Inconsistency in relativity paper

[js1]

I recently decided for some reason to re-read some of the first part, the kinematical part, of Einstein’s initial paper on relativity, “The Electrodynamics of Moving Bodies”, in which he develops the conceptual framework of special relativity and derives the Lorentz transformations. I must have done it more carefully than the first time I looked at it many years ago, because I noticed what seems to be a rather glaring contradiction, not in the derivation of the Lorentz transformations themselves, but rather in Einstein’s interpretation of them, which he gives in section 4, “The Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Clocks.”

In this section he expresses the traditional and well verified interpretations now known as space contraction and time dilation. He first describes the modified length values applicable to the moving frame, where the values of a length unit moving with velocity v along the axis of the length unit, measured in the stationary reference frame, give a contraction according to the standard Lorentz formula.

He then gives the Lorentz transformation for time, and eliminates the position dependence of the expression by substituting x=vt, where v is the relative velocity, and x,t are the values observed in the laboratory reference frame, at rest. He then gives the well-known retardation of √(1-v^2/c^2), which he reduces to the first approximation of ½ (v^2/c^2), a value which as far as I know has been repeatedly verified, and accounts for the change in observed decay rates in unstable nuclei and particles at high velocities. So far, no surprises.

But then he makes an astonishing (to me) remark about “peculiar consequences” of this derivation. He gives an example, which seems to me to differ in no fundamental way from any previous one used in the arguments by which he derived the Lorentz transformations, where two synchronized clocks are at rest some distance from each other. One is then moved at velocity v towards the other until it reaches it. At this point, Einstein gives us a new formula for the time retardation, one different from the Lorentz transformation, one for which he gives no mathematical or logical justification, which is the value of the Lorentz retardation at velocity v, multiplied by the time of transit of the moving clock to the position of measurement at the laboratory clock (!). He seems, for some reason I don’t understand, to have integrated the Lorentz value over the time path, to take from t1 to t2, ∫(v^2/c^2)dt, giving a retardation of ½ t(v^2/c^2), one which I don’t believe is ever observed.

Einstein then extends the example to that of two clocks starting at rest at the same point, and the moving clock traveling by any closed path back to the starting point, then showing a cumulative retardation given by this new formula. He proceeds to give a physical case of a clock stationed at the equator giving such a retardation when compared to a clock at the pole.

But this would mean that a satellite clock in stationary orbit at the equator, if compared to a resting clock at the pole, would show at the launch a value more or less that given by the Lorenz transformation, something on the order of 10^-10, but after a year that retardation would have increased by a factor of 3.15x10^7, the number of seconds in a year, and would increase by a similar amount each year thereafter! If that were the case, satellite clocks would be essentially unusable over much of the earth’s surface, and as far as I know is completely inaccurate.

I am at loss to understand what was Einstein’s thinking in this remark. It makes no sense to me whatsoever. Obviously I must be missing something.

 

[bcrowell]

Can you give a section number and a phrase to search for? There is a copy of the paper here: http://fourmilab.ch/etexts/einstein/specrel/www/

 

[js1]

yes, as I said in the post, it is in section I, part 4, at the end of it. The part I refer to begins with "From this there ensues the following peculiar consequence."

 

[Jonathan Scott]

... But this would mean that a satellite clock in stationary orbit at the equator, if compared to a resting clock at the pole, would show at the launch a value more or less that given by the Lorenz transformation, something on the order of 10^-10, but after a year that retardation would have increased by a factor of 3.15x10^7, the number of seconds in a year, and would increase by a similar amount each year thereafter! If that were the case, satellite clocks would be essentially unusable over much of the earth’s surface, and as far as I know is completely inaccurate.

I am at loss to understand what was Einstein’s thinking in this remark. It makes no sense to me whatsoever. Obviously I must be missing something.

I don't know how you get this. I can't see anything wrong with what Einstein says here. The time difference is linear in t. The longer the interval, the greater the difference.

 

[Frame Dragger]

The good news is that he knew he was missing something, and didn't think he was correcting Einstein. That's a better start than most get. Welcome to PF btw js1!

 

[atyy]

The equation is only correct up to 4th order.

There is an incorrect prediction by Einstein in that paper about clocks at equators and poles. The prediction is logically correct within the framework of special relativity, but physically incorrect because in real life general relativity must be taken into account. Satellite clocks take general relativity into account. In some cases, the special relativity correction is canceled by the general relativity correction (colloquially speaking only, actually GR includes SR in some sense), but I don't remember which ones. DaleSpam had a good link some time ago from some guys in Australia, I think.

