## Minkowski and Spacetime

Herman Minkowski was one of Einstein's teachers at the ETH (Eigenossiche Techniche Hochschule) in Zurich. He was never particularly impressed with Einstein, finding him disruptive and seeming uninterested in the material. While Einstein did well in theoretical Physics and Mathematics, he was not impressive either in the experimental courses, or the other courses.

But Einstein's papers of 1905 came like a bombshell to him and to everyone in the Physics universe. In each case, the approach was radical, and completely unorthodox, but clearly to be taken very seriously. Minkowski immediately began to study especially the Relativity papers and in 1908 produced a highly significant reinterpretation of what Einstein had done.

### Rotations

Let us first consider ordinary space and rotations in that that space. One can analyse the location of objects within space by specifying three numbers, usually designated as length, width and height. These are three so called axes, often designated by x,~y,~z. These three numbers are sufficient to locate any point in space one would care to discuss. There is however an arbitrariness to these labels of location. The first is where to place the point x=0, y=0 and z=0. It is clearly arbitrary, but to make sense of what say x=5, y=7 and z=-13 means one has to specify where x=0, y=0 and z=0 is. The second point is that although the x,~y,~z denote distances in perpendicular directions, there is no specification as what directions they refer to. Thus you could take x to be measurements along the long side of the room, y to be measurements along the short side, and z to be the height. Or x could be the height, y the long side distances, and z the short side. Or x could be along one of the diagonals, y the horizontal direction perpendicular to that, and z the height. Or an infinity of other possibilities.

However, one of the things that is independent of how you "coordinatize" the world, one thing remains that same, and that the distance from one physical place to another. No matter how one defines the x,~y,~z, that distance should be the same.

To simplify things let me work in two dimensions instead of three, like the surface of a sheet of paper. One can define the x and y coordinates by picking an arbitrary two perpendicular directions, and then measuring off distance on whatever units one wants (eg meters). Let us take two of them and call them x y and another one x' y'. If we choose the x=0, y=0 to designate the same point as x'=0, y'=0, then one says that one of these systems is a rotation of the other. None of these systems of coordinates is better than the other. Sometimes one might be more convenient than the other, but it is purely convenience and not that physics cares which one we choose.

The distance between two points labelled by x1,y1 and x2, y2 will be given by Pythagoras's theorem
distance squared = (x2-x1)squared +(y2-y1) squared.
But this must be the same as
(x`2-x`1)squared +(y'2-y'1)squared.
because both of these expression designate the same distance. Distance is a quantity which is independent of how one expresses the location of the points that one wants the distance between. Note that in general x2 and x'2 are different, y2 and y'2 are different, and similarly for the first point, but the distance is the same. The coordinates are a useful way of specifying things, but has no real meaning on its own. But distances are physically meaningful.

Minkowski realised that all of Einstein's work could be summarized by a generalisation of the above. Instead of regarding space as an entity on its own, and time represents a movement though this space, let us regard time as also a coordinate just like the three spatial coordinates are. Time and space together are a way of giving a location to a point, and event. They tell us both where and when something happened.

This in itself is nothing radial. People had been drawing diagrams showing how things moved in space and in time for many centuries. It was the next item that was revolutionary. Minkowski said that one could define a distance, in the 4-dimensional space-time, just as there is a distance in space. And this distance is something that is independent of how one defines the coordinates. This distance however has a peculiarity, namely that the equivalent of Pythagoras theorem has a minus sign in it. If one has two points labelled by t1,x1,y1,z2 and t2,x2,y2,z2 which designates two events (where and when) then the distance between them is given by
distance squared = (x2-x1)squared +(y2-y1)squared +(z2-z1)squared - (c(t2-t1))squared
Notice the different sign, the - sign, in front of the time part of this distance. He argued that all of Einstein's special relativity could be summarized in this formula. Physics was about these special kinds of distances. While the coordinate difference was a sort of distance, it really had not significance of itself. Those kinds of distance were called either spatial coordinate distances or temporal distance. But the distances given by that weird Pythagoras's formula were proper distances, and it was proper distances that were physical.

Notice that that weird minus sign means that the distance squared can be either positive (if the spatial terms dominate) negative (if the temporal term dominates) of zero. One usually thinks that if two points have zero distance between them, that they are the same point. But in this case, that is not true. Let us imagine that y1=y2 and z1=z2, then the distance squared is
(x2-x1)squared -(c(t2-t1))squared=0
or
(x2-x1)squared= (c(t2-t1))squared

One possibility for this is x2-x1=c(t2-t1) of that the spatial distance between the points, x2-x1 is the velocity of light (c) times the time difference. But this is just what would happen if something was travelling at the velocity of light. The distance would equal the velocity of light times the time. Thus two points in spacetime which have a zero proper distance between them are two points which cold be connected by the path of alight ray travelling between them. Zero proper distance then means not that the two are the same points, but that the two points could be connected by a light ray.

