7. Getting in Synch

25 September 2013

The constancy of the speed of light imposes certain characteristics on objects moving inertially through another frame, as seen from that other frame:

  1. Time dilation
  2. Mass increase
  3. Lorentz-Fitzgerald contraction
  4. Relativity of simultaneity (RoS)

Particle accelerators have demonstrated time dilation and mass increase, but contraction and relativity of simultaneity, which by their nature aren’t easy to show, are necessary as complements to the first two effects in inertial frames.

Because a rotating frame is not inertial, and because the speed of light is neither constant nor a limit in a rotating frame, the contraction and RoS may not be required. RoS is the easiest to determine, so we’ll start with it.

Here’s Einstein’s train again, Lorentz-Fitzgerald contracted and with time-dilated clocks, seen as it passes the observer:

The clocks are synchronized in the train frame, but show RoS to the observer watching it go by, with the locomotive clock earlier than the caboose clock.

We now put a train on a circular track and make it long enough for the locomotive to come up behind the caboose. Once we get it running at relativistic speed, conditions on board are equivalent to those at the rim of a rotating disk.

The crew was having a lottery drawing in the caboose at noon, and the deadline for time-stamping the tickets was 11:50. If we assume relativity of simultaneity, we see the problem immediately.

The locomotive engineer can see the winning number drawn in the caboose and still have time to mark his ticket. Knowing the future and/or affecting the past aren’t allowed, so relativity of simultaneity can’t apply to a rotating frame.

We therefore have ordinary, everyday, plain vanilla simultaneity on the train. That requires synchronized clocks, but how can we synchronize them? The Sagnac experiment showed that we can’t use Einstein synchronization.

We can’t synch all the clocks across the face of a rotating disk because no two points in a rotating frame have the same acceleration or velocity. However, all the points at any given radius do have the same speed, and thus have the same time dilation. A light pulse from the disk center will reach all points at a given radius, such as the rim, simultaneously in the non-rotating frame of the disk center. Thus all points at that radius will be initialized together, and will run at their dilated rate in synch.

The slow transport method might also work here, as without RoS, there is little to suggest that the conductor walking the length of the train wouldn’t remain in synch with the caboose as he set clocks in the cars.

It is interesting to note that a rim-riding observer would see all clocks both interior to him and exterior in the non-rotating frame running at faster rates, quite the opposite of the inertial case where other clocks are seen to run more slowly.

The case against Lorentz-Fitzgerald contraction is more difficult, but with RoS, we’ve already knocked out one of the props it requires.

5. One for Martin Gardner

3 November 2011

18th Century Oxford astronomer James Bradley observed a small displacement in the position of stars over the course of the year. In 1727, he determined that this was caused by, and in the direction of, the motion of the earth around the sun and a fixed speed of light rays. He further deduced from this that the speed of light was 183,000 miles per second. Displacement of light rays due to motion of the observer is called stellar aberration.

An often used analogy is a man with an umbrella in a rain shower. If he stands in one spot, he holds the umbrella directly overhead to keep dry.However, if he starts walking, he must tip the umbrella ahead to keep the raindrops off.The faster he moves, the more he must tip the umbrella from vertical.

Non-relativistic pluvial aberration

The geometry is simple for slow speeds, but should his motion be relativistic, we would need to use the aberration formula Einstein derived in his 1905 paper.

We have a different analogy to demonstrate aberration, one that can show both slow speed and relativistic aberration. We place two boats on a lake, our sailboat ‘A‘ doing 6 knots eastward, and a motorboat ‘B‘ on a roughly southwesterly course at 10 knots, and we’re going to collide at point ‘C‘. The arc is at 10 knots.

The ‘true’ collision angle (Φ phi) off our course is DCB, but we see the motorboat coming at us along line AB, an apparent angle of DAB. Setting up a simplified trig formula for these angles, we find the true angle DCB is atan(sinΦ/cosΦ). The apparent angle (α alpha) DAB has our sailboat’s speed added to the cos leg, atan(sinΦ/(cosΦ+v)).

The late Martin Gardner was a mathematical recreationist who often wrote about interesting mathematical oddities, puzzles, and games in his books and in his Scientific American column. He especially liked ruler and compass solutions to complicated problems, where you construct with only arcs and circles, perpendicular and parallel lines, and a limited number of angles.

The above analogy is a ruler and compass construction. Here, then, is a ruler and compass solution to finding relativistic stellar aberration for any speed and true stellar angle.

