Relativity-5: Galilean and Einsteinian relativity

(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.

Relativity-4: Measuring time and space

(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

Relativity-3: The elusive neutrino

(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

Relativity-2: The CERN-Gran Sasso experiment

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

The nice feature about the experiments involved in the recent reports of faster than light neutrinos is that the basic ideas are so simple that anyone can understand them. It involved producing neutrinos at the CERN laboratory in Switzerland and detecting them at the Gran Sasso laboratory in Italy. By measuring the distance between the two locations and the time taken for the trip, one could calculate the speed of the neutrinos by dividing the distance by the time.

The measured distance was about 730 km so if we take that as the exact value, and if the neutrinos were traveling at exactly the speed of light (299,792 km/s), the time taken would be 2.435022 milliseconds (where a millisecond is one-thousandth of a second) or equivalently 2,435,022 nanoseconds (where a nanosecond is one-billionth of a second). What the experimenters found was that the actual time taken was 60 nanoseconds less than this time, which seemed to require the neutrinos to be traveling slightly faster than the speed of light. Since the existence of faster than light particles has never been confirmed before, this would be a major discovery and so the search is now underway to see if this conclusion holds up under close scrutiny.

If the experimental results are at fault and the effect is spurious, this must arise from errors in the distance measurement and/or the time measurement. Although the time difference that produced the effect is very small (60 nanoseconds out of a total travel time of over 2 million nanoseconds constitutes only about 0.0025% of the total time), the experimenters say their time measurements are accurate up to 10 nanoseconds, much less than the size of the error needed to resolve the discrepancy, thus ruling that out as the source of error. Similarly, if the actual distance were less than the measured distance by just 18 meters, the effect would again go away. The experimenters used GPS technology to measure the space and time coordinates of the events and say that their experiment can measure distances up to an accuracy of just 0.2 meters, making that too an unlikely source of any error. As for the possibility of some kind of random statistical fluctuations causing the effect, the number of neutrino measurements they have taken over the past two years exceed 16,000, which makes that highly unlikely as the source of error.

So why is there still skepticism? It is because the very feature of neutrinos that makes this experiment so conceptually simple is also what makes it so difficult to rule out what are called systematic errors. These are artifices of the experimental setup that can bias the results consistently in one particular direction, unlike random errors that can go either way and can be reduced by repeating the experiment a large number of times, as was done in this case. Unearthing systematic errors is difficult and time consuming because it depends on the esoteric details of the experimental set-up. What some other groups will now try and do is identify possible sources of systematic errors that the original experimenters did not consider, while others will repeat the experiment with different experimental set-ups, measuring the time and distance using different techniques so that the likelihood of systematic biases pushing the results in the same direction is reduced. Yet other groups will examine if any of the side effects that would automatically accompany faster than light travel are also seen. It is this kind of investigation for replicability and consistency that characterizes science.

But getting back to the original experiment, the reason that neutrinos are good for measuring velocities that may exceed the speed of light is that they usually travel at speeds close to or at the speed of light. If a particle has zero mass (as is the case with ‘photons’, the name given to particles of light), then according to Einstein’s theory of relativity, it must travel exactly at the speed of light. If it has a mass, however small, it can approach the speed of light but never attain it because to do so would require an infinite amount of energy. But it takes less energy to accelerate lighter particles to high speeds than it does heavier particles.

In the case of neutrinos, we have not been able to directly detect them having any non-zero mass as yet. All we have been able to do so far is put a small upper limit on the amount of mass it can have, which is 2 eV/c2 which is about 3.5×10-33 kg. (By comparison, the particle with the smallest mass we know, the electron, has a relatively huge mass of 511,000 eV/c2.) It had long been assumed that the mass of the neutrino was exactly zero. But it turns out that there are three kinds of neutrinos and that they may oscillate from one kind to another as they travel through space, and the postulated mechanism for such oscillations require that they have non-zero mass. The purpose of the CERN-Gran Sasso experiment was to actually look for such oscillations, and it just so happened that it turned up the evidence that neutrinos may be traveling faster than light, completely shifting the focus of attention. Such accidental discoveries when looking for something else are not uncommon in science, the discovery of X-rays being one of the more famous examples.

