by Steve Bryson
We have been using, as a starting point for this course, the (God given) principle that "The speed of light is a constant!!!". We have derived the form of the Lorentz Transformation from this principle in combination with the insight that we live in a four-dimensional spacetime. This principle has other implications, and here I want to examine one of them.
If the speed of light is constant for all observers, then by moving you can never go faster than the speed of light, as this would mean that in some way you 'caught up' with the light. You can hardly do this if the light is always going the same speed of light relative to you. Thus in principle you can never go faster than the speed of light. It has nothing to do with engineering requirements or some other technical aspect of how you move.
In spacetime diagram terms, we express this by saying that your world line can never have a tilt greater than that of the world line of light. In our special coordinates this means that your world line can never have a tilt greater than 1.
The principle that gives you this can actually be seen by looking at how the coordinate axes of a moving coordinate system tilt in your spacetime diagram. The faster the motion, the closer to each other the coordinate axes get, until at the speed of light they meet. The moving time and x axes merge into one. This is referred to by physicists as a coordinate singularity. It is not always a bad thing to have these singularities, but it does indicate that something funny is going on. In this case, it is felt that starting with two separate dimensions and collapsing them down into one is something that the world does not do.
The first real objection to all this relativity stuff is not the twin paradox, but rather the problem of addition of velocities. Consider this scenario: I am flying along at 75% of the speed of light relative to Earth and I launch you in a rocket at (relative to me) 75% of the speed of light. Then, you may immediately ask, wouldn't you be going at 75%+75%=150%=one and a half times the speed of light relative to Earth? This of course contradicts the principle that you can never go faster than light in any coordinate system, and therefore indirectly contradicts the principle of relativity!
The resolution lies in looking carefully at how velocities in different coordinate systems add. It turns out that you do not simply add the two velocities together. Instead you use a somewhat more complicated formula. If v1, v2, and v3 are your velocity relative to Earth, my velocity relative to Earth, and your velocity relative to me then the formula for the addition of velocities is:
In the above example, you end up going 96% of the speed of light relative to Earth. If you look at this formula, you will find that it can never have a value greater than the speed of light so long as each velocity is less than the speed of light. It turns out that this funny formula for addition of velocities comes very naturally from treating motion as a rotation in spacetime.
If we cannot go faster than light, then we cannot get places as fast as we might like to go. As Alpha Centauri is 4.3 light years away, I cannot get there in less than 4.3 years Alpha Centauri time (even though I may experience a shorter travel time). Thus if I found out that there was going to be a convention of relativity teachers there next year, no matter how hard I tried I could not make it. I would arrive there too late to participate. In particular, I could not causally effect what happened there. I could not even send a message, for light would take 4.3 years to get there, and nothing goes faster than light. This is an event in spacetime that I can have nothing to do with. There are many such events.
It is easy to characterize these events. They are all events that are far enough away from me so that I would have to travel faster than light to get to them. In spacetime diagram terms, they are all the events such that if I drew a line from me to them (that is to say from my world line at the time I start to travel to these events) the tilt of that line would be greater than one. This also covers all the events that I could send messages to, as the world line of the medium of the message must also have a tilt less than one.
All this also applies to my past (events with my time coordinate less than zero), as they cannot communicate with me faster than the speed of light.
On our spacetime diagram, it is thus easy to tell what parts of spacetime we can have anything to do with. To find out if you can have interaction with an event, draw a line from your origin to that event, and then if that line has a tilt less than one you and the event can talk. (It can talk to you if it is in your past and you can talk to it if it is in your future.) The boundary between events that you can communicate with and those you cannot is the two lines with a tilt of one (one in each direction). These two lines are the world lines of light going in two directions, and these lines (this boundary) are called the light cone of the event. It is called a cone because in three dimensional spacetime it actually forms a cone.
We call the set of events that we can communicate with inside the light cone and those events that we cannot communicate with outside the light cone.
Those events that have my time coordinate greater than zero and are in my light cone are truly in my future, no matter how I move. Similarly, those events with time coordinate less than zero are in my past. Thus even though my time measurements of events are relative to my state of motion relative to them, their being in my future or past is not. This is safeguarded by the speed of light being constant for all observers.
