Relativity and Cosmology

The Opera experiment made headlines around the world last September with their announcement that neutrinos sent from the CERN laboratories to Gran Sasso appeared to be moving at superluminal speed. After repeated claims of retests that gave the same results, it is now accepted that it was an instrumentation error that caused the anomaly.

In the aftermath, Antonio Ereditato (left), spokesperson of the Opera collaboration, announced on March 30 that he stepped down. He is no longer leading the Opera experiment.

This happened after a workshop was held at the Gran Sasso laboratories, where the various experiments reported their findings and discussed them (no details released yet). Following the workshop, the Opera collaboration is reported to have voted on removing Ereditato from the leadership position. The motion did not pass, but the voting showed that the collaboration was split, and this may eventually have led Ereditato to step down.

Question is, was it wrong from him to push the announcement of the headline-grabbing results too early? Or is it perhaps good that he pushed the slow-grinding wheels of science a little? Tommaso Dorigo thinks the latter:

"Let us instead try to educate the public on the fact that what happened to Opera's superluminal neutrino claim is good science: we study an effect, find something unexpected, and then try to kill the effect with all our means by studying it in more detail and with all the other tools we have available. What survives this kind of treatment is usually only real, trustable effects."

-J

We know that the isotropy of the speed of light is one of Einstein's postulates of special relativity. Let us attempt to confirm this by determining the one-way speed of light with a cunning apparatus.

The cross hairs to the right represent a setup with four observers, p, q, r and s, each a distance L from the gun at the center. This gun is designed and tested to simultaneously shoot identical high-speed pellets isotropically in all four directions. Put this whole lot in the vacuum of space, far from any gravitating body and at rest relative to the cosmic microwave background (CMB) radiation.^[a] Also, stabilize the setup as to not rotate relative to the distant stars.

Shoot one set of pellets. As a pellet strikes a target at each observer, their clocks are automatically set to the same time. The four clocks are now perfectly synchronized, without any worries about the speed of light.

Each observer now sends a time-stamped laser pulse to her opposite number (p to r, r to p, etc.) Knowing the distance 2L between opposite corners, it is reasonable to assume that they will get a light travel time of 2L/c seconds and hence isotropic one-way light-speed of c. Next, gently accelerate the whole structure in the direction p->r until it has a constant speed v relative to the CMB, as measured by the change in CMB redshift.

Each observer repeats the one-way light-speed test and they find the following respectively: light took 2L/(c-v) seconds to travel from p to r, 2L/(c+v) seconds from r to p, while the other two directions still took 2L/c seconds. This means that the effective speed of light was c-v in the p to r direction and c+v in the r to p direction.

The clocks have all suffered the same gentle acceleration, so their time keeping and sync should not have been affected at all. So, have we shown that the speed of light is not isotropic and that the one-way speed depends on the laboratory's velocity relative to the CMB? Was Einstein mistaken?

-J

[a] This means that the CMB average temperature (or redshift) is observed to be the same in every direction.

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Recall the prior Blog's dialogue on the Starship Enterprise bridge, where Captain Kirk asked Spock and Scotty to go and resolve their differences in interpreting the test results (which according to Scotty implied that the two Vulcan clocks run faster than the ship's clock). Spock disagreed.

So off they went to the ship's CR4.

"Mr. Scott, our bridge clock was present at both flyby events, so you effectively measured the invariant spacetime interval between the events: Δτ = √[Δt² - Δx²], in geometric units.⁽¹⁾ In the bridge frame Δx=0, so Δτ = Δt = 1333 ns, the elapsed time that you found.

"In the frame of the two clocks, we know the distance between the clocks was Δx' = 1000 ft. Since the spacetime interval is the same for all inertial frames, it is very easy to show that Δt' = √[Δτ'² + Δx'²] = 1667 ns. This is the elapsed time that you recorded from the readouts of the Vulcan clocks.

For every frame of reference, the farther two events are separated in space, the more they are also separated in time⁽²⁾ - simple and logical. No apparent time dilation or length contraction".

A thoughtful Scotty replies:

"I thought we have to use the Lorentz transformations, which have all the elements of clock synchronization, time dilation and Lorentz contraction mixed together".

Spock: "These are favorite Human abstractions, Mr. Scott. This is how people frequently come up with contradictory conclusions".

Frowning, Scotty counters: "But Mr. Spock, I was taught those things in Starfleet Engineering School. Are you now telling me they are in effect useless?"

Spock: "Not useless Mr. Scott; just overly complex and confusing in this case. Do you agree that the spacetime interval solution is clear-cut?"

Scotty: "Ay, Mr. Spock, I see that. It does look like mathematical sleight of hand, though. I would like to test this in more situations".

Spock: "I do not understand 'mathematical sleight of hand'. We may come back to that later, but right now we must report back to the bridge".

-J

(1) Where c ~ 1 (ft/ns), meaning distance and time are effectively measured in the same units.

(2) This true for clocks synchronized by the Einstein-method and spacetime intervals where the time separation is larger than the space separation ('time-like' intervals).

We know that two ideal clocks permanently at rest relative to each other can be synchronized by means of the Einstein method.⁽¹⁾ Simply stated it means: measure the distance (d) between the clocks (Alpha and Beta); send a light pulse from Alpha to Beta at time t; when that pulse reaches clock Beta, set its time to t+d/c.

An equivalent method is to start with the two clocks in close proximity, set both to the same time and then slow-transport either Alpha or Beta (or both) to end up a distance d apart. Their synchronization can then be verified in both directions, using the above method, ensuring that they are indeed "Einstein-synchronized".

