Testing Lorentz Invariance 

One way of testing Lorentz invariance is to examine the characteristic excitation frequency of certain ions moving at high speed. According to Lorentz invariance, the resonant frequency of the ions (in the lab frame) as a function of their speed v, with everything else being equal, is 

_{} 

To test this, we can produce a beam of ions with speed v (we choose units so c = 1), and subject those ions to laser light from the front and back (i.e., parallel and antiparallel to the direction of the ions), of frequency n_{p} and n_{a} respectively relative to the lab frame, as depicted below. 


The laser frequencies are adjusted until both lasers are exciting the ions at their characteristic frequency. (This condition is detected by the presence of the Lamb dip, to be discussed later.) Focusing on just a single ion, the situation is as depicted in the spacetime diagram below, where x and t and standard inertial coordinates in the lab frame. 


(The experiment can be performed at different times of day to change the orientation relative to the Earth’s motion, verifying the isotropy of light speed relative to the standard inertial coordinates in the lab frame.) The time intervals between periodic pulses in this double resonant condition with ion speed v are related by 

_{} 

so the frequencies in this tuned condition with ion speed v are related by 

_{} 

Therefore we have 

_{} 

Now, if the resonant frequency of the ions was invariant, i.e., independent of speed, the product of the laser frequencies in the tuned condition would vary as a function of v in proportion to 1/(1v^{2}). On the other hand, it follows from equation (1) that the resonant frequency varies with speed in accord with Lorentz invariance if and only if 

_{} 

for every speed v. Thus by confirming that the product of the laser frequencies in the tuned condition always equals the same value for any speed v, we confirm that the resonant frequency varies in accord with Lorentz invariance. Of course, to perform a valid test, we must ensure that no extraneous fields or other influences alter the conditions of the ions. The only difference in their condition must be their state of motion. 

At this point it’s worthwhile to review in more detail how experiments of this kind are actually performed. In practice the beam of ions consists of ions with a range of velocities, distributed around the nominal velocity. The parallel laser (from behind) when operating at any given frequency will excite the subset of ions whose precise velocities cause the Dopplershifted laser light to precisely match their resonant frequency. As the frequency of the laser is varied, the number of ions that respond – and hence the fluorescence given off by the beam – varies in proportion to the velocity distribution of the beam. Suppose we adjust the parallel laser until achieving maximum fluorescence. (It isn’t necessary to for this to be the exact maximum, it’s just convenient to get near the maximum fluorescence for a good signaltonoise ratio.) Then we turn on the antiparallel beam, and vary its frequency. This increases the total fluorescence, as this laser excites the ions at various velocities. This is illustrated in the figure below, for the condition when the parallel laser is tuned to the ions at the center of the velocity spread, and the antiparallel laser is tuned to the ions at a slightly higher velocity. 


Now, as we tune the antiparallel laser to be in sync with lower velocities, the total number of excited ions (and hence the total fluorescence) will increase, because we are adding the ions excited by the parallel laser to the increasing number of ions excited by the antiparallel laser. However, when we tune the antiparallel laser to the same subset ions that are already being excited by the parallel laser, there will be an abrupt drop in the number of excited ions by roughly one half. As a result, whenever the two lasers are tuned to excite the same part of the velocity profile (not necessarily at the maximum of the profile), we will get a dip in the fluorescence, as depicted below for the case when n_{p} has been set for the center of the profile and we vary n_{a}. 


The sharp drop in fluorescence when both lasers are tuned to the same population of ions is called the Lamb dip (not to be confused with the famous Lamb shift), named after the physicist Willis Lamb. Once this condition has been identified, by measuring the fluorescence of the ion emissions, we know the two lasers are both in tune with the resonant frequency of the ions of a single velocity, and hence (according to the relativistic Doppler shift formula) we have n_{p}n_{a} = n_{I}(0)^{2}. This has been confirmed experimentally to very high precision. 