Similarly, those parts of spacetime that I cannot communicate with are not relative to my state of motion. Thus spacetime is split into sections:
For us, living and experiencing a three-dimensional space in time, the idea of velocity is a natural measure of motion. Velocity is given (in terms of coordinates) by x/t, where x is the distance traveled and t is the time it took to travel that distance. In spacetime, however, x/t is not the velocity but the tilt of the world line. What can we say about a spacetime version of velocity? In other words, how do we generalize the idea of velocity to four dimensions?
Let us look more closely at the definition of velocity as x/t. This is taking the distance and dividing by the time--it assumes a difference between space and time, giving time a privileged position. The whole idea of thinking in spacetime is to say that time is not an inherently different direction. Thus, time should be in the top of the equation as well as x. In other words, we want to think in terms of velocity in the time direction as well as in the x (or y or z) direction. This is treating space and time as the same kind of things.
What do we put in the bottom of the definition then? It should be something that matches our sense of velocity as x/t, so that we recover our usual sense of velocity for small velocities. In particular, for us in our own coordinate systems whatever we put in the bottom should be equal to our time coordinate, yet it is not our time coordinate. What do we know that does this?
The spacetime distance does this. Recall from last time that the spacetime distance s from our origin to a point on our world line is just our time coordinate t of that event. This is the same as saying that in our reference frame the spacetime distance is just the elapsed time (= the time coordinate).
Thus we take as the spacetime velocity direction/(spacetime distance). So the time velocity is given by t/s, the x velocity is x/s, and so on. How is this related to the three-space velocity (=x/t) that we are used to?
First, in our reference frame, s=t for us, so our spacetime velocity in the t direction = t/t is exactly one (we are always moving through time at a speed of 1 (=the speed of light) relative to ourselves!).
For velocities of objects moving relative to us with (three-space) velocity v = x/t, we have that their spacetime velocity in the x direction is given by
x/s = x/t times t/s
= v times the spacetime velocity in the t direction.
So we only need to know the spacetime velocity in the time direction. As is shown in the first part of the technical handout, this is given by:
This is going to be very important in the next section.
We need to get a little technical here. There are these two things that physicists like to think about quite a lot. They are called energy and momentum. These two quantities are special to physicists because in most interactions of objects they are conserved. We speak of the observed facts of conservation of energy and conservation of momentum.
I wish to talk about conservation of momentum for a while. Momentum is a way of measuring the amount of motion that a body has. By saying that momentum is conserved, I mean that if you take a physical system, add up the momentum of all it's parts, then later no matter how the momentum of the individual parts have changed, the momentum of the entire system will be the same. This makes momentum very dear to physicists as it allows them to calculate all sorts of things about the interactions in the system.
In old fashioned three-space plus time Newtonian physics, momentum, denoted by p, was given by the formula p = mv where m is the mass and v is the (three-space) velocity of the object we are interested in. For low velocity interactions (the only kind that physicists saw until about 60 years ago) this quantity was indeed conserved.
When relativity came on the scene, however, it became clear that p = mv will not be even vaguely conserved in high velocity collisions. Indeed, it wasn't really conserved in the low velocity interactions--the violation of the conservation was just too small to measure. By doing a rather complicated analysis, physicists (lead by Einstein) calculated that there is still a conserved quantity, and it is given by:
This looks a lot like the old fashioned formula for momentum, with the mass replaced by the quantity
This quantity gets bigger the faster we go. This lead people to talk of the mass of an object increasing as the object went faster and faster. This lead to great confusion.
It is no wonder that it lead to confusion! Here the physicists were insisting on defining momentum in terms of the three-space velocity! Look at the new formula for the momentum above. It is just the mass multiplied by the spacetime velocity in the x direction. So what can be more natural than to take as the definition of momentum mass times the spacetime velocity (a four-dimensional object) rather than mass times the three-space velocity (a three dimensional object in time)?
