by Steve Bryson
We have discovered that time can naturally be thought as a fourth dimension, and that if you are moving relative to me then your time axis appears rotated relative to mine. This will mean that you and I will measure the time coordinates of an event differently. We now ask several questions:
What happened to your space (x) axis?
How is this rotation related to rotations in three-space?
How does this all fit in with "The speed of light is a constant!!!"?
What are the quantitative implications of this rotation (exactly how do our measurements differ)?
It will turn out that the first three questions will be answered together, and the last question will only be answered in a technical appendix. We will here be concentrating on the first three questions.
Our path to an answer to these questions will be to define what happens to the x-axis so that the speed of light will be the same in all coordinates, rotated or not. This will force us to treat the rotation of coordinates due to motion differently from rotations in three-space. In order to do this we need to be more clear about how to represent the motion of light on our spacetime diagrams.
Think back to our point moving steadily from left to right in our spacetime. On our spacetime diagram it's world line looked tilted, and the faster it moved the more it was tilted:
We need a way to measure the tilt of the line, in order to translate the demand that 'the speed of light is a constant!!!' into our spacetime diagram. We will measure the tilt of the line in the following way. If the line passes through the center (origin) of our coordinate axes, then here is how you measure the tilt: Pick any point on the line. On our spacetime diagram the point (an event) will have coordinates given by the time and the x position. Then simply divide the x coordinate by the time coordinate:
Then we have a numerical measurement of how tilted the line is. It is defined in such a way that a line at 45 degrees will have a tilt of 1, as then the t coordinate will equal the x coordinate.
Comments: The tilt as I am defining it is the inverse of the slope of a line that you learned about in high school geometry. I am using the inverse in order to be more intuitive--the greater the speed of motion the greater the tilt. In fact, as speed is distance traveled divided by the time it took to travel, the tilt as I've defined it is the speed of the moving point. (For those of you who know the difference, it is actually the velocity, as it carries the proper sign and so indicates direction as well as speed.) Finally, some of you may be bothered by the fact that the time coordinate and the x coordinate are in different units (seconds and inches, for example). As we are only interested in comparing tilts rather than calculating them, This is OK so long as we don't change the units we are using. In any case, we will soon start measuring time in units of distance anyway.
So from now on we can think of the tilt of a line as given by it's velocity. We are now in a position to think about light.
We can now ask "what does a beam of light look like on our spacetime diagram?" This is a subtle enough problem that it deserves it's own section.
This question already gets us in trouble because a beam is a three-dimensional object. In order to put light on our spacetime diagram we need to think in terms of a flash of light that acts like a point in our one spatial (x) dimension. So for now, when I say 'light', I mean a point moving at the speed of light. Think this: That point on the x axis lit up by the passage of a flash of light.
So let's put this moving point of light on our spacetime diagram. What does it look like? Like any point moving in a spatial direction, it looks like a tilted line on our spacetime diagram. Let's say it's moving from left to right and goes through the origin of our coordinates. Then the only thing we need to know is the tilt of the world line. This is given by it's velocity, and if we are measuring time in seconds and the spatial part in inches, then the velocity of light is about 11,802,852,680 inches/second. This is quite a tilt! Here's what it looks like on a spacetime diagram.
It's tilted so much that we cannot, on any diagram at a reasonable scale, distinguish light's world line from the x axis. Yet light's world line does not coincide with the x axis, it is just very tilted.
We need to think about the motion of light, and this way of measuring tilt does not help us. There is a way out. We can use our freedom of defining our units of measurement to make the tilt more reasonable. After all, we do not have to measure things with inches and seconds! We could use miles and hours, meters and minutes, or furlongs and fortnights. It turns out that no matter what units we choose, if we choose units that are in our everyday experience the speed of light (=tilt of light's world line) will be very large. This is simply because compared to things in our everyday experience light is very fast.
So if common units do not help us, let us make up our own. At the risk of being conservative, I wish to go on measuring spatial coordinate in terms of inches. I will now choose a new unit of time so that the tilt of light's world line is a reasonably visualizable one. Now I can choose this tilt, as I am making up my time unit. What tilt do I want? What can be more reasonable than a tilt of 1? I do this by simply taking my old time unit and multiplying it by the speed of light, measured in the old units. Lets say that my old units were inches and seconds:
New time unit=seconds multiplied by 11,802,852,680 inches/second.
What kind of unit is this? It is seconds multiplied by a whole lot of inches divided by seconds. The seconds cancel out, and the new time unit is 11,802,852,680 inches! We are now measuring both time and spatial directions in inches, though with a very different scale in the two directions. Let us adjust the scale on the x axis, so that one x unit is 11,802,852,680 inches. We are still measuring spatial distances in inches, but now one mark is not one inch, but 11,802,852,680 of them.
What is the tilt of light's world line? Because light travels at 11,802,852,680 inches/second, it will travel 11,802,852,680 inches in one new time unit (I defined the new time unit so that this would be true). Thus in the new units, the speed of light is exactly 1, and so it's tilt is exactly 1:
Let us give the new time unit a new label that will constantly remind us of how it is defined. It is defined as seconds multiplied by the speed of light. Now I am getting tired of writing 'the speed of light' and 11,802,852,680 inches/second again and again, so let me denote the speed of light by the letter c. Then my new time coordinate will be denoted by ct. In this way we will never forget how this new unit was defined.