It’s important to note that the precise frequency and direction of the photons comprising the fluorescence are not critical, since the fluorescence is being examined only to detect the laser settings that produce a Lamb dip, signifying that the parallel and antiparallel lasers are in sync with a subpopulation of the ions moving at one specific velocity. The procedure does not need and is not designed to measure the frequency or direction of the fluorescence. We merely need to arrange our detectors to catch a representative sampling of the emitted photons from the target population of ions. It doesn’t even matter precisely which specific subpopulation of ions is in tune with the lasers. By following the procedure described above (first setting n_{p} to pick out one subpopulation of ions, and then adjusting n_{a} to the point of a Lamb dip), we are assured that these tuned values of n_{p} and n_{a} are in resonance with the characteristic frequency of the same subpopulation of ions in terms of the rest frame of those ions. It follows, if Lorentz invariance is valid, that the product n_{p}n_{a} should be the same, regardless of the velocity of the ions. Thus if we perform this procedure at two different velocities (one of which may be zero), the ratio of those products should equal 1. 

The optimum frequency, in the lab frame, for viewing the ion emissions will depend on the angle of incidence. Often such experiments place photodetectors along side the beam, so they are gathering photons at an angle around 90 degrees from the direction of the beam. For photons at exactly 90 degrees in terms of the standard inertial coordinates of the lab frame, the received frequency would be simply n_{I}(0)/g where g = (1v^{2})^{1/2}. However, this would not be the frequency of maximum intensity, because the angular distribution of the emissions is uniform in the rest frame of the ions, which implies (by aberration) that the distribution of angles of the emissions is swept forward in terms of the lab frame coordinates. The median emission would be the ions at 90 degrees to the beam in terms of the ion frame, and these photons will arrive blueshifted at the detectors, with a frequency of n_{I}(0) g. So, if filters are used to block noise and scattered light, it is helpful to chose the bandwidth to allow photons in this range of frequencies. We should stress, however, that the photodetectors are not performing a measurement of the emission frequencies, they are merely gathering fluorescence to detect the Lamb dip condition for tuning the lasers. The measurement of the ion frequency is performed by the lasers. 

Incidentally, such experiments are sometimes described in a different but equivalent way, making use of standard coordinate systems. Since all experiments have consistently shown that every physical entity and process is Lorentz invariant, including mechanical inertia, it follows that standard inertial coordinate systems are related by Lorentz transformations, and hence the resonant frequency of ions (shown to be Lorentz invariant) free of external disturbances always has the value n_{I}(0) when expressed in terms of comoving standard inertial coordinates. Letting n_{0} denote this constant, and letting n_{a}^{*} and n_{p}^{*} denote the frequencies of the lasers relative to the rest frame of the ions (accounting for the Doppler shift from the lab to the ion frame), we have at the fully tuned condition 

_{} 

Now we consider the quantity 

_{} 

What value do we expect this quantity to have? The answer depends on whether we believe in relativistic time dilation and Lorentz invariance. If we don't, we would apply the classical Doppler formula (with no relativistic timedilation) to give the relations 

_{} 

On this basis we would expect to have 

_{} 

On the other hand, if we believe in relativistic timedilation and Lorentz invariance, the frequencies of the laser beams in the rest frame of the ions are 

_{} 

so we expect to have 

_{} 

All such experiments to date have shown R = 1 to within experimental error, consistent with the relativistic Doppler formula, including the time dilation factor, in perfect accord with Lorentz invariance. 

