(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.

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

The reactions to the reports of the CERN-Gran Sasso discovery of possibly faster-than-light neutrinos open a window into how science operates, and the differences in the way that the scientific community and the media and the general public react whenever a result emerges that contradicts the firmly held conclusions of a major theory.

The initial reaction within the scientific community is almost always one of skepticism, that some hitherto unknown and undetected effect has skewed the results, while the media and public are much more likely to think that a major revolution has occurred. There are sound reasons for this skepticism. Science would not have been able to advance as much if the community veered off in a new direction every time an unusual event was reported.
[Read more…]

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

In the previous post in this series, I said that Einstein’s claim that the speed of light must be the same when measured by all observers irrespective of how they were moving led to the conclusion that the rate at which time elapsed must depend on the state of motion of the observer. But if time is not an invariant entity, then we need to be more precise about how we measure it for observers in relative motion to one another so that we can better determine how their measurements are related.

What we now postulate is that associated with each observer is a grid of rulers that spreads out into all space in all directions. At each point in space are also a clock and a recorder. It is assumed that all the rulers and clocks of all the observers are constructed to be identical to each other, the clocks are properly synchronized, and the recorders never make errors. When an event occurs anywhere at any time, the location and time of that event are those noted by that recorder who happens to be exactly at the location of the event and who notes the ruler and clock readings located at the place at the instant when the event occurred. This rules out the need to make corrections for the time that elapses for the light to travel from the location of the event to the recorder.

If there is another observer who is moving with respect to the first, that person too will have her own set of rulers and clocks and recorders spread out through all space, and the location and time of an event will be that noted by her recorder using her rulers and clocks at the location where the event occurs. This set up seems rather extravagant in its requirement of infinite numbers of rulers and clocks and recorders but of course all these rulers and clocks and recorders are merely hypothetical except for the ones we actually need in any given experiment. The key point to bear in mind is that the location and time of an event for any observer is now unambiguously defined to be that given by that observer’s ruler and clock readings at the location of the event, as noted by the observer’s recorder located right there.

What ‘Einstein causality’ says is that if event A causes event B, then event A must have occurred before event B and this must be true for all observers. If one observer said that one event caused another and thus the two events had a particular ordering in time, all observers would agree on that ordering. Thus causality was assumed to be a universal property.

What we mean by ’causes’ is that event B occurs because of some signal sent by A that reaches B. So when the person at B is shot by the person at A, the signal that caused the event is the bullet that traveled from A to B. Hence the clock reading at event A must be earlier than the clock reading at event B, and this muust be true for every observer’s clocks, irrespective of how that observer is moving, as long as (according to Einsteinian relativity) the observer is moving at a speed less than that of light. The magnitude of the time difference between the two events will vary according to the state of motion of the observer, but the sign will never be reversed. In other words, it will never be the case that any observer’s clocks will say that event B occurred at a clock reading that is earlier than the clock reading of event A.

But according to Einstein’s theory of relativity, this holds only if the signal that causally connects event A to B travels at speeds less than that of light. If event B is caused by a signal that is sent from A at a speed V that is greater than that of light c (as was claimed to be the case with the neutrinos in the CERN-Gran Sasso experiment) then it can be shown (though I will not do so here) that an observer traveling at a speed of c²/V or greater (but still less than the speed of light) will find that the clock reading of when the signal reached B would actually be earlier than the clock reading of when the signal left A. This would be a true case of the effect preceding the cause. The idea that different observers would not be able to agree on the temporal ordering of events that some observers see as causally connected would violate Einstein causality and this is what the faster-than-light neutrino reports, if confirmed, would imply.

Note that this violation of Einstein causality occurs even though the observer is moving at speeds less than that of light. All it requires is that the signal that was sent from A to B to be traveling faster than light.