Next: The elusive neutrino

Relativity-1: Going backwards in time

Part of the reason that recent reports of the detection of neutrinos traveling faster than the speed of light aroused such excitement is because of claims that such a discovery would overthrow Einstein’s venerable theory of relativity and that if you could send a signal faster than the speed of light, you could go backwards in time. Are these claims true or simply overheated? If true, what exactly was overthrown? And what does it mean to ‘go backwards in time’ anyway?

My initial reaction to the faster-than-light neutrino report was one of skepticism, saying that I would wait and see if the result held up but was not hopeful that it would. I did not give my reasons for this pessimism and reflecting later, I thought I should because understanding what was claimed (and why) serves as a good vehicle to understand the elements of the theory of special relativity as well as how science works., so the next series of posts will deal with these questions. (I was overdue for a series of posts on a single topic anyway.)

Let’s look first at the ‘backwards in time’ claim. There is a simple (but wrong) way of interpreting this and a more subtle (but correct) way.

To see the simple way in which something traveling faster than the speed of light can cause things to appear to go backwards in time, think of a situation in which a man fires a gun at another man but with the bullet traveling faster than the speed of light. Nothing requires the shooting of people to understand this phenomenon but this is the customary example that is used, perhaps because a bullet is the fastest object that most people can think of (although it is still much slower than the speed of light) combined with the fact shooting someone is so dramatic and final that reversing the process seems impossible, kind of like Jesus rising from the dead.

Suppose the shooter is at point A and the person hit is at point B 10 meters away. Suppose you are standing right next to the person at B. If the bullet travels faster than the speed of light, what will you see? Remember that we ‘see’ something only when the light from that event enters our eyes. Since the speed of light (at 299,792 km/s) is beyond anything we are familiar with from our everyday experiences, let’s greatly slow things down by assuming that it travels at (say) 1 m/s and that the bullet travels at (say) 2 m/s.

You will see the gun at A firing 10 seconds after it fires because the light from that instant will take that much time to travel the 10 meters to reach you. But one second after the gun is fired, the bullet will have traveled two meters towards B (and you), and light emitted by the bullet at that point will take only 8 more seconds to reach you. In other words, you will see the bullet at the 2 meter point 9 seconds after the gun is fired, which is one second before you see the gun firing. Similarly you will see the bullet at the 4 meter mark 8 seconds after the gun fires, at the 6 meter mark 7 seconds after the gun fires, at the 8 meter mark 6 seconds after the gun fires, and the bullet entering the person at B 5 seconds after the gun fires. Put it all together and what you see first is the person at B being hit (five seconds after the gun fires) and then in the next five seconds will see the bullet emerging from the victim and traveling back and entering the gun.

This no doubt looks like is going backwards in time. But this example is not what is meant by going backwards in time according to the theory of reelativity. After all, the victim was in fact hit five seconds after the gun was fired so there is no actual reversal of the ordering of the events. What you saw is more like watching a film run backwards, which is not really going backwards in time. This effect is an illusion, an artifice caused by the fact that light takes time to travel and your special location next to the victim. Had you observed the whole sequence of events while standing next to the shooter at A, you would not have noticed anything unusual because you would have seen the gun fire right at the beginning, the bullet at the 2 meter mark after 3 seconds, at the 4 meter mark after 6 seconds, at the 6 meter mark after 9 seconds, at the 8 meter mark after 12 seconds and hitting the person at B after 15 seconds. Everything would have seemed normal.

What this example does illustrate is that specifying the time at which an event occurs by the time noted by an observer is not satisfactory because it depends on where the observer is situated relative to the events. (For example, the bullet was observed at the 2 meter mark at 3 or 9 seconds after the gun was fired depending on where you were standing.) We will also see later that in addition to the location, the state of motion of the observer (if you were observing the events from a moving train, for example) also affects the time at which they see events.

It is in trying to unambiguously pin down exactly when something happens that we arrive at a deeper understanding of Einstein’s theory of relativity and what we really mean by going backwards in time.

Next: The CERN-Gran Sasso experiment