.

Why? Because the two effects are totally uncorrelated: if the Earth were to spin around in 23 hours instead of 24 (which it did, earlier in its history) the accelerated aging of special relativity would be increased, with no corresponding increase in the general relativity slowing effect. If, as Cecil claims, these 2 effects are equal and opposite, what we have is nothing short of a miraculous coincidence – one which will not persist for long, as the Moon’s tidal drag continues to slow the Earth’s rotation. Only if the two quantities can be shown to be algebraically equal, and not merely written off as numerically equal, does this answer make sense.

Any chance of this being updated or corrected?

2

Some portion of the shape of the Earth is due to its rotation. If the Earth rotated in 23 hours instead of 24, then there would be a larger equatorial bulge. People at the Equator would be moving faster and be farther from teh center of the Earth.

So the effects are correlated. I don’t know if they cancel exactly. Got any calculations handy? <grin>.

3

You bet. Here ya go:

This is not so clearly wrong.

4

Jon: The Earth and Moon exert a drag on one another because of tidal forces. This drag slows their rotations and causes the Moon to spiral slowly outward in its orbit – and it’s been going on for so long that the Moon is now tidally locked to the Earth, showing us the same face at all times. Point is, when the Earth and Moon were formed, the Earth WAS spinning faster, in as little as 20 hours, and as the Earth cooled and hardened, it was that larger bulge that was frozen in place. I’m not sure the Earth is flexible enough for the size of the central bulge to reduce as its rotation slows, and unless that bulge is variable, the 2 effects cannot be correlated.

As for calculations:

For small velocities relative to the speed of light, the general relativity time dilation is given by:

t/t0 = 1/(1 + gh/c^2)
(Reference: http://www.phys.virginia.edu/classes/252/general_relativity.html)

Using Encyclopedia Brittanica’s numbers for the size and mass of the Earth:

Radius of Earth at equator: 6378136 m
Effective gravity at equator (including centripetal acc.) = 9.7641 m/s^2

Radius of Earth at poles: 6357136 m
Effective gravity at poles: 9.863 m/s^2

Speed of light © = 300,000 m/s

(Reference: http://britannica.com/bcom/eb/article/4/0,5716,108974+10+106190,00.html)

Given a proper choice of contour integral, the expression for general relativistic dilation gives:

t/t0 = 1/(1 + g(equator)h(equator)/c^2 - g(pole)h(pole)/c^2)

Plugging in the values above gives t/t0 = 1.0000047, which amounts to a total difference, over a 75-year lifetime, of 3 hours, 5 minutes, and 23 seconds – that’s how much older you’d be because of living higher up in Earth’s gravitational field.

For special relativity:

t/t0 = 1/Sqrt(1 - v^2/c^2)

Plugging in the value for v at the equator gives t/t0 = 1.0000012, which is 47 minutes 20 seconds over 75 years – that’s how much younger you’d be because of all the spinning the equator does.

So the net effect looks to be an increase in aging of 2 hours, 18 minutes, and 3 seconds for living at the equator vs. living at the poles. So my advice would be to live at high latitudes, you’ll live longer – but not much =)

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The column (including Slug Signorino’s illustration) can also be found on pages 183-184 of Cecil Adams’ book “The Straight Dope Tells All”.

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PaulT: “Speed of light © = 300,000 m/s”

I don’t know if this carried through into all of your calculations, but c is much closer to 300,000,000 m/s. That SR dilation certainly seems a tad high, since I got a gamma of 1.0000000000012. I didn’t do the rest, because I’m just not that familiar with GR.

7

That’s what I get for using constants out of my head – that’s the speed of light in km/s. Nevetheless, that doesn’t change the fact that they don’t cancel out, and certainly doesn’t undermine my logical argument against Cecil’s answer.

Can’t seem to edit my earlier post, so here are the correct numbers:

The general relativity result is only one million times too large: the actual number should be around 1.11 x 10 ^ -2 seconds, a much more believable answer.

The special relativity result is exactly as Achernar stated, gamma = 1.0000000000012, so over 75 years, special relativity effects would contribute 2.85 x 10 ^ -3 seconds – still a smaller number than the general relativity effect, so the net effect is still negative – people at the equator age slightly slower than those at the poles.

