Special Relativity and Common Sense
In this little paper I will try to reconcile some of the unintuitive consequences of Einstein's Special Relativity Theory with our ordinary senses of time and space.
An Apparent Problem
Suppose that Joe is kayaking on a big whitewater river, and is surfing a wave in the upstream direction. He feels as if he's moving very fast, but, in fact, he's holding a position right above the rock which produces the wave. At other times, still pointed upstream and tilted forward at the same angle, he's carried rapidly backward on a wave in deeper water. He may then feel as if he's not moving when he really is. Suffice it to say that, without looking at the distant shore (which he's too busy to do), it's hard for him to tell whether he's moving or not.
Sam, in another kayak, lacks the coordination required for surfing, but he's strong enough to paddle upstream, overcoming the current. At other times, he paddles slowly, allowing his boat to be taken downstream even though it's still pointed upstream. In circumstances like these, kayakers really do get confused as to who's moving in which direction and who isn't.
The situation becomes much more complex when we apply the relativistic concepts of time and speed to the paddling of Joe and Sam. Brian Greene, in The Elegant Universe, gives one of the best accounts of relativity for the non-physicist (such as myself), and I will borrow here his example of a light-clock. This device consists of two small mirrors fixed on a bracket so as to face one another. Light bounces back and force between the two mirrors, but Greene considers only one photon. Each round trip between the mirrors counts as a tick of the clock, and what he calls the world's most impractical clock could, in theory, be used by Joe and Sam. Let's suppose that they have light-clocks fixed next to the compasses on their boats, and that each has the amazing perceptual ability to watch an individual photon in either clock and count round trips.
Sam, believing himself to be exactly compensating for the current, sees that Joe is slowly passing him, and is therefore working his way upstream through the rapids. The light-clock on Joe's boat differs from, say, a ping-pong ball bouncing beteen the mirrors. If a ping-pong clock were started at rest, the ball would be left behind if the clock were to be put in motion. If it were started when the clock is already moving at a constant speed through a vacuum, the ball, having its own forward momentum, would keep up with the clock. However, a photon is not a ping-pong ball, and one might expect the one in Joe's clock to be left behind as he moves forward.
But most things are moving with respect to most other things. Even if Sam is stationary with respect to the river bank, he and the bank are moving rapidly with respect to the sun and moon, and, given continental drift, more slowly with respect to, say, the North Pole. If Joe's photon is to be left behind, so ought Sam's. Indeed, if photons are to be left behind because their clocks are moving with respect to something or other, it would be impossible for there to be light clocks.
In fact, the notion of a light clock makes intuitive sense. This is the sort of thing we're willing to take as a premise in our reasoning. It follows that both Joe and Sam can see that Joe's clock keeps running. Sam concludes that Joe's photon is following a slightly diagonal course in order to keep up. Since it then has farther to go from one mirror to another, Joe's clock, from Sam's perspective, is very slightly slow. In a similar way, all of Joe's actions, keeping time to his clock, will be very slightly slow.
At this point, Joe paddles faster and Sam gets confused. If Joe's faster paddling makes his boat go faster, then his clock will run slower, and he'll go slower. How will he get through the rapids?
This is one confusion that can easily be laid to rest. Joe's boat does go faster when he paddles faster, with the consequence that his clock, and everything else, will run slower. But this difference will be so tiny as to have no practical effect. Joe's speed won't be quite what we would calculate it to be if we didn't take account of this phenomenon, but it will easily be enough to take him through the rapids.
Let's now move to Joe's perspective. Having made his way through one set of rapids, he now thinks that his paddling is only "keeping him on a standing wave". He knows that Sam sometimes lets his boat be taken backwards, and he concludes that Sam is now the one who's moving. Hence, Joe, following the same reasoning, will conclude that Sam's clock, which isn't leaving its photon upstream, is running very slightly slow.
By this time, it's clear that it doesn't really matter who's moving relative to the bank and who isn't. Joe and Sam are certainly moving in the opposite direction relative to one another, and, from the perspective of either, the other's light clock must be running slow.