Minkowski then said that Einstein was saying that clocks would measure the proper distance along any path through spacetime that they traversed. Ie, plot the path (the location x.y,z at every time t), and calculate the length using the weird Pythagoras's expression for every little line segment of that path and add them all up. This expression was what clocks would measure. Clocks are like odometers for spacetime. One of the features of odometers is that the distance they measure depends on the path that they follow. If you drive from Vancouver to Toronto, and you go via the Trans Canada or you take a side trip to Austin, Tx, the length of your trip will depend on which path you take. Similarly with clocks. If you travel on a wiggly path from the same place but at different times, the distance measured by the clocks will be different. If you travel in a straight line (Ie, you stay at rest at the same point so that only the time changes, then the distance squared will be just -(c(t2-t1)) squared. (usually for timelike curves, curves where the distance squared is negative, one reverses the sign and talks about "proper time" rather than proper distance.) If however you follow a wiggly path, then you will in addition get a contribution from terms like (x2-x1) and the proper time squared will be
(c(t2-t1) squared -(x2-x1)squared.
Notice that that spatial part makes the proper time squared less. A straight path from one point to another whose separation is timelike has longest, not the shortest proper time between the two points. For spatial curves the straight path is the shortest distance. For temporal spacetime paths, the straight line is the longest time. Since your body's aging is a form of clock, if you just stay at home, you will age more than if you move around (especially if you move around at near the velocity of light.)

This is often called the "Twin's Paradox". There is nothing paradoxical about it, any more than it is paradoxical that driving from here to Toronto is a different distance if you drive via the Trans Canada than it is if you drive via Huston.

Of course the paradox comes in if you believe in Newton's time-- time being uniform everywhere, and is independent of anything that physical things do. Many many many people have that Newtonian prejudice so deeply ingrained that they find it impossible to give up. For them Relativity sounds non-nonsensical. But looked at from Minkowski's point of view, it is almost trivial.

## Spacetime Diagrams We can draw the geometry of the spacetime by giving graphs of the various aspects of the geometry.

In this figure we draw the diagram of the original spacetime with the positions (as distances from x=0) and times. In this case we are assuming that we have a object at rest at x=0. It is at rest, so it will remain at x=0 for all times. In addition consider another object whose path is in blue. It is travelling with uniform speed in the x direction. The larger the time t, the further away it is from the object at rest. I am going to assume that the units of both time and space are chosen so that the velocity of light is one unit of space in one unit of time. (Eg, measure distances in light seconds. It define 300000km as being one ls (light second) unit. Then light travels one 1 ls per second. We plot the diagram in terms of ls units for space and seconds for time.

The moving object is travelling at a speed is less than that of light.

Now assume that one has a traveller travelling with the object (eg the object is on a train which is travelling at the speed of the object and there is a passenger physicist on the train.) He now sets up other clocks which do not move with respect to him, and synchronize the clocks with the usual procedure-- ie by sending light signals to do so, and subtracting off the light travel time to find out when the same time as the time when the light signal was sent out was. The horizontal blue light represents the line in space and time which is, for this travelling physicist, the same time as t'=0. Note that the synchronization is not the same as that for the non-travelling observer.

consider the piece of moving path of length Dt'. By the weird Pythagoras thm for spacetime, the piece D't squared= Dt squared - Dx squared. Ie, Dt' is smaller than Dt. The moving clock ticks more slowly than the stationary clock from the viewpoint of the stationary physicist.

Now lets look at a stationary ruler. This is graphed by the shaded region. If a ruler is moving, there is a crucial feature of defining its length, namely one needs to figure out what the location is of each end of the ruler at the same time Since what "at the same time" means is different for the two observers, this highlights one of the problems with the lengths. For the moving observer, the length of the stationary ruler is given by the intersection of the rod with the line which represents "the same time" for the moving rod. Then the length of the measuring stick as determined by the moving physicist will be DX' squared= DX squared - DT squared. The moving observer will just say that the length (DT')of that rod (which he sees as moving) is shorter than the observer at rest will say it is (DT). This is both because the concept of simultaneity is different and because distances behave that weird Pythagoras's law.

In trying to understand Special relativity, many people have come up with to them seem to be paradoxes ( and which some of them use to try to argue that special relativity is wrong and ridiculous).

### Rod in the barn. A pole vaulter has a long pole and he has a barn in which he wants to store the pole. Now, if he stands beside the barn with the pole, the pole is longer than the barn and so will not fit. He therefore races toward the barn with the pole toward the open door. From the point of view of the physicist at the barn, his pole is shorter because it is moving. If we assume it is moving fast enough (close to the velocity of light) the pole for him is shorter than the barn. Thus as the pole vaulter runs into the door, one can close the door with the pole entirely inside the barn. But from the point of view of the pole vaulter, it is the barn that is shorter, and there is no way that pole can fit in the barn. This is supposed to be a paradox. Of course the the two people's notion of simultaneity is different. A crucial feature of the above story is that when the barn physicists says that pole fits in the barn, he means that at some instant of time, both ends of the pole are inside the barn. But for the pole vaulter, the notion of simultaneity is different. When "the same time" is means different things to him. We can draw a diagram of the problem. The red shaded region is the barn, and the grey is the pole. The pole is travelling fast. (near the speed of light). The shaded region of the pole and of the barn cross each other. There is clearly a line which crosses the two such that both ends of the pole (where the line of simultaneity crosses first the one end of the barn, then the one end of the pole, then the other end of the pole and then the other end of the barn.) In this notion of simultaneity, the pole is clearly inside the barn at some time. In another notion of simultaneity, the ends of the pole are never inside the barn at the same time. There is no contradiction. Just a difference in what "at the same time" means.