See DaleSpam's post #200 https://www.physicsforums.com/showthread.php?p=1921487

 

[yuiop]

But this would mean that a satellite clock in stationary orbit at the equator, if compared to a resting clock at the pole, would show at the launch a value more or less that given by the Lorenz transformation, something on the order of 10^-10, but after a year that retardation would have increased by a factor of 3.15x10^7, the number of seconds in a year, and would increase by a similar amount each year thereafter! If that were the case, satellite clocks would be essentially unusable over much of the earth’s surface, and as far as I know is completely inaccurate.

It is known that GPS satellites that require a high degree of time keeping accuracy to be useful, have a correction factor programmed into their on board clocks that allows for relativistic time dilation effects so that the orbiting clocks effectively tick at the same rate as clocks on the Earth surface at the equator.

This correction factor to first order is essentially:

$$\sqrt{\left(1-\frac{2GM}{Rc^2}\right)} \sqrt{\left(1-\frac{v^2}{c^2}\right)}$$

Now the first factor is gravitational time dilation and for the satellite the orbital radius R is greater than the radius of the Earth and this effectively speeds up the satellite clock rate relative to the surface clock. The second factor is the velocity time dilation factor and the orbital velocity of the satellite is higher than the rotation velocity of the surface of the Earth and this effectively slows down the satellite clock rate relative to the surface clock.

The gravitational effect on the time dilation of the satellite is greater than the velocity time dilation effect and so uncorrected the orbiting satellite clock ticks faster than a surface clock. In practice they apply the relativistic corrections to the satellite clock before launch so that when the satellite clock is sitting alongside the surface reference clock, the satellite clock is ticking slower. When it is launched into orbit, relativistic effects speed up the satellite so that it is running at the same rate as the surface clock.

You raise an interesting question about how clocks at the equator compare to clocks at the pole. It turns out that when you take both the gravitational and velocity time dilation effects into account, the two effects cancel each other out for clocks at sea level anywhere on the Earth. The Earth is not a perfect sphere, but slightly oblate (fatter at the equator) and the additional radius of the equator causes a speed up the equator clocks due to gravitational time dilation that cancels out the slow down due to velocity time dilation. Is this a coincidence? The answer is no. When the Earth was much hotter and its constituent matter could effectively flow and reshape itself. Imagine it started as a perfect sphere, then the rotation would mean clocks at the equator would run slower than clocks at the poles, but the material effectively flows from where clocks run fast to where clocks run slow (which is another way of expressing the law of gravity) so matter flows from the poles to the equator resulting in the oblate shape of the Earth we have today.

When Einstein wrote his 1905 paper he was only considering velocity time dilation and it would only be possible for him to take gravitational time dilation into account after he had formulated General Relativity, which came much later.

 

[js1]

Thanks for the comments. But rather than resolving the issue, they seem to me rather to underscore the problem, which is the two different formulas for time retardation which Einstein presents in his 1905 paper. The first, the Lorentz transformation, is that which is widely used and and tested. It is the second part of the corrective factor given here by kev for the adjustment of satellite clocks. But the second formula, which Einstein claims would be applicable to satellite clocks (by the example he uses) is the Lorentz factor multiplied by t, the time of transit of the satellite in orbit after it was launched, a cumulative effect which so far as I know has never been observed.

The second issue which has been implicitly raised, is the difference in the nature of time alteration in General and Special Relativity, a difference which does not seem to be appreciated here. The first is the effect on the moving clock due to the effects of the gravitational field during its transit, in relation to the stationary clock. It is a physical effect, and is cumulative over the path of transit. The second, at least as it is developed by Einstein initially in the 1905 paper I referenced, is purely the effect of the difference in measurement by a stationary observer of a moving inertial frame due to the conditions specified by Einstein in his paper, namely the equivalence of inertial reference frames and the constant and finite velocity of a light pulse necessary to make the measurements.

The difficulty I raise is that Einstein then contradicts this view and the Lorentz formula for the time measurement variation. He proposes a second formula, one for which he gives no rationale, namely the Lorentz formula multiplied by the time of transit of a moving clock put in motion relative to the first after the motion is started. The only significant difference in this example from those previously used in the paper is the acceleration of the moving clock from rest in the stationary frame to the inertial velocity, v, a difference which seems minor and would be necessary in most tests of the Lorentz formula anyway. And in fact the effect of it appears to have been Shown by Sherwin^1 to be negligible.