How, then, can Joe's clock run slower than Sam's while Sam's clock is running slower than Joe's?
This seeming paradox can be sharpened in the following way. One doesn't have to count ticks, or anything else, to see that the clocks are behaving in different ways. Suppose now that Sam and Joe are exactly even with each other as they pass. From Sam's perspective, the angle with which Joe's photon hits the mirrors of his clock will not be a right angle. And vice versa. But both Joe and Sam will perceive their own photons hitting their mirrors at right angles. We can thus state the problem without referring to time in an explicit way. Is an angle a right-angle or not?
This way of thinking naturally suggests a common experience. A penny looks circular when viewed at right angles, but elliptical otherwise. And this is a matter of perspective, not of psychology. We can imagine a naive soul asking, "Well, is it circular or not?"
In the present case, there is, of course, another jump. An angle, even when viewed from the same angle, will be assigned a different number of degrees because of relative motion. If we could "stop time", it would be contradictory for the angle to be both a right angle and not one. But, since we can no more stop time than we can repeal altitude, we can't really keep time out of consideration.
Let's now try another analogy. In an ordinary two- dimensional Mercator projection, the islands of Iceland and Java are about the same length in the east-west dimension. But, when we add a dimension and go to a globe, Iceland is seen, more accurately, to be much smaller. Moreover, any two- dimensional map will have similar distortions.
The addition of a third dimension overturns equalities which were previously thought to hold in a two-dimensional map. It would thus be surprising if the addition of a fourth dimension, whether we think of it as spatial or temporal, did not upset previously held three-dimensional equalities. One of Euclid's axioms was that all right angles are equal, and we must now add the proviso that they not be in relative motion with respect to one another.
Does this sort of thinking make us more comfortable with the situation of Joe and Sam on the river? I think we still have a sense of mystery, but, perhaps, are no longer so mystified that there is a mystery. Let's try another analogy, this one from ancient history.
In Plato's Theatetus, it is seen that he and Socrates, and their friends, were concerned that water in a bucket could be pronounced cold by a person sitting in a warm room and not cold by someone coming in from the outside. The answer was that 'cold' is a relative term with an implicit standard of comparison. The underlying form is not, 'x is cold', but, 'x is colder than y'. When the two people seem to disagree, they aren't really. They are comparing the same object, not to a single other object, but to two different objects.
The moral here is that seeming contradictions can be resolved by adding another term to the predicate. This, of course, is very much in the spirit of relativity. The added term in the relation would seem to be "from the perspective of z". So, Joe's photon-angle is a right angle from Joe's perspective, but not from that of Sam. Hence, Joe's photon is bouncing more slowly than Sam's from the perspective of Sam, but not from the perspective of Joe.
There is no general problem with relations of three or more places. We understand 'x gives y to z' and 'x sells y to z for w' readily enough, and we can even form images of people handing things to one another as instances of those relations. So far so good. But what is perspective?
Perspective is ordinarily a spatial relation, as when one sees a building at a particular angle from a particular distance. With Special Relativity, all that matters is the speed of the object relative to the observation point, although spatial relations will be required to calculate that speed. This, in itself, is not an unacceptable bending of the term "perspective". However, the concept of perspective seems to somehow involve the concept of observation, and this is a human factor which must be further examined.
Before discussing further the role of observation in perspective, we can eliminate mere belief. Suppose that Sam, who happens to be stationary relative to the river bank, believes himself to be going backwards. Further, his perception, which turns out not to be foolproof, indicates that his own photon is tracing a diagonal course. None of this makes any difference. One's own photon is always going straight up and down from one's own perspective, and it doesn't matter what one believes about one's relative motion, or even what one senses. This is worth pointing out because the possible relevance of observation in perspective need not bring into play the almost random psychological phenomena associated with belief. The projection of a tilted penny on one's retina really is elliptical, regardless of one's beliefs about the shape of the penny. It's relative motion, not beliefs about relative motion, that count.