(If the observer herself can travel faster than the speed of light (which is far less likely to occur in reality than having an elementary particle like a neutrino doing so), then one can have other odd results. For example, if the speed of light is 1 m/s and I could travel at 2 m/s, then one can imagine the following scenario. I could (say) dance for five seconds. The light signals from the beginning of my dance would have traveled 5 meters away by the time my dance ended. If at the end of my five-second dance, I traveled at 2 m/s for 5 seconds, then I would reach a point 10 meters away at the same time as the light that was emitted at the beginning of my dance. So if I look back to where I came from, I could see me doing my own dance as the light from it reaches me. So I would be observing my own past in real time. This would be weird, no doubt, but in some sense would not be that much different from watching home movies of something I did before. It would not be, by itself, a violation of Einstein causality since there is no sense in which the time ordering of causal events has been reversed.)

So the violation of Einstein causality, not the theory of relativity itself, is really what is at stake in the claims that neutrinos traveling at speeds faster than light have been observed. This is still undoubtedly a major development, which is why the community is abuzz and somewhat wary of immediately accepting it is true.

Next: What could be other reasons for the CERN-Gran Sasso results?

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

In the previous post in this series, I posed the situation where, seated in my office, I observe two events on the sidewalk outside my window and measure the locations and time of two events and deduce the distance between them and the time interval according to the rules for using my own ruler and watch. Now suppose another person is moving with respect to me (say in a train that passes right by where the two events occur) and sees the same two events as I do and measures the locations and times of the two events and deduces the distance and time interval between them using her ruler and watch. Will her measurements agree with mine?

When it comes to location and distance measurements, it is not hard to see that the two results will be different. When I take ruler readings of the two events, the ruler is not moving compared to the two events. But because the person in the moving train’s ruler will be moving along with her in the train, the ruler readings of where the two events occurred will be affected by her motion. After the person in the train takes the reading on her ruler at the location where event A occurred, by the time the later event B occurs, she and her ruler would have moved along with her train and so the ruler reading for event B would be different from what would have been obtained if the ruler had been stationary. So the locations and the measured distance between the two events based on her two ruler readings will be different from those based on my two ruler readings.

What about the time interval between events A and B? It used to be thought that even though the two observers used different clocks and they were moving relative to each other, as long as the clocks were identical and synchronized properly, the two observers would at least agree on this because it seemed so commonsensical that time was some sort of universal property, independent of the observer measuring them or her state of motion. Time measurements were said to be invariants.

These relationships between the location and time measurements made by observers moving with respect to one another were first postulated by Galileo. It is now known as ‘Galilean relativity’. Galileo used these relations to show why, even though the Earth was moving quite fast through space (a seemingly absurd idea at that time), a ball thrown vertically upwards would fall back down to the same point from where it was thrown, and not be displaced because the Earth had moved during the time that elapsed. This everyday observation had previously been used to argue that the Earth must be stationary but Galileo turned it around to show that it was consistent with the Earth moving.

But one consequence of the assumption that time is an invariant is that if you measure the speed of light (by taking two events, one consisting of light being emitted at one point and the other of it being detected at another point and dividing the difference in ruler readings between the two events by the time interval between the events), you would get different values for two observers in relative motion to each other, since the distances traveled (i.e., the differences in the ruler readings) would be different for the two observers but the time interval would be the same. In other words, the measured speed of light was not an invariant but depended on the speed with which the observer was moving.

What Einstein postulated (based on several reasons that I will not get into here) was that the speed of light was the same for all observers. In other words, it is the measured speed of light that is an invariant, the same for all observers irrespective of how they are moving. One important consequence of this is that the elapsed time between two events is no longer an invariant, and depends on the observer. Time is no longer a universal property but depends on who is measuring it. The difference in measured times is tiny for the normal speeds we encounter in everyday life, which is why we don’t perceive it. But it does leads to things like the celebrated ‘twin paradox’ where if you have a pair of identical twins, one remaining on Earth and the other going in a rocket at high speed to a distant star and returning, the traveling twin would have aged much less than the one who stayed home.

Needless to say, this caused some consternation and it took some time for people to be persuaded that this seemingly bizarre result was correct. What Einstein did was force us to be more precise about how we measure the location and time at which events occur, so that we can meaningfully compare the results of different observers viewing the same events.

Next: Measuring time and space more precisely.