This means that our spacetime 4-momentum has four parts, one for each direction in spacetime: mct/s, mx/s, my/s, and mz/s. In terms of the three-space velocity, they are given by the formulas (where here we are only considering motion in the x direction to that the y and z parts are zero):
We see that the space part of the 4-momentum is just the correct-that is to say conserved- 3-momentum from above. What is the significance of the time part of the 4-momentum? If you are a mathematically sophisticated person, you try to approximate the formula for the time part of the momentum. If you do this you will find that you get that for low velocities this will be equal to:
mc2 + 1/2 mv2
The second term here is just the usual three dimensional expression for the energy of motion from old fashioned Newtonian mechanics. The first term is completely new. It has the units of measurement that physicists use to describe energy (mass times distance squared divided by time squared) and it is added to the non-relativistic expression for energy. Thus it is natural to assume that this is the new relativistic measurement of the energy of a body.
This is what physicists do. They define the energy of something as the amount of 4-momentum that body has in the time direction. What is the value of the energy of something that is not moving relative to you? That something will have velocity v equal to zero, so the energy E is given by the formula
E = mc2
This is one of the very few formulas that almost everyone knows, though most people do not know what it means. It does not mean that energy and mass are the same thing. After all, if an object moves, it also gets the energy of motion. What it does say is that an object of a given mass has a certain (very large) amount of energy inherent in it. Actually manifesting this energy as energy is something that we have to go into the realm of quantum field theory to see.
The reason that this inherent mass of objects was never before known is that physicists only know how to measure changes in energy. This energy due to mass never changes, so it was never seen.
For now, just let me say that we do indeed know how to convert a certain amount of matter into energy. The sun shines by converting electrons and anti-electrons into light energy. In fact, the sun converts about 4.75 million tons of electrons and anti-electrons into energy every second! (This amounts to about 4x1041 electrons per second. The Sun has enough electrons to do this for many billions of years.) This is where the tremendous amount of energy that the sun puts out comes from. This is also done in any explosion of a hydrogen bomb. It is also done rather often in an experimental setting Berkeley and Stanford (as well as many other laboratories). Because the speed of light squared is a very large number, a little bit of mass is worth a whole lot of energy.
Conversely, energy does have mass according to this formula. In this sense, an object does indeed get more massive when it moves, but this gain in mass is the 'weight' of the energy of motion of that object. It is not what physicists mean when they refer to the mass of an object. What they are referring to can be best thought of as the mass-energy of an object at rest. This turns out to have a very nice geometric interpretation.
All of the above discussion on energy and momentum was unpleasantly abstract. Does it have a more intuitive description in terms of objects in spacetime? Yes it does!
The spacetime 4-velocity defined above is truly a spacetime velocity as it measures your rate of travel along your world line. You have a certain coordinate (in whatever coordinate system you are using) which falls on your world line, and this coordinate changes as your time coordinate changes.
Now in your coordinate system, the time coordinate is the only coordinate that changes-- your x coordinate is always zero in your coordinate system. As stated above, your spacetime velocity is always 1 and is in the time direction. If you are moving relative to me, however, you have some spacetime motion in both the x and the time directions. I can compute these components of your velocity from the formulas for spacetime velocity in terms of your 3-velocity. It turns out that your spacetime velocity will also have magnitude 1 in my coordinate system. In fact, it will always have magnitude 1 in every coordinate system, though the components will be different.
Now the spacetime 4-momentum is mass times the spacetime velocity. This is something called a vector. You can think of a vector as an arrow in our space. The length of this arrow will always be equal to m, the mass, and the direction of the arrow will be along the object's world line. Then when the object is motionless relative to you, the object's 4-momentum vector will always point along it's world line. It will point parallel to your time axis, so it will be the time component of the 4-momentum. There will be no x (or y or z) component.
Now set the object in motion relative to you. It's world line will lilt, and it's 4-momentum vector will also tilt, keeping the same length (= the mass). Then you will see both a time and an x component of the 4-momentum.
Thus the significance of the mass of an object is, for us, that it is the length of the 4-momentum vector.
One thing about spacetime that is simply impossible to draw on a spacetime diagram is that as an arrow tilts, the components both get bigger. Thus as the object moves, both the time component (energy) and the x component (3-momentum) of the 4-momentum get bigger.