A Hypothetical Solution
Let's now follow Joe and Sam as they shed their helmets, PFDs, and wetsuits to deploy on to a nearby golf course. On the second hole, Joe misses a short putt and exclaims, "Shit! I coulda made it!"
Sam, a little testy because of Joe's hot-dogging it on the river, replies,
"Whaddaya mean, coulda made it? It didn't even rim the cup."
It should be noted here that Sam's reply, which amounts to a demand for explanation, is perfectly appropriate. Joe implicitly admits as much when he says,
"I was thinking of that terrific ender I had at Great Whirlpool, and I didn't move the putter straight through. That's all you have to do to make such a short putt."
Of course, it's not literally true that one has to do only one thing to make a putt, but one can see the explanation being sketched out. Sam, already having heard enough about Joe's kayaking prowess, might protest the explanation,
"Nuts! If you had a sudden pain in your balls, I could see. But you're just a bad putter."
The argument here is a purely empirical one of the ordinary sort about causes and effects. A different action might have had a different effect. It might then be claimed that a sentence such as, "I could have made it" actually has a meaning something like,
"If I had concentrated mentally in a certain way and moved the putter in a certain way and ..... and, the ball would have fallen in the hole."
The list of 'ands' would be fairly long, and would vary with the context and the speaker, but it would presumably be finite. The claim would then be that a categorical statement with no 'ifs' can be analysed into one or more hypothetical statements. These are counterfactual in the sense that Joe didn't concentrate and the ball didn't drop. We have him saying what would have happened if he had done certain things he didn't do.
Is this what Joe, or anyone in his situation, would really mean?
John Austin, in this case, argued to the contrary. So, earlier, did Isiah Berlin. Their view was that no set of hypotheticals can be made to do the work of a categorical statement. Austin said that "I could have..." doesn't mean "if anything" and simply attempts to state a fact.
This is the sort of philosophical dispute that never gets settled because, among other things, it presupposes different notions of the meaning of "meaning." A lot of people don't like hypothetical and counterfactual statements, but I have extensively argued elsewhere that, whatever we do, we can't avoid them. Let us therefore see how an analysis in those terms would apply to the present case.
Joe's assertion that he could have made the putt isn't a necessarily true statement of logic or mathematics, and hence might be false. Perhaps Joe couldn't have made it. The polluted river water might have had an after-effect which affected his vision and made it hard for him to line up a putt no matter how hard he concentrated. There is evidence that can be brought to bear, but confirmation is usually more difficult than it is for simple predictions.
Now, back to the river. The paradox we began with was that Sam's clock seemed to be running slower than Joe's while Joe's was running slower than Sam's. Under a hypothetical analysis we will get,
(1) If one is in Sam's position relative to Joe, one will observe that Joe's photon is not hitting his mirrors at right angles, so that his clock is running slow.
(2) If one is in Joe's position relative to Sam, one will observe that Sam's photon is not hitting his mirrors at right angles, so that his clock is running slow.
These propositions certainly do not contradict one another, or even give us a sense of paradox or mystery, and they are closely related to the difference in perspectives mentioned earlier. There is, however, a logical difference. When we spoke of an object, x, being y from perspective z, we were speaking of a three-way relation with no explicit hypothetical element. For example, 'x is between y and z' asserts a three-way relation to hold in a categorical way. By introducing the hypothetical element explicitly, we have begun to explain what "perspective" might consist in.
There is, in fact, a scientific tradition of using hypothetical statements to explicate concepts such as temperature. One can say, for example, that the temperature of a liquid will be ninety degrees centigrade in exactly ninety minutes. That is taken to mean:
(3) If it is measured at that time, the temperature reading (on agreed standard instruments) will be ninety.
This prediction can, of course, be confirmed or disconfirmed.
One can also say that the liquid was the same temperature ninety minutes previously, even though it was not being measured at that time. The claim would then be:
(4) If it had been measured at that time, the temperature reading would have been ninety.
