Space and Time in Special Relativity
As I have been receiving very little feedback recently, I would like to ask my readers to answer, after reading this, if a) those who are not physicists found it understandable, and b) those who are physicists found it a passably good explanation of the issues.
So, let’s get started. The most important new feature about space that arises in Special Relativity (as opposed to Newtonian physics) is that space and time are no longer separated entities but part of a single entity, the much-heard about "space-time continuum". [Trivia question of which I do not know the answer: Does the use of this phrase in popular geeky culture, specially in phrases such as "to disrupt the space-time continuum", trace back to Doc's use of it in Back to the Future, or did it exist previously to the movie, perhaps coming from Star Trek?] But what does it mean exactly to say that space and time are joined in "space-time"? Many people think that it means only that time is the fourth dimension, that objects move through time as well as through space, and that to locate precisely an event one needs to give the full "space-time coordinates", that is four numbers three of which specify position in space and the fourth position in time. All of this is true but not very exciting, it is not the distinctive feature of Einstein's Special Relativity (it is true also in Newtonian physics) and it is not the reason why it is worth talking of "spacetime" as a single entity. The reason for this is rather that time can be "mixed" with spatial dimensions in the same way spatial dimensions can be mixed among themselves.
Imagine you are standing, looking forward with one arm pointing to each side. Your position defines a front-back direction, and a left-right direction which is orthogonal to the other one. ("Orthogonal" is math jargon for "perpendicular" -a strange case in which the jargon word is shorter and easier to say than the common one!) The distinction between these two directions is clear and real -but is relative to your point of view. For another person who is standing like you but rotated in some angle with respect to you, the front-back direction will be different from yours. For example if he is rotated a small angle to the right, then his "front" direction is a combination of your "front" direction and your "right" direction.
In just the same way, observers who are in different states of motion have different "time" and "space" directions. Suppose now you are standing like before, and the other person is not rotated but in motion, moving forwards with respect to you. Your standing defines in the 4-dimensional world a "time" direction and a "front" spatial direction (as well as the other two space directions which do not concern us now). For the other person, there is also a "time" direction and a "front" spatial direction, but these do not correspond to yours. His "time" direction is a combination of your "time" and your "front" directions, in much the same sense as the rotated person's "front" is a combination of your "front" and your "right" directions.
There are more analogies between both cases. If you and a friend standing just by your side but rotated with respect to you calculate the distance to something, you will find the same answer even though you "split" it differently among the different directions. Perhaps what is 5 meters forwards for him will be pehaps 3 meters to the right and 4 meters forwards to you, but the "real distance" (5 meters in this case) is the same for both of you. Pythagoras' Theorem ensures that if dx and dy mean distance forwards and distance sideways for you, dx' and dy’ mean distance forwards and sideways according to him, and ds is the "true distance", then it will be true that dx^2 + dy^2 = dx’^2 + dy’^2 = ds^2. (Make a drawing if you are not convinced. In the particular example before dy' was 0, but it need not be. Of course, "^2" means "squared".) You and your friend see a different forward distance and a different sideway distance to the point you are looking at, but the total distance ds comes out the same. If we allow your friend to be also tilted in the vertical direction, and we look at points that are not in the same plane as yours, the formula generalizes to Pythagoras' theorem in three dimensions, dx^2 + dy^2 + dz^2 = dx’^2 + dy'^2 + dz'^2= ds^2, where dz is the vertical distance according to you, and dz’ the vertical distance according to him. The important thing is that there is an absolute "spatial distance" which can be split differently in a forward part, a sideways part and a vertical part for different observers, but that comes out the same for every observer.
As long as they are not moving, that is. Let's go back to the case when your friend is moving forwards with respect to you. Suppose you and your friend have both rulers and clocks, to measure both spatial and temporal distances between events. Then between any pair of events, you will measure a certain spatial distance ds (which comes from the three separate spatial distances in the three directions as we saw) and a temporal interval dt. Your friend will measure a different spatial distance ds, and a different temporal interval dt’. But both of your measurements will add up to the same "absolute spacetime distance" DS, defined by the formula: ds^2 – dt^2 = ds’^2 – dt’^2 = DS^2. The only difference with the usual Pythagorean formula is a – replacing a +. (After all, some difference exists between time and space!) But what matters is that, just as "forward distance between points" and "sideways distance between points" are not absolute concepts but depend on which way the observer is looking (but the "spatial distance between points" does not depend on that), it turns out that "spatial distance between events" and "temporal interval between events" are not absolute but depend on how the observer is moving (but the "absolute spacetime distance" does not depend on that). This mixture of time and space is the defining characteristic of the Special Theory of Relativity.
It is easy to see from this how other celebrated fact about relativity emerges, the fact that the speed of light is an absolute. The formula above for DS^2 subtracts a time squared from a distance squared; this would not be possible if we were not measuring time and space in the same units. To use the same units for space and time, there must be a velocity which is absolute and serves as conversion factor. This velocity is the speed of light. The formula for DS^2 is correct if we measure for example time in seconds, and distance in light-seconds, defined as the distance traveled by light in a second. (If we were measuring time and space in the usual units, like meters and seconds, the formula would be DS^2 = ds^2 – c^2 dt^2, the appearance in it of the speed of light c showing that it is an universal constant of nature.)
There is one last question my friend made to me, and which I have not answered so far. OK, there is a difference between time and space in that time has a – in a formula where the three spatial dimensions have a +. How does this relate to the ordinary distinction between space and time, to facts such as “we can go backwards in space but not in time”? What other differences between space and time follow from this sign difference? Do ALL of them come from it?
I think that the answer to the last question is yes: all differences between time and space must ultimately be traced to this sign difference. But that does not mean I can spell out a complete answer for the other two questions, partly because the conceptual status of our ordinary notion of “flow of time” is not very clear; it is something related to philosophy and psychology as well as physics. But some of the differences between time and space can be cleared by physics and made more intuitive than just a minus sign. A following post will explain these issues. After that, we’ll be ready to get into General Relativity.