1. Sherwin, “Some Recent Experimental Tests of the 'Clock Paradox'”, Phys. Rev. 129 no. 1 (1960), pg 17.

 

[Fredrik]

The second issue which has been implicitly raised, is the difference in the nature of time alteration in General and Special Relativity, a difference which does not seem to be appreciated here. The first is the effect on the moving clock due to the effects of the gravitational field during its transit, in relation to the stationary clock. It is a physical effect, and is cumulative over the path of transit. The second, at least as it is developed by Einstein initially in the 1905 paper I referenced, is purely the effect of the difference in measurement by a stationary observer of a moving inertial frame due to the conditions specified by Einstein in his paper, namely the equivalence of inertial reference frames and the constant and finite velocity of a light pulse necessary to make the measurements.

This isn't a difference between special and general relativity. Both theories say that a clock measures the proper time of the curve in spacetime that represents its motion. (The only difference is that spacetime in SR is always Minkowski spacetime while spacetime in GR can be some other solution of Einstein's equation). You're talking about comparing what a clock measures with the time coordinate assigned by some coordinate system. That's an issue that comes up in both theories, and does so more often in SR only because there's a natural way to associate a global coordinate system with each inertial observer in flat spacetime.

 

[yuiop]

The time dilation factor (dtau/dt) that I gave earlier

$$\frac{d\tau}{dt} = \sqrt{\left(1-\frac{2GM}{Rc^2}\right)\left(1-\frac{v^2}{c^2}\right)}$$

is the GR time dilation outside a non rotating body. When the observer is very far from the gravitational body and R tends towards infinity spacetime becomes flat and the first term inside the square root reduces to unity and the equation reduces to the SR time dilation factor. This is because far from a gravitational body spacetime becomes flat and SR becomes valid. When doing calculations near a gravitational body such as on the surface of the Earth then if you do not use the first term that takes gravity (or the curvature of space) into account then you will get incorrect results and that is why Einstein gets an incorrect result for his comparison of a clock at the North pole to one at the equator. Since he had not yet discovered GR, we should not be too critical of him for making that error. Of course, there are other factors that influence time dilation, but they are very minor compared to the main two factors I mentioned for the case of the Earth and its satellites.

You seem to be asking why the time coordinate Lorentz transformation as described here http://hyperphysics.phy-astr.gsu.edu/Hbase/relativ/ltrans.html#c2 appears different from the time dilation factor desribed here http://hyperphysics.phy-astr.gsu.edu/Hbase/relativ/tdil.html#c2. One is the transformation of the time at a single event, while the other is the transformation of the interval between two events.

 

[JesseM]

Thanks for the comments. But rather than resolving the issue, they seem to me rather to underscore the problem, which is the two different formulas for time retardation which Einstein presents in his 1905 paper. The first, the Lorentz transformation, is that which is widely used and and tested. It is the second part of the corrective factor given here by kev for the adjustment of satellite clocks. But the second formula, which Einstein claims would be applicable to satellite clocks (by the example he uses) is the Lorentz factor multiplied by t, the time of transit of the satellite in orbit after it was launched, a cumulative effect which so far as I know has never been observed.

Einstein was just talking about special relativity time dilation in flat spacetime, and this type of time dilation can be derived directly from the Lorentz transformation. Consider a clock that is at rest in frame S for some period of time, so we pick two events on its worldline with a time separation of T in frame S (which must be the time elapsed on the clock itself between these events), and want to know the time T' between the same pair of events in another frame S' which sees the clock moving at speed v. So, if you place the origin of S at the first of the two events, so that event has coordinates x=0, t=0, then the second event must have coordinates x=0, t=T in S. Using the Lorentz transformation, in frame S' the first event has coordinates x'=0, t'=0, while the second event has coordinates:

x' = gamma*(x - vt) = gamma*(-vT)
t' = gamma*(t - vx/c^2) = gamma*(T)

So, the time interval T' between the two events in the S' frame must be T' = gamma*T, which is just the time dilation equation. Of course this derivation only works for a clock moving inertially (constant velocity = constant speed and direction), but if a clock is moving at constant speed but in a non-constant direction in frame S', you can treat that as the limit case of a bunch of very short constant-velocity segments each lasting dT of proper time, in which case the same argument indicates each segment lasts dT' = gamma*dT in S', and since the speed v is constant in S gamma is constant too, so the total time elapsed in S' between two events on the clock's worldline is still just gamma multiplied by the time T that elapses on the clock itself. This idea of treating a continuously curved path as the limit of a path made up of a bunch of very short constant-velocity segments is what Einstein was talking about at the end of section 4 of the 1905 paper when he said "It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide" and "If we assume that the result proved for a polygonal line is also valid for a continuously curved line ..."