These two clocks were of Vulcan origin and Starship Enterprise performs a flyby at a (mild) speed of 0.6c. When adjacent to clock Alpha, Scotty records the time on its display and also the ship's clock's time. He does the same when adjacent to clock Beta, so that he could compare the elapsed time between the two flybys, as given by clocks Alpha and Bravo and by the ship's clock. To his surprise, Scotty finds that the elapsed time between the flybys according to the two clocks is larger than what the bridge clock has shown.⁽²⁾

At the debriefing Captain Kirk requests: "Right Mr. Scott, did we learn anything about the Vulcan clocks in this experiment?

Scotty: "Ay, Captain, we found that Vulcan clocks tick faster than our bridge clock".

Spock: "How can you say so, Mr. Scott? All Vulcan clocks conform to the United Federation of Planets standards of time. What is more, Starfleet physics predicts exactly the opposite. The Vulcan clocks were moving relative to our ship, so to us they will appear to tick slower than the bridge clock".

Scotty: "Sorry Mr. Spock, but I measured it and as Kepler once said: to measure is to know."

Spock: "Captain, Mr. Scott never measured the rate of the Vulcan clocks".

Kirk cuts them short: "Scotty, I spot a difference of opinion here. Mr Spock, will the two of you please settle the issue off the bridge and then report back".

If you were Spock, how would you have resolved the disagreement?

-J

Notes

1. "Permanently at rest relative to each other" implies free-fall in zero (or at least extremely weak) gravity and not too far apart.
2. Take c as one foot per ns, the distance (d) between the clocks as 1000 ft in the Vulcan frame. Clocks Alpha and Bravo will then show an elapsed time of 1667 ns between the flybys, while the bridge clock will record the elapsed time as 1333 ns.

The Higgs ("God-particle") has apparently been found and weighed.⁽¹⁾

Although slightly out of my field, this is tantalizing news from CERN. There are reasonably clear indications that it has a rest-mass/energy of mc² = 125±1 GeV. It's heavy, but they call it a "relatively light Higgs".

The full implications for particle physics, cosmology and relativity must still sink in, but I think there are many - so watch this space.

-J

(1) http://www.science20.com/quantum_diaries_survivor/firm_evidence_higgs_boson_last-85478#comment-91551

Einstein's clock synchronization method is based on the principle that, in every inertial frame, the one-way speed of light in vacuum is the same as its average round-trip speed.^(a)

In synchronization terms, it means the following. Let clocks A and B be at rest in an inertial frame; measure the coordinate distance d between two clocks; send a time-stamped light pulse from A to B; at the instant of reception, add the propagation delay (d/c) to the time stamp and set clock B accordingly. Clocks A and B are now 'Einstein-synchronized' in that inertial frame.

The relativistic principle and hence Einstein's method of clock synchronization is a convention, not an absolute truth.^(b) This may sound like relativistic heresy, but there are other valid ways of synchronizing clocks - it is just that none works as well as Einstein's.^(c) This Blog article attempts to show why by means of the Sagnac effect.

The figure (right)^(d) is largely self-explanatory. The two red arrows represent two signals moving in opposite directions at speed k relative to the blue ring. If the ring is non-rotating, each signal will circumnavigate the ring in time t = 2Πr/k, as measured by a laboratory clock.

In Newtonian mechanics, this is also true if the ring is rotating, because in the lab frame, the difference in the distances that the signals have to travel in the two directions exactly cancels the apparent faster or slower speed of the signals (u = v ± k), depending on signal direction. Signals traveling in opposite directions arrive at the same time and remain in phase; hence there should be no Sagnac effect. This Newtonian result is in conflict with observations of the Sagnac effect.

In Einstein dynamics, the principles are the same, accept that we cannot simply add the speed of the signal to the speed of the ring, i.e. u = v ± k does not work. We must use the relativistic speed addition equation u = (v±k)/(1±kv/c²). This gives a definite difference in the time for signals circumnavigating the rotating ring in opposite directions: Δt = 4Πrν/(c²-v²), as measured in the inertial laboratory frame.^(d,e)

This time difference is independent of the speed of the signals (k) and just depends on the radius and the rotation rate of the ring. Hence, in contrast to classical Galileo/Newton theory, Einstein's theory does predict the Sagnac effect and it agrees with all experiments performed so far. The Sagnac effect is a very simple proof that the Einstein clock synchronization method works. But, why is it the best scheme?

The answer is simple: by convention, it forces light to propagate at the same speed in both directions around the ring. All other clock synchronization schemes imply that light moves at different speeds in the two directions. As we have seen, in Newton mechanics this results in zero Sagnac effect. In others, having time dilation and Lorent contraction, like Lorentz ether theory (LET), it results in horribly complex equations for the observed Sagnac effect.^(f)

Einstein clocks are cool...

-J

PS: see reply #29 below for a summary of the topic.

Notes:

(a) More precisely stated: the observed one-way speed of light in vacuum is constant and isotropic in every inertial frame.

(b) A very good discussion of the conventionality of relativity can be found in this Wiki and also in this Blog entry.

(c) One alternative clock synchronization is due to Selleri, which has preferred inertial frame, but with time dilation and Lorentz contraction in any other frame. It is discussed in "The relativistic Sagnac Effect: two derivations", section 3.5. The paper contains a complete mathematical treatment.

(d) I borrowed the graphics from a physicsinsights.org article, which is excellent for a introductory discussion of the Sagnac effect and shows the calculations involved. It is a lot more accessible than "The relativistic Sagnac Effect: two derivations" in (b) above.

(e) Because v² « c² in the usual Sagnac interferometers, v² is normally ignored and just Δt = 4Πrν/c² is used. This is precisely double the Einstein clock synchronization offset between the transmitter and the receiver for each direction.

(f) For an example, see eq. (23) of "The relativistic Sagnac Effect", referenced in endnote (c). Note that the equations given there are for time as measured by a ring clock and not a lab clock, as used above, but there is only a factor γ difference.