Speeding things up in the diagram example, A is now our starship doing .6 c, and B is a light ray from a distant star that we see when we get to point C. Line BC actually represents the angle of the field of light rays that we fly through getting to point C. The arc is scribed at lightspeed, 1.0 c.

Einstein’s equation covers aberration at all speeds, from snails to starships. The relativistic correction in his formula increases the forward sweep, sharpens the apparent angle off the bow when you get up to relativistic speeds. To sharpen the apparent angle DAB, we must either shorten the vertical line BD, or lengthen the horizontal line DA. We can do that by dividing DA by the inverse of the Lorentz factor (0.8 for .6 c).

The question now becomes, where can we find the inverse Lorentz factor for any given speed? Not coincidentally, if we construct a vertical from our starship to the lightspeed arc, the intersection E is the inverse factor

for any speed, snail through starship. Now to scale up the line DA.

We construct a line from D through the point E, giving us triangle DAE. Dividing AE by the inverse Lorentz factor (essentially itself) yields a length of 1.0. By similar triangles, if we scale AE up to 1.0, the new base will be line DA scaled up by the same factor.

We extend line DE to intersect (point F)a constructed parallel line 1.0 away from the base, then drop a vertical to the base line, intersecting at point G. DG is now our corrected base length, and angle DGA is the relativistic stellar aberration angle for that speed and true stellar angle.

This construction holds for any starship speed and for a star at any angle, including ‘behind’ the starship.

Now that we have aberration in the toolkit, we can go for a ride on a rotating disk.

4. Keeping the Beat

30 October 2011

Clocks ticks keep track of the natural passage of time in whatever frame they are in. However, the starting point for counting those ticks may be arbitrary. Unless clocks at different locations are synchronized, they are useful for only local activities.

Clock setting in SRT is a critical but straightforward task in inertial frames. Einstein proposed one method in his 1905 paper. Send a light pulse from the master clock to a distant slave clock, where it initializes that clock and is reflected back to the master. The round trip time is measured, then the second clock is notified to adjust its setting by half the round trip time. This is called Einstein synchronization.

A second method involves bringing the master and slave clock together and synchronizing them, then taking them back to their intended locations. This is called ‘slow transport’ because the movement speed is very much less than c, so slow that time dilation is negligible.

As we watch from the train frame, the train conductor walks from the caboose to the locomotive, synchronizing clocks in the cars as he goes. He returns to the caboose, and all clocks he passes agree with his.

uncontracted train

Einstein synchronization would work as well, had he a clear path through the cars for his light pulse.

But we’ve noted from the lab frame that the moving train exhibits Lorentz contraction, mass increase, and time dilation. It also exhibits relativity of simultaneity, with clocks at the locomotive reading earlier than clocks at the caboose. The conductor has synched his trains clocks, but from the lab frame we see a spread of times, with events happening first in the caboose, then flowing up to the locomotive.

Seen as the contracted train passes, the clocks in the caboose read later than the locomotive's.

The train is travelling at .866 c, and one time unit long in its own frame, here Lorentz contracted by the Lorentz factor of 2. The animation has clock rates slowed for clarity, but relative times are to scale for .866 c.

All well and good for straight line travel, but Ehrenfest is about going around in circles. How do we synchronize clocks going .866 c on the rim of a disk?

You can’t use Einstein synchronization. Sagnac demonstrated that the time of travel around the circumference of a rotating disk depends on the direction of the light beam, co- or contra-rotation, so the speed of light measured in a rotating frame is not constant, much to the consternation of relativists. The reason is obvious from the fixed (non-rotating) frame – the starting/ending point has moved while the light pulse is travelling around. Within the rotating frame, that point hasn’t moved, but that points out a difference between inertial and rotating frames. Unlike inertial frames, where any frame is as valid as another, in rotation there exists a ‘preferred’ frame – the inertial frame in which the center of rotation is fixed. Rotation rates, the frame’s equivalent of ‘speed,’ combine with simple addition, not with the Einsteinian ‘addition of velocities’ formula, and there is no lightspeed-like limitation on rotation rate.

The rotating frame is therefore not relativistic, though it may retain some relativistic traits like the time dilation and mass increase demonstrated in particle accelerators. Given that, can we use ‘slow transport’ to synch clocks?

We know that clocks at different radii run at different rates, so synchronization of clocks is radius dependent. We’ll deal with clocks on the disc rim moving at .866 c.