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

To get a better grip on what is involved in the theory of relativity, we need to think in terms of ‘events’, things that occur instantaneously at a point in space and which every observer will agree happened and is unique. An example of an event might be me clapping my hands once. That occurs at one place in space (where my hands meet) at one moment in time (the instant they make contact) and all observers will agree that I did indeed clap my hands. Of course, actual events will be spread out over a region of space (my hands are quite big objects) and over a small but extended interval of time (the period during which my hands are in contact while clapping) but we can imagine idealized events as things that occur at a single point in space at a single instant in time. Specifying an event also uniquely specifies a location and a time since only one event can occur at any point in space at a particular time.

Suppose we have one event A that takes place at one place at one time (say a neutrino created by a nuclear reaction at CERN) and another event B that takes place at another place at another time (say the detection of the arrival of that same neutrino at the Grand Sasso laboratory). Einstein causality says that since event A caused event B, event A must take place before event B. Even if the neutrino were to travel at a speed greater than the speed of light, all that would do is reduce the time difference between the two events, not reverse their order, as was noted in the example given in the first post in this series. So why is this event seen as such a sensational development?

The answer lies in the fact that Einstein causality is believed to hold true for every observer who sees the same two events, irrespective of the state of motion of the observer. And the existence of faster than light neutrinos means that even though we on Earth will continue to see event A before event B, there are observers who are moving relative to us who will see the neutrino being detected at Gran Sasso before it was created at CERN or, more bizarrely in the case of the shooting example, that the bullet will emerge from person B and seem to travel back into the gun of person A. And unlike in that earlier example, this will not be due to an illusion due to the accident of where the observer happened to be located.

To understand how this can happen, we need to go more deeply into the question of how we measure the location and the time of events and how they differ for observers moving with respect to one another. Location and distance measurements seem pretty straightforward and we do it all the time when we measure the length of something. We simply hold a ruler along the line joining the two events, take the ruler readings at the locations of each of the two events, subtract the smaller reading from the larger, and the resulting number gives us the distance between the two events.

As for the time interval between two events, we can look at our watch when we see event A occurring and note the reading, then look again when we see event B occurring and note the reading, and once again subtract the smaller reading from the larger. The resulting number gives us the time that lapsed between the two events. There is a slight complication here in that it takes time for light to travel from one place to another so the actual time at which event A occurred would be a little earlier than when we see it. But since we know the speed of light, we can take that into account. All we have to do is measure the distance between where we are and the location of event A and divide that by the speed of light to get the time taken for the light to reach us. We then subtract that time from our watch reading to get the ‘true’ time at which the event A occurs. We can do the same thing for event B.

For example, in the earlier example, if you were standing next to the victim at B, you would have seen the bullet at the 2 meter mark 9 seconds after the gun fires. If you had been standing next to the shooter at A, you would have seen it 3 seconds after the gun fired. If you correct for the time of travel for the light to reach you from the bullet at the 2 meter mark, the bullet would be said to be at that point one second after the gun was fired, irrespective of where you were standing. So the time of an event can be specified uniquely in the case of different observers who are not moving with respect to the events.

What if the observer is moving, though? The question that Einstein pondered is the following. Suppose I, seated in my office, observe two events on the sidewalk outside my window and measure the distance between them and the time interval according to the above methods using my own ruler and watch. Now suppose another person is moving with respect to me (say passing by in a train) and sees the same two events as I do and measures the distance and time interval between them using her ruler and watch. Will that person’s measurements of the distance and time intervals agree with mine?

It is the answer to this question that determines whether we live in a world in which Galilean relativity rules or one in which Einsteinian relativity rules.

Next: Galilean and Einsteinian relativity

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

Neutrinos are very elusive particles that are produced in nuclear reactions. They interact hardly at all with anything, which enables them to penetrate anything easily. In any given second, tens of billions of neutrinos are coming from the Sun and passing though each square centimeter of our bodies and the Earth without doing anything, and heading off into the vast empty reaches of space on the other side. As a result of its extremely low interactivity with matter, it is hard to measure their properties, even basic ones like mass, because measurement involves getting the measured object to interact with the detector so that we know something about it. The existence of neutrinos was first postulated in 1930 as a theoretical device to explain missing energy in certain nuclear reactions but its elusive nature meant that it took until 1956 for direct experimental detection of their existence.