Amusingly, a letter appeared in the European Journal of Physics C in 2010 claiming that all such experiments have been misinterpreted, and that if interpreted correctly they actually show a violation of Lorentz invariance. More specifically, the EJPC letter contends that Lorentz invariance would actually imply R = 1 + v^{2}, so the experimental result R = 1 represents a violation of Lorentz invariance. The letter reaches this strange conclusion by first asserting that, in all such experiments ever performed, the characteristic frequency of at least some of the ions relative to their rest frame – when the ions are moving at various speeds in the apparatus – is different at different speeds, commenting that "the transition frequency for the moving ions can get shifted … due to external fields and charged particles causing Stark and Zeeman effects". But of course it is essential for such experiments to preclude any external disturbing effects (or place bounds on their magnitude), because otherwise we are no longer testing the effect of velocity, we are testing the Zeeman effect. Furthermore, the very paper cited by the EJPC letter for the possibility of Zeeman effects states 

Despite the use of linearly polarized light, the Zeeman effect might not only cause symmetric broadening but also a net frequency shift when the ion beam becomes polarized by optical pumping. However, for the fields present near the photomultiplier, this shift is estimated to be below 5 MHz and can be neglected at the present level of accuracy. 

Obviously the EJPC letter has misrepresented its source, and is effectively claiming that all experimenters have botched the experiments by allowing disturbing influences into the apparatus (above the levels they reported), invalidating any possible test of Lorentz invariance. Needless to say (or so one would have thought), any valid experiment must either avoid such effects, or else correctly account for them in their analysis. After all, Lorentz invariance signifies that the characteristic frequency of a certain ion, in terms of the standard inertial rest frame coordinates of the ion, is always the same, independent of the velocity of the ion (i.e., the characteristic temporal measure of fundamental physical processes is the proper time of special relativity), but only in the absence of any disturbing effects. This is precisely what these experiments are designed to test, and this obviously requires that we either avoid or else properly account for any physical disturbances that affect the natural frequency of the ions in their own rest frame (if the effects vary with velocity). 

After the one misleading comment on the possibility of Zeeman effects, the EJPC letter never mentions them again, nor does it offer any attempt to quantify the magnitude of these effects, nor to dispute the experimenter’s claimed upper bound. Instead, the EJPC letter presents a different (and rather bizarre) line of reasoning to arrive at its conclusion. It bases its reasoning on a comment made in the description of one particular experiment of this kind, in which emissions from the ions are observed transversely (more or less) through a filter centered on the nominal resonant frequency n_{0} in the lab frame. (The paper also reported that the filter had halfwidth of 10 nm.) In that particular experiment, the nominal resonant frequency of the ions in the rest frame of the ions corresponds to a wavelength of 548 nm, and the velocity of the ions was 0.064c, so the emissions at 90 degrees (in the lab frame) are expected to have a wavelength of about 549 nm in the lab frame, taking into account the transverse Doppler shift. This is still quite close to the center of the filter, but more importantly, the photodetectors are not limited to gathering photons at precisely 90 degrees in the lab frame. In fact, as explained above, the frequency of the photons emitted from the ions at 90 degrees in the ion frame (which would be the middle of the intensity distribution) would be not n_{0}/g but rather n_{0} g, corresponding to a wavelength of about 547 nm in this particular experiment. To gather photons over this whole range of incidence angles and frequencies, a filter centered in n_{0} with a halfwidth of 10 nm is quite reasonable. 

Recall that the halfwidth of a filter is the difference between the two wavelengths where the transmissivity is one half. For a Gaussian profile, this implies that the filter had 97.3% of its maximum transmissivity at both 547 nm and 549 nm, so it is essentially transparent to all the fluorescence emitted from the ions at their resonant frequency. It’s also worth noting that the optical lenses for the PMT accepted a range of incidence angles of 90 ± 16 degrees, which more than covers the optimum range of viewing angles (noting that the ions at 90 degrees in the ion frame would be at only about 3.6 degrees from perpendicular in the lab frame). In a subsequent experiment performed with ion speeds of 0.338c, a BG39 filter was used, with peak transmissivity around 515 nm (which happens to be the wavelength of the median of the intensity distribution), with good transmissivity even up to 584 nm, so again the filters were well suited to gather fluorescence emitted from the ions at the expected frequency. The optical lenses and range of incidence angles of the PMTs for the highspeed experiment were not specified, but, as we stressed previously, the experiment does not need and is not designed to measure the direction or frequency of the fluorescence. Only the intensity of the fluorescence is used to identify the Lamb dip when the lasers are in tune with the ions. The purpose of the filters is merely to limit noise and scattered light from the lasers. 