There are three scenarios mentioned earlier.
1. The circumference contracts, yielding a reduced value for pi.
2. The measuring rods contract, yielding an increased value of pi.
3. Lorentz contraction doesn’t occur in rotation, keeping pi at 3.14…

The Lorentz factor for .866c is 2, so in case 1, the rim circumference will contract to pi*r, in case 2, contracted measuring rods will yield 4 pi*r, and in case 3, we have the non-relativistic normal value of 2 pi*r.

In the earlier example above, we’ve seen how synchronized clocks in a contracted train frame appear evenly unsynched in the lab or fixed frame, and that this spread of time comes as part of contraction.

Let’s assume both contraction and valid slow transport, and have a rim-riding conductor travel around the circumference setting clocks.

He synchs his watch with a master clock on the rim, then as he travels in the co-rotation direction, we see his watch and the clocks he sets running progressively earlier than the master clock. Here is a significant difference between rotating and non-rotating frames. Where in an inertial frame the train will run off toward infinity and the spread of time will have no consequences on board the train, in the rotating disk, the conductor will eventually come upon the starting point, the locomotive will be nudging the caboose.

In accordance with our formula of v/c time unit change per contracted distance unit, the conductor will now find his watch either 2.7 or 10.9 time units earlier than the master clock, depending on the case 1 or case 2 scenario. If the conductor reverses course and returns the way he came, travelling contra-rotation, he will find all clocks he passes still in synch with his watch, and arriving back at the master clock, again still in synch. Should he continue contra-rotation for another circuit, he will find his watch unsynched the same 2.7 or 10.9 time units, only now unsynched later instead of earlier.

This conflicts with Einstein’s simultaneity relationship, where if clock A is synchronized with both clock B and clock C, then clocks B and C will be synchronized with each other. This means that either clocks cannot be synchronized in the rotating frame, or that the time spread doesn’t occur, and therefore we cannot have Lorentz contraction.

Of course, rim clocks can be synchronized on the disk. A light pulse from the center of the disk will reach all rim clocks simultaneously, and no reflected pulse is required, or indeed even useful, since the center and rim clocks run at different rates. Indeed, if the master clock is placed on the rim, a center clock may be synchronized to its dilated tick rate, and negatively (earlier) offset by the round trip light time, so that all rim clocks may continuously read and synch to the master by observing the center clock.

Because assuming either contraction/time spread scenario produces non-sensical results, we look at slow transport. To produce anything other than good synchronization around the rim would require the slowly transported clock to drift one way or the other from the continuously synched center clock, and then the only clock out of synch would be our transported clock. Again, a non-sensical outcome.

So now that we have our rim clocks running in synch, let’s put that in our tool kit and look at Einstein’s take on stellar aberration.

3. Bouncing Along

20 June 2011

Time Dilation

We’ll start with time. Here’s a clock.-

Think of an hourglass, then replace the top and bottom with facing mirrors, and the sand with a rather large, glowing, and improbably slow photon bouncing between the mirrors once every time unit. We have one, and others have identical clocks in other reference frames.

It’s night time, and we’re sitting on the laboratory porch watching our clock. A darkened train passes by, another inertial reference frame, moving along at .866 c, a speed that has a Lorentz factor, γ, (gamma) of 2.0, i.e. half the clock rate, half the length, double the mass according to Einstein. Someone on the train has an identical clock, and instead of the vertical line our clock describes, we see the train photon tracing an up and down zig-zag path.

The train photon takes one time unit in the train frame, but the diagonal path we see it taking is longer than the vertical path our own photon takes. Since the rule is that everyone sees a photon traveling at the same speed, the time passage we see between reflections is longer than our own clock’s time unit. Time passes more slowly in the train frame. (Better and more detailed explanations are available on the web)

The odd thing is that the people on the train see our clock tracing out the same diagonal in the dark, so they think our clock is running more slowly than theirs. Their train runs off into the night, and we all are content to let the other go with their slow running clock, because we’ll never see them again, nor will they see us so it doesn’t matter how out of sync our clocks become. This is important in the paradox, and we’ll consider it again.

Lorentz-FitzGerald Length Contraction

So clocks run slowly in the other frame. Should that be enough to make everyone see light travelling the same speed, or do we need to apply the Lorentz-FitzGerald equation to length, too?

Let’s buy another clock, upscale this time, with two photons, the second running horizontally. Note that the photons coincide when they strike the corner reflector.