### Coin in slot Consider a slot in an infinitely thin piece of metal. and an infinitely thin coin. The coin is travelling at near the speed of light with respect to the slot. The coin is thus considered to be short, and it can gradually drift downwards while travelling forward such that the coin can go through the slot. But look at the problem from the point of view of the coin. It is now the slot that is shortened, and the coin cannot go through the slot. The problem with this is that it is assumed that the "drifting downward" is presented as if it is an entirely innocuous thing. It is not. It means that the actual velocity of the coin is not purely horizontal but has a vertical component. It is in the direction of the velocity that things shorten, not perpendicular to the motion.

When we go to the frame of the coin, it is the length in the direction of motion ( which is not parallel to the coin or the slot) which changes, The coin gets longer in that direction but not in the perpendicular direction. Thus from the coin's point of view, it is that component that is now longer. Not only does the coin change its "length" but it also seems to rotate. Similarly the slot gets shorter only in the direction of motion. It also tilts from the coin's point of view. The coin now goes through the slot, not parallel to the slot, but threading the slot like a pin through a hole. For the coin to go through the slot, the part of the coin perpendicular to the direction of motion must go through the perpendicular part of the slot. But perpendicular directions do not change. Thus if the coin gets through in one frame it does in the other as well.

## Fields

One of the results of Special Relativity is that things cannot travel faster than light in vacuum. If they could, the length squared along some piece of the path is spacelike. It's slope is less than that of light in the x-t diagram. Now, by having an observer travel quickly enough that piece of the curve will travel backward in time in for that observer. One should then have another particle that will travel back toward the place the first particle originated, and again backward in time, arriving back at the same spatial point but in the past. One will have time travel. While of course it has never been proven that time travel is impossible, it is strongly believed it is impossible. Closed timelike curves make physics extremely difficult. But this makes something else very difficult. Consider Newton's three laws. The first says that particles travelling without any external forces goes in straight lines with constant speed. The transformations for different observers always convert straight lines to straight lines. Now the speed of a particle is the length of the velocity, which is the change in position with time. The first is which time should one use. Since the time as seen by any observer is different, in order to make things more independent of the observer, it would make sense to use the proper time instead of the coordinate time as the time to use in dynamics. The speed is thus the change in distance in a unit proper time. But the distance along the curve IS the proper time. So the speed is the proper time in a unit proper time. It is 1. So the speed, defined in terms of the proper time, and the distance along the curve, is always 1. So, the particle certainly does always travel in straight lines at constant speed. (it always travels at constant speed). The First law is OK.

The second law says F=ma where a is the change in velocity per unit time. The speed is always the same, so all accelerations must only change the directions. This means that all forces must be such that the force does not have any part of it which points along the direction of travel in the 4-dimensional spacetime. The force must always to perpendicular to the direction of travel in the 4-D spacetime (all forces in relativity are "centrifugal"). But this does not contradict Newton. It just says that the types of forces are restricted. Because these forces are 4-D not just spatial, this limitation in forces does not really restrict much. The forces in space are really not limited in any way. So the second law is OK. A bit strange, but OK. It is the third law which is problematic. This says that if body A exerts a force on body B, then B exerts on on body A which is equal and opposite. But when. After all A's force on B can change in time. the force of B on A at which time is supposed to equal and opposite which force of A on B? This turns out to be an impossible problem. There is no way in which the third law can be valid, unless the force is a contact force. Ie, special relativity would seem to imply that we have to go back to Descartes-- all forces are contact forces. This destroys for example Newton's theory of gravity. But there is another way out, and that is Fields. One has the example of Electromagnetism, which of course was what led Einstein to Special Relativity. The electromagnetic field is created locally by charges. Any change in the Electromagnetic field, such as that done by the motion of a charge, will travel away from the origin at the velocity of light. Those changes in the electromagnetic field can then influence particles which are not in contact with the first one. This has some vague similarity with Descartes's picture, that there is something ( for him it was actual matter, composed of atoms) which body A has to first influence, and it then changes and influences particle B.

Eventually it turned out that any particles at all are very difficult to make consistent with special relativity, so that by the 1930's it was realised that it was all fields. All matter is described by fields all the way through, so that now, there are just fields in fundamental physics. Sometimes of course thee fields interact with each other in ways that look like particles (especially with quantum mechanics for which fields have particle like aspects under certain situations). But the equations which we solve are field equation. And furthermore, the interactions between the fields are local-- are contact interactions, which implements the third law in the context of field theories. (Of course what does acceleration mean when there are not any particles that travel along paths? And what do straight lines mean when there are no lines?) Thus not only did special relativity change what one means by time and space, but even what one means by particles. There are approximations in which one has the field such that the places where the value of the field is large are all clumped together. In that case the field theory can result in those clumps travelling along approximate straight lines and one can apply Newton;s laws as approximations.