(By the way, I have reconsidered my earlier advice about moving to high latitude – and I would suggest that if you’re really serious about adding valuable seconds to your life, you should move to Mercury instead – it may not spin as quickly as the Earth, but it’s much lower inside the Sun’s massive gravitational field, and it orbits faster to boot!)

8

One correction to my last post:

I said:

Of course, I meant to say they age more quickly than those at the poles – the general relativity effect of being further out overwhelms the special relativity effect of being faster.

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[quote]
the Moon is now tidally locked to the Earth … as the Earth cooled and hardened, it was that larger bulge that was frozen in place.

[quote]

The Moon is tidally locked, the Earth is not. The Earth may have some frozen bulge, but it’s certainly not the only source of bulge. Since the Earth is not tidally locked, the position of the tidal bulge varies over time.

Maybe picking nits, but of course the Earth is flexible enough. Deflection happens under the action of any force, no matter how small. The question is whether the effect is significant or even noticable.

The sources I’ve found all claim that the “centrifugal force” effect is the major contributor to the bulge, but I’m unable to find a proof.

As to your calculations, the input numbers (except for c!) appear to be about right. The GR time dilation equation I can’t check (the link is borken, and I can’t seem to find an equivalent link). I’m not convinced you’ve accounted for all the effects. There’s only a factor of four difference between your final calculations of the two effects; an error or invalid approximation could wipe that factor out.

But I’m quite willing to believe the two effects don’t cancel each other out; my disagreement is with your claim that there’s no correlation between the Earth’s rotation and the size of the effects. All my sources indicate that the difference between equatorial radius and polar radius is mostly due to the current rotation of the Earth, and I just don’t buy your “tidal lock” argument.

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Your points are well taken, Jon.

I never meant to say that the Moon’s tides were the cause of the equatorial bulge – I said that the tides were causing the slowing of the Earth’s rotation, which slowing would throw off Cecil’s precise counteraction between the two effects. If we disagree on the cause of the Earth’s present bulge, I can understand why you find this argument unconvincing – however, I would direct you to the Encyclopedia Brittanica article above, which points out that the bulge is larger than we would expect from hydrostatic fluid models. If we accept that the Earth was hotter (and thus more easily deformed) in its early days, when it was also spinning faster, then there’s no reason that some of the extra bulge might not be a remnant of that dizzier time. And if even the TINIEST bit of the bulge is a remnant, then Cecil’s argument won’t hang together – the bulge remnant would introduce an offset, and lead us back to the same problem, where an amazing coincidence is required in order for these 2 effects to cancel out.

Consider this: what if the Earth had a slightly different density, like the other plantes do? The gravitational field would change, and the general relativity effect would change with it, but the radius could remain the same, and the special relativity effect would remain the same. At some level, these are independent quantities, and I can’t believe that they just HAPPEN to come out equal and opposite, when they could easily have been different.

11

New physics grads, you can’t tell 'em anything. :slight_smile:

If you’re going to insist that everything come out exactly to the last decimal point, we’re never going to get anywhere.

How about if I just explain away that factor of four, to first approximation?

First, (and the most bedeviling part), was your figures for effective gravity. My textbooks, and that Britannica site, seem to give different values (Br1,Br2,Br3,Br4) than yours:

Gravity at equator (including centripetal acc.) = 9.7803 instead of 9.7641 m/s^2
Gravity at poles: 9.832 instead of 9.863 m/s^2

That takes care of just about half of the factor of four.

The other half is hidden in your “proper choice of contour integral.” That is the proper result for gravitational attraction, but it is not for centripetal force, which you included in your equatorial gravity figure. For that, it comes out to half, also. This makes a difference in the overall calculation because the centripetal force dominates the gravity difference between the equator and the pole.

So, they’re essentially equal, as Cecil said.

Cute problem, thanks. Welcome to the Straight Dope.

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Well, stranger things have happened. It would indeed be a coincidence if they happen to be equal and opposite. But I submit that they are not independent.

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You’re not addressing the real problem, though, which is that the general and special relativity effects are not dependent on the same characteristics of the Earth. If the Earth were the same size, but had half the mass it has, the general relativity effect would be drastically lower, but the special relativity effect would remain the same. So if Cecil’s answer IS true, and the effects precisely cancel each other, then that’s only true for a planet with exactly our planet’s charracteristics, and has no bearing on whether inhabitants of a planet IN GENERAL would age slower or faster at the poles.