Now let’s put the clock aboard the train, still running to the right. In our lab frame, the horizontal photon at 1.0 c will be chasing the right mirror at .866 c. Without contraction, the vertical photon will have bounced several times before the horizontal photon’s ‘slow’ overtake of .134 c catches the right mirror and bounces. Since both horizontal and vertical photons must get back to the corner reflector at the same time in each frame, an uncontracted train frame doesn’t do it. We find that contracting the horizontal frame by the Lorentz factor, 2.0 in the case of .866 c, combined with the time dilation, gets the photons striking the corner mirror together. What about the far mirrors?

Let’s horizontally contract the train clock and step along the paths of the two photons. This diagram is to scale for .866 c, and overlapping snapshots have been removed for clarity, including the one at lab time 2.00, where the vertical photon bounces. The time is lab time; the dilated train time would be half that.

The horizontal photon strikes the right mirror and bounces at lab time 3.72, not the 2.00 time of the vertical photon. That’s a problem.

It also has to travel the entire distance back to the corner reflector in 0.28 units of lab time, but with the corner reflector moving rightward at .866 c and it moving leftward at 1.0 c, we have a closure rate of 1.866 c. The photon makes it right on time.

These snapshots were taken from the lab frame as the train passed. The continuous paths of the two photons are marked in red and green. At any point, the two paths are of equal length in the lab frame, regardless of how far along it appears they’ve travelled in the train frame.

Whoa, how do we allow a closure rate of 1.866 c when nothing can travel faster than light? That’s the closure rate in the lab frame, but the lab frame still sees the photon at 1.0 c. The train frame will also see the photon moving at 1.0 c, because we must use Einstein’s addition of velocities formula, where the velocities combined is then divided by the velocities multiplied plus 1.

That squares away the corner reflector bounces, but what about the discrepancy at the right mirror?

A digital clock at that bounce would read 1.0 on board the train, which would be 2.0 in lab frame time, the same 2.0 we found at the top mirror bounce. The return bounce at the corner reflector reads 4.0 lab time for both photons, which would be the expected 2.0 train time. Let’s take a closer look at the right mirror bounce.

The train time at the right mirror is different from the train time at the corner reflector, which we zeroed when the two photons departed. The only way to account for the discrepancy at the right mirror is for lab frame observers to see a spread of times along the length of the train, with earlier times toward the locomotive and later times cabooseward. On a long-enough train, the conductor in the caboose would have the on-board lottery results before the engineer in the locomotive even picked his numbers. (This is encompassed in the concept of relativity of simultaneity, which has many explanations on the web if you want more info.)

The spread is like time zones in the U.S., or more properly, like sundial time, a continuous change as you look from east to west. In our close up, the amount of change is .866 time units over a stretch of 1.0 distance units, not coincidentally v/c time units per Lorentz-contracted distance unit. Another piece of the puzzle we’ll use later on.

This non-intuitive spread of time aboard the train frame gives us a way of regarding the contraction we see. The definition of length is the distance measured between two point at the same time in the same frame. For a train 100 meters long in its own frame, in our lab frame the locomotive has not yet hit the point 100 meters ahead of where the caboose is now, and when it does, the caboose will have moved forward from that 100 meter trailing point.

The train observers will also see this spread of clock time in our own lab frame, and this is what allows both frames to regard the other frame as contracted.

Lorentz contraction, then, is as much a matter of time as of length, so getting our clocks set right is important. We’ll hit that next.

2. Rolling Toward Relativity

19 June 2011

Non-Relativistity

Before getting into the details, here are some general comments on the nature of the paradox and its treatment.

A basic function of SRT is to tell us the ‘where’ and ‘when’ of objects in one inertial reference frame as observed from a different inertial reference frame. If we have a train speeding past us at 86 mph, using grade school math we can calculate for any given instant where Seat 2A in Car 9 is, relative to our position. For a train going 86% lightspeed, using junior high math, we can do the same. If the method we use to calculate that position does not yield simple, rock-solid x,y coordinates, then we haven’t solved the problem.

A common characteristic of many papers on the paradox is the explicit or implicit assumption that SRT, and specifically Lorentz contraction, applies in the case of a rotating reference frame. It does not.

SRT is designed for inertial reference frames. It can cover straight-line acceleration, an example of ‘addition of velocities’ that Einstein addressed in his original paper. An object in an inertial frame cannot exceed the speed of light, even if enough power were available to propel it that fast. The simple reason is that the inertial frame itself may not exceed the speed of light. This is because at lightspeed, Lorentz contraction reduces all distances in the direction of travel in the frame to zero.