To arrive at its odd conclusions, the EJPC letter disregards all this, and assumes (incorrectly) that every experiment has used a filter of zero width, so that only photons of precisely n_{0} could pass, and it assumes (incorrectly) that only photons with an incidence angle of precisely 90 degrees in the lab frame are detected. (Both of these assumptions are not only incorrect, they are absurd, because a filter of zero width would pass zero signal, as would a filter with zero angular range, and the highspeed experiment didn’t even use a filter centered on n_{0}.) Then, on the basis of all these erroneous and absurd premises, the EJPC letter asserts that since the experimenters clearly observed a fluorescence profile, they must have been viewing photons emitted by ions with wavelength n_{0}g in the ion frame. This would obviously be a violation of Lorentz invariance, or else evidence that the experiment was botched by allowing extraneous effects into the apparatus. But of course this is not what has been observed in these experiments. As explained previously, all such experiments have consistently shown that the frequency of the ions in the lab frame is n_{0}g, and hence the frequency in the ion frame (expressed in terms of standard inertial coordinates) is n_{0}. The observed fluorescence is perfectly consistent with these findings. 

To round out its absurd line of “reasoning”, the EJPC letter argues that, since it infers from the imaginary zerowidth and zeroangle (and zero signal!) filter that the ion frequency must be n_{0}g in the ion frame (due to some unknown extraneous effects in the apparatus), the Doppler formula would lead us to conclude that the product of the frequencies of the tuned lasers ought to be n_{0}^{2}g^{2}, and so the expected value of R, defined as the product of the laser frequencies divided by n_{0}^{2}, should be g^{2}, which is about 1 + v^{2} rather than 1. But of course the experiments have all yielded a value of 1, so the EJPC letter argues that these findings represent a violation of Lorentz invariance, i.e., that the relativistic Doppler formula is invalid. But of course this makes no sense at all, because even if we accepted the fantasized zerowidth filter, with the physically impossible zero angular range, and even if we accept the counterfactual proposition that the filter frequency is misplaced, and even if we accept the counterfactual proposition that the experimenters allowed unmodeled disturbances into the apparatus to give the ions a range of resonant frequencies outside the bounds that the experimenters explicitly placed on such effects, and even if we accept the proposed redefinition of R as the ratio of the product of laser frequencies to the square of the ion frequency in the lab frame (rather than the ion frame) – even if we accept all this – the EJPC letter’s “reasoning” still would make no sense, because the “experiment” would be reduced to a tautology. Under these premises, equation (2) is still valid, regardless of how the ion frequency varies with velocity. Dividing through equation (2) by n_{I}(v)^{2} as the EJPC letter proposes (since this is the frequency of the ions in the lab frame, as “measured” transversely to the beam) and calling this ratio “R”, we get “R” = 1/(1v^{2}), regardless of the function n_{I}(v)! The EJPC letter asserted that, under all its absurd and counterfactual premises, this is the value of “R” we should expect for Lorentz invariance, but it failed to note that under those premises this is the value of “R” we would expect for any physical hypothesis. In other words, under the premises proposed in the EJPC letter, the experiment could not possibly yield any other value of “R”, regardless of how the ion frequency (in the lab frame) varies with speed. The “experiment” proposed by the EJPC letter is a useless tautology. And of course the fact that the actual experimental results yielded a value R = 1 falsifies the premises of the EJPC letter. 

It may also be worth mentioning that such experiments need not even make use of the nominal frequency n_{0} in their evaluations. For example, one experiment determined the values of n_{p} and n_{a} for two different nonzero velocities, v_{1} = 0.064 and v_{2} = 0.030, and found that 

_{} 

to very high precision, consistent with Lorentz invariance. Of course, if v_{2} = 0 this expression reduces to the previous expression for R, with the numerator being n_{0}^{2}, which is to say, the square of the effective resonant frequency of the ions in their rest frame. 