I recognize that this was not the question that was literally asked (but then again, the question didn’t take gen rel into account at all, so Cecil was already generalizing).

RM, your criticism of my calculation may well be correct (you’ll notice that my original point did not depend on numerical calculation, but on the implausibility of 2 unrelated effects HAPPENING to cancel each other out).

I approximated g at the poles and equator using Brittanica’s values for the product of G and the mass of the Earth, and the radii given for the equator and poles, and factoring in the centripetal acceleration at the equator. I think I simply mis-multiplied the product – not surprising, since all I have to work with here is an adding machine =) Another source of error would have been the difference between the gravitational field of a sphere and that of a geoid, which would have been much more difficult to figure. As for the contour integral, I must admit that you are probably correct – I had assumed that a path could be found that would allow me to use the numerical averages, but that’s only true if the functions are linear, and as you pointed out, centripetal acceleration is a cosine function. So path choice would be more difficult than I imagined.

But that doesn’t answer my question about density – is this really a case of numerical coincidence? Usually when two things coincide, there’s a theoretical reason for it, like when Maxwell found out that his EM waves moved at exactly the measured speed of light – can there be a consistent theoretical explanation for these 2 effects cancelling (if in fact they do), or are we doomed to accept it as a chance occurrence?

14

The earth does adjust fairly quickly to the slowing down. If it didn’t, the earth’s mantle would be too viscous to allow mantle convection. That was actually one of the strongest arguments against plate tectonics–until Goldreich and Toomre showed that the excess bulge was not an artifact of the slowing of the earth.

However, when we talk about the surface of the earth, we generally refer to sea-level. In order to make the comparison valid, we want to have both the polar observer and the equatorial observer at the same height above sea-level. Water, of course, is much more fluid than the earth’s mantle. It flows very quickly. Its surface is essentially an equipotential surface.

So, your polar time traveler and your equatorial time traveler are pretty much at the same gravitational/“centrifugal force” potential. That is one of the reasons that things “HAPPEN” to cancel.

15

I guess that does explain it. I had done a search of the Britannica site, trying to find your values. You can find mine on that second link Br2, in equation (14). Phi corresponds to latitude. It’s pretty easy to evaluate, since the sine values are zero for the equator, and one for the poles. A gal is 10^-2 m/s^2, so-named in honor of Galileo.

16

Wow, RM, good point about sea level. But let’s suppose that we do put our observers at sea level, one on a ship at the equator and one at the North Pole, on the pack ice. Now you’re right that that puts them at equal potentials, as far as total fictitious forces are concerned (since gravity, in general relativity, is just as ‘fictitious’ a force as centrifugal force, since both can be removed by proper coordinate choices). So if they’re at equal potentials, why would there be any general relativity effect at all?

And as for special relativity, we may be able to eliminate that too: special relativity allows conversion between inertial reference frames, but clearly NEITHER the North Pole nor the equator is an inertial reference frame, since objects at rest there do NOT remain at rest – they fall down.

So is it possible that Cecil’s answer is right, even though his reasoning is wrong?

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Naw. His reasoning is ok (to be honest, I had the same first reaction that you did), it’s just that he doesn’t go into the nasty details.

Another thing, he likes to vet his answers on sci.physics. Ask Marilyn should be so wise.

18

Some years ago, I heard about a group of elderly people, some of them purportedly close to 130 years old, living in some mountainous area of what was then the USSR (sorry, can’t remember more details). My theory at that time was that since they were further away from the center of the earth (because of being high in the mountains), they were spinning faster with the earth than those people on lower ground and my simplistic understanding of the theory of relativity is that as one approaches the speed of light, time slows. Is this pretty much the same thing that is being discussed here, and how far off base am I with this? I never could get anyone to seriously listen to me before (could be because my majors were psychology and music).

19

Spider Woman, I’m pleased that a non-Scientist would think of something like this. Congratulations. :slight_smile: Having said that, No. There is an enormous difference in degree. You could spend 80 years at Mach 3 (much faster than anyone on the Earth is going) and you’d live to be about 1 second longer. Anything noticable, like 50 years, is absolutely out of the question.

You’re right in recognizing the similarity - it is more or less the same thing that people in this thread are talking about. But what they’re talking about is certainly no way to live longer - it’s more of a curiosity than anything.

20

Kind reply, didn’t make me feel like an idiot.