A rotating reference frame is not an inertial frame, and is not subject to the lightspeed limit. Every rotating frame, regardless of its spin rate, has a radius beyond which the frame itself exceeds ‘c’. For many SRT reasons, physical objects may not remain fixed in the frame beyond that radius, but the frame itself has no such limitation. The rotating frame is non-relativistic.

The question here is whether or not objects within that radius, maintaining fixed positions in the rotating frame, are subject to all the effects of SRT. We’ll briefly cover a few ingredients toward the answer, but bigger and better explanations of SRT are available on the web.

Lorentz and FitzGerald

It’s been known since Newton’s time that light has wavelike properties, and since waves must travel in a medium – sound through air, ocean waves through water – inquiries were made into the characteristics of light’s medium, named the ‘ether’ for convenience.

Michelson and Morley performed an experiment in 1887 that would have measured the motion of any ether, or our motion through the ether. They didn’t find a thing. More experiments indirectly determined that light always travels the same speed, regardless of the motion of the observer or source. This didn’t quite fit Newton’s mechanics. The Lorentz contraction hypothesis helped somewhat.

Lorentz contraction is more properly titled Lorentz-FitzGerald contraction, as Dutchman Hendrik Lorentz and Irishman George FitzGerald independently came up with the equation for transformation between reference frames.

The equation comes from a rearrangement of the Pythagorean theorem, and the c part makes the velocity a unitless fraction of lightspeed, much like Mach number in aviation.

The equation held that if length were variable, not absolute, that was how it would vary. Whether they believed it anything more than an ad hoc, empirical tool is doubtful, both because it’s a ridiculous notion to begin with, and because simply accepting the variability premise can quickly lead to the full set of SRT effects, and with a little more math, the equivalence of energy and matter, E=mc.

Einstein came up with the contraction hypothesis, too, but he did it the hard way, starting with Maxwell’s electrodynamic equations and moving electrons, and coming up with mass increase, time dilation, stellar aberration, addition of velocities, and a lot of other answers to questions that hadn’t been asked (and don’t apply to the Ehrenfest paradox). Here’s the paper
On The Electrodynamics Of Moving Bodies.

Let’s look at some SRT effects next.

1. MerryGoRounds and Record Players

2 December 2009
rotating disc

High Speed Non-Relativity

The Ehrenfest Paradox is the single (arguably) remaining unresolved paradox in Einstein’s Special Relativity Theory. It proposes that relativistic Lorentz contraction around the circumference of a rotating disc or cylinder will alter the value of π.

Einstein’s 1905 Special Relativity Theory (SRT) holds that an observer in any sort of inertial motion will measure a fixed speed of light, ‘c’. This immediately produces three effects for observers in different inertial reference frames.

  1. Time will appear slowed down in the other frame – time dilation.
  2. Mass in the other frame will appear increased.
  3. Length in the other frame will appear shortened in the direction of travel – Lorentz contraction.

Further examination of momentum and energy produces Einstein’s signature equation, E=mc.

Both time dilation and mass increase are observed in particle accelerators. Lorentz contraction has yet to be demonstrated, since anything large enough to have a measurable length at relativistic speeds is too heavy to accelerate that fast. Mathematically, though, it follows from the two observed effects.

In 1909, Paul Ehrenfest proposed a paradox that has physicists tied in knots to this day. Einstein found one solution that led him to consider non-Euclidean geometry, curved space, which led him on to General Relativity (GRT) in 1916.

Briefly, then, here is the paradox -

If the circumference of a rotating disc travels at a relativistic speed, the circumferential length should contract in the direction of travel. The radius of the disc travels sideways, not in the direction of its length, so its length should be unchanged. π (Greek letter pi) is the circumference divided by twice the radius, but circumference here has contracted, therefore the value of π has decreased, and varies with the rim speed.

That was Einstein’s conclusion, but here’s another take on the paradox: Measuring rods placed around the moving rim will contract, requiring more of them to go all the way around. Thus the circumference now increases, and the value of π also increases.

Opposite results, equally viable. A cynical take would be that the contraction of the rods equals the contraction of the rim, therefore the value of π remains the same.

We’ll take a different approach here, looking at the nature of contraction itself.


Follow

Get every new post delivered to your Inbox.