Relativity-8: General relativity

(For previous posts in this series, see here.)

To understand the role of Einstein’s general theory of relativity, recall that the original OPERA experiment claimed that they had detected neutrinos traveling faster than the speed of light. This posed a challenge to what is known as Einstein’s theory of special relativity, proposed in 1905, which said that the relationship between the clock and ruler readings for two observers moving relative to one another would be different from the ones given by the seemingly obvious relationships derived by Galileo centuries earlier. According to Einstein’s theory, it is the speed of light that would be the same for all observers, while clock readings could differ, and that Einstein causality (the temporal ordering of any two events that are causally connected by a signal traveling from one to another) would be preserved for all observers. One inference that followed from Einstein causality is that no causal signal can travel faster than the speed of light, and this was what was seemingly violated by the OPERA experiment.

But Einstein had a later and more general theory that he proposed in 1915, called the general theory of relativity, that included the effects of gravity. He showed that clock readings were not only affected by the speed with which the clock was moving, they were also affected by the size of the gravitational field in which the clock found itself. This is the source of what is referred to as the ‘gravitational red shift’ that enters into cosmology that causes the light emitted by distant stars and galaxies to be shifted towards larger wavelengths as they escape the gravitational field of those objects on their journey to us.

To understand what is going on, recall that when we measure the elapsed time between two events, what we are really doing is measuring the number of clock ticks that occur between the events. According to general relativity, the stronger the gravitational field, the slower the rate at which a clock ticks. The slower the rate at which a clock ticks, the less time that it records as having elapsed between two events.

So, for example, since we know that the Earth’s gravitational field decreases as we go up, this means that if we take two identical clocks, one on the floor and the other on the ceiling, the one on the floor would have fewer ticks between two events than the one on the ceiling, even if both are stationary. So the clock on the floor would ‘run slower’ than the one on the ceiling and hence the time interval measured between two events measured by clocks on the floor will be less than that measured by clocks on the ceiling.

In the OPERA experiment, the time measurements were made using GPS satellites. These are whizzing by at both high speeds (about 4 km/s) and high altitudes (about four Earth radii). Typically, the signals are handed off from one satellite to another as they appear and disappear over the horizon and the transition is almost seamless and produces such small errors that we do not notice it. But the OPERA experiment requires such high precision that they arranged to do the experiment during the transit time of just a single satellite so that even that source of error was eliminated.

Because the rate at which clocks run depends upon the size of the gravitational field, one has to make corrections to allow for the fact that the time readings given by clock readings of the satellites will be different from the time readings given by clocks on the Earth, and so one needs to make extremely subtle corrections to the GPS time stamp to get the correct clock readings on the Earth. This is why much of the attention has focused on this aspect. It is not that the OPERA experimenters overlooked this obvious feature (such general relativistic corrections are routinely made by GPS software in order to make the GPS system function with sufficient accuracy) but whether they have made all the necessary corrections to the extremely high level of precision required by this experiment.

Carlo Contaldi at Imperial College, London has suggested that the clocks at CERN and Gran Sasso were not synchronized properly due to three effects, one of which is the fact that the gravitational field experienced by the satellite is not the same at all points on its path since the Earth is not a perfect sphere. He says that the errors that would be introduced are of the size that could produce the OPERA effect. (You can read Contaldi’s paper here.)

Ronald A. J. van Elburg at the University of Groningen has argued that subtle effects due to the motion of the detectors with respect to the satellite could have shifted the time measurements at each clock on the ground by 32 nanoseconds in the directions required to explain the 60 nanosecond discrepancy. (You can read van Elburg’s paper here and reader Evan sent me a link to a nice explanation of this work.)

The OPERA researchers (and some others) have challenged some of these explanations and said that they will provide a revised paper that explains more clearly all the things they did.

There have been no shortage of ideas and papers pointing out problems and possible alternative explanations for the OPERA results. Sorting and sifting through them all before we arrive at a consensus conclusion will take some time.