Here, we have the counterfactual conditional which can be confirmed or disconfirmed only with indirect evidence, and the assignment of a probability to (4). This sort of interpretation has always made people nervous, but no very satisfactory alternatives have been found.
In fact, the case of past unmeasured temperatures reflects a much older debate, that centering around the notorious tree falling unobserved in the jungle. The hypothetical account, usually known as "phenomenalism", would amount to something like:
(5) If anyone had been in the "right" position in the jungle, he or she would have observed the crashing of the tree.
Of course, there are no "standard" ways of detecting tree- falling, as opposed to temperature. Thus, we would have a whole family of statements along the lines of (5). Moreover, it is part of phenomenenalism that (5) and its siblings would have to be expanded to the point that all its terms refer only to sensations. But, even in the case of (4), we would ultimately have to talk about the sensations of people reading temperature gauges.
Needless to say, phenomenalism is not universally accepted. But it does represent the logical extreme of the sort of hypothetical analysis which can be used to counter some of the Special Relativity "paradoxes."
Some Limiting Cases
When we move beyond the fairly ordinary cases of Sam and Joe and the tree falling in the jungle, some special problems arise.
Some hundred years ago, a Dr. Crippen, a physician practicing in the North London suburb of Camden Town, found it necessary to murder his wife. After thoroughly washing his hands and sterilizing his instruments, he cut her throat with his scalpel when she was asleep. He then cut her into little pieces, and flushed them down the toilet. According to the standard phenomenalist analysis, the claim that he had done so would be something like:
(6) If anyone had been in position, he or she would have observed a Crippen-like figure severing the head of a woman, etc.
However, as has been pointed out by Sir Freddie Ayer, Dr. Crippen would not have proceeded with his work, or even murdered his wife in the first place, if someone had been, say, taking tea some twenty feet away. It has been suggested, half humorously, that, if one had had the body of a mouse, one might have been able to watch the doctor without putting him off his game.
In the case of Special Relativity, Brian Greene points out that, when muons are accelerated to very high speeds, their rate of decay is radically slowed. That is, their other processes, including decay, are slowed along with their clock speeds. But this is relative to the position of the observer sitting beside the accelerator ("his perspective").
From the muon's perspective, the observer is going very fast in the opposite direction, and his clock speed, and his actions, are similarly slowed. In fact, the observer's senility and ultimate decomposition is slowed just as much as that of the muon from the other perspective. The question now is whether a muon, not noted for its ability to make observations, can have a perspective.
Greene complains, perhaps jokingly, about "excessive anthropomorphism" in the case of muons, and it is probably true that scientific theories can be formulated without addressing such questions. But, for those of us trying to reconcile common sense with relativity, they cannot be avoided.
If it seemed a stretch to talk about being a mouse in order to observe Dr. Crippen, talk of being a muon appears to go beyond all bounds. The situation, however, isn't quite that bad.
First, the antecedents of hypothetical counterfactual statements don't have to describe physical possibilities. Geographical and climactic conditions would have prevented anyone then alive from observing a snowfall near the South Pole in the year 1585. Even Sir Francis Drake would have respectfully declined such a request from Queen Elizabeth.
Second, one doesn't really have to imagine being a muon. One has only to imagine being stationary relative to the muon (which can also be thought to be "stationary" if it makes one feel better) while other things are speeding away at nearly the speed of light. As before, it doesn't matter who's speeding away from whom.
Of course, physical impossibilities remain. The observer, sitting with a white jacket and a clip-board, cannot be accelerated to nearly the speed of light. Still, in defense of this general position, it should be pointed out that we could hardly get through an ordinary gossip session concerning our friends and relatives without talking about physical (including mental) impossibilities.
In conclusion, it looks as if we've substituted one kind of difficult thinking for another. The hypothetical analysis does resolve the original feeling of contradiction. Instead, we have to take seriously possibilities that are just barely imaginable. Is that an advance?