While the fact that neutrinos interact hardly at all with matter makes them hard to detect and discern their properties, this same elusiveness make them attractive candidates for measuring speed. This is because once produced they ignore everything in their path and travel in a straight line with constant speed so that measuring the distance traveled and the time taken does give you the speed. Even light is not as good for this purpose because both its speed and its trajectory are affected by the matter it passes through, as we all experience when we see how distorted things look when seen through glass prisms or bowls of water. Even slight changes in the density of the atmosphere can affect the path of light, which is the reason why we see mirages. So if you use light, the path taken by it in going from one point to another may not correspond to the straight geometric line distance connecting the two points that can be calculated once we know the coordinates of the two points, and so calculating the distance traveled by the light is not simple. But in the case of neutrinos, the path taken is dead straight and thus the geometric straight-line distance between two points will be the actual distance traveled by the neutrinos.

Another advantage is that the speed of neutrinos, unlike that of light, is unaffected by the medium it travels through. When light passes through glass or water, its speed is reduced which is the cause of the distortions we observe. As another example, take the light coming from the Sun. This light is produced as a result of nuclear reactions that produce both photons (particles of light) and neutrinos, among other things. But because the Sun is such a dense gas, it slows down light considerably and the photons produced at the core of the Sun can take as much as tens of thousands of years merely to reach the surface of the Sun, a distance of roughly 700,000 kilometers. Once there, it can travel freely in the vacuum of space to cover the remaining150 million kilometers to the Earth (over 200 times the radius of the Sun) in just over eight minutes. Neutrinos that are also produced in the core, however, travel almost as fast within the Sun as they do in the vacuum in space because matter is almost invisible to them. So if a neutrino and a light photon are produced in the same reaction in the core of the Sun, the neutrino will reach us long before the photon does.

Supposing the CERN-Gran Sasso experimental result holds up and the neutrinos are in fact traveling faster than the speed of light. Does this mean that Einstein’s theory of relativity is completely overthrown? No. Einstein’s theory does not rule out particles traveling faster than the speed of light. Such particles, known as tachyons, have always been allowed by the theory but we have never confirmed their existence so far. There have, however, been various false alarms in the past, which is part of the reason for the skepticism about the present claim.

What Einstein’s theory says is that if a particle has zero mass, then it travels at exactly the speed of light but if it has non-zero mass, then its speed can approach the speed of light but cannot attain it. Particles can approach the speed of light ‘from below’ (these are the normal particles we have experience with that always have speeds less than that of light,) or ‘from above’ (they always have speeds greater than that of light, and these are called tachyons that we have never shown to definitively exist), but neither can cross the barrier of the speed of light to the other side. So the existence of faster-than-light particles would not overturn Einstein’s theory of relativity completely since that theory always allowed for their existence, but would still be a momentous discovery because it would be a completely new phenomenon.

So does this mean that the existence of tachyons can be easily absorbed into existing knowledge? Not quite. The problem with the existence of tachyons is what it does to something known as ‘Einstein causality’, which is something that is connected to the theory of relativity, but is in addition to it. What this says is that if two events are causally connected, (i.e., one event causes another) then the cause must precede the effect. Going back to the commonly used bloodthirsty example, if person A fires a gun and the bullet enters person B, Einstein causality says that the firing of the gun by A must occur before the bullet enters person B because one caused the other. This seems eminently reasonable but we have to bear in mind that it is an assumption that is based on experience and, like all such assumptions, is subject to empirical scrutiny. If faster-than-light particles exist, the theory of relativity says that Einstein causality can be violated. i.e., effects can precede causes. It is this possibility, sometimes referred to as ‘going backwards in time’, that boggles the mind.

So how does the existence of tachyons violate Einstein causality? In the first post in this series, I gave an example where there seemed to be a situation of going backwards in time but said that this was not really so, because that was an illusion that arose due to the fact that we were dependent on when light from an event reached the observer.

To better understand what constitutes violations of Einstein causality, we have to get into the subtleties of what we mean by measuring distance and time, and this lies at the heart of the theory of special relativity. What Einstein did was make our understanding of how to measure distances and time more precise and operational, and in doing so altered our fundamental understanding of those two seemingly mundane concepts.

Next: Measuring time and space