All this was, admittedly, very dry and technical, but it has left us with a very easy way to visualize the motion of light on our spacetime diagram. Light's world line is tilted at 45 degrees. We now have a new way of saying "The speed of light is a constant!!!": "Light's world line has a tilt of 1 (is tilted at 45 degrees) in all coordinate systems!!!". This second phrasing will give us a precise way of finding it's implications.
We can now turn to the question of how your x axis changes when you are moving relative to me. We will use the demand that the world line of light have a tilt of 1 in everyone's coordinate system.
Let us recall the example at the end of the last handout, where you were moving relative to me, and there was a third point called her moving relative to both of us. From our two points of view our spacetime diagrams looked like this:
Now let's see what happens if we replace her by light. Then according to our demand, the world line of the light must at the same tilt in our two spacetime diagrams:
The tilt of your and my world line are different in the two diagrams but the tilt of the light's world line remains the same. This is different from the above example.
How can this be? To me, your world line makes a smaller angle with light's world line than my world line does. Yet in your diagram, your world line makes the same angle with that of light as mine did in my diagram. This is a seeming paradox.
Let us look more closely at the idea of tilt. Very picturesquely, the tilt of a line measures how close that line is to one axis compared to how close it is to the other axis. By how close, I mean the angle between the line and the axis. To say that a line has a tilt of 1 is to say that the angle that line makes with one coordinate axis is the same as the angle it makes with the other. Thus if as your world line tilted in my spacetime diagram taking your time axis with it your x axis tilted towards the world line of the light by the same angle as your time axis tilted down, the world line of the light would still make the same angle with each axis! The tilt of the light's world line would still be 1:
If your x axis tilted towards the world line of the light the same amount as your time axis tilted down (also towards the world line of the light) due to your motion, then I would see that the world line of the light has a tilt of 1 in your coordinate system. In other words, in order for light to have the same speed for all moving observers ("The speed of light is a constant!!!"), the x axis rotates backwards from what would be expected given the rotation of the time axis.
This type of rotation is called The Lorentz Transformation or Lorentz Rotation.
Rotations:
This answers the question 'How is the rotation due to motion related to spatial rotations?'. It is related, but different.
You may have many objections to this. Remember that we are trying a point of view, and we will find that this point of view matches well with experiment. One of the first objections that comes up is that if you are moving relative to me, your coordinate axes are not perpendicular from my point of view. This is correct, but in your point of view they are perpendicular, and so there is really no problem.
We thus now see that both your time axis and your x axis rotate from my point of view if you are moving relative to me. This means that you will measure the time and spatial coordinates of an event differently from me.
A summary of where we are is called for here.
1) We have found that if you are moving relative to me, your world line is tilted compared to my world line in spacetime, that is to say from my point of view.
2) We have, from the principle of relativity and the laws of electricity and magnetism that the speed of light is the same for all observers, moving or not. This means that the world line of a light beam will be tilted the same from all points of view.
3) We recognize that your time axis, that is to say the direction in spacetime defined by your time coordinate, always coincides with your world line, and so is tilted relative to mine. This means that you and I will give different time coordinates to an event.
4) In order for the world line of light to have the same tilt in both our diagrams and to be consistent, as your time axis tilts towards the world line of light, your x axis also tilts towards the world line of light by the same amount.
5) This means that you and I will give different spatial coordinates to an event.
And so we have that if you are moving relative to me, you and I will give different coordinates to the same spacetime event.
Thus the space and time coordinates of an event are relative to our state of motion.
This is the sense in which this is a theory of relativity.
As I mentioned in the first session, the Special Theory of Relativity is also a theory about what is absolute in spacetime.
Though space and time taken separately are relative to our state of motion, the spacetime of which they are parts does not depend on our state of motion (or anything else about how we observe it).
Or, to quote Hermann Minkowski,
"Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union between the two will preserve an independent reality."
Spacetime is that union. How do we see this? What aspect of spacetime can we use to appreciate that it is spacetime that is left absolute? In order to find a thing to fit this, let us think about rotations.
The Lorentz transformation is, as I said above, a rotation in spacetime. It is, admittedly, different from the rotations we are used to, but it is (especially when expressed mathematically) the same kind of animal. Let us therefore think a bit about more 'normal' rotations.
Here is a purely spatial rotation of x and y axes:
We can superimpose the before and after pictures to get a way to compare the two points of view:
Now the x and y coordinates of a point are going to depend on whether they are measured from the unrotated axes or from the rotated axes. To say it another way, the distance of the point from each axis depends on whether or not the axis is tilted. The coordinates of the point are relative to the orientation of the axes--it is as simple as that.
There is, however, something that is the same when measured with both coordinates: The distance of the point from the center of the axes. It doesn't matter how you tilt the coordinates, how far the point is from the center does not change. Why is this so? Because you did not move the point or the center when you rotated the coordinates. If neither point moved, how can the distance between them change?
The simplest way to express this distance is by measuring it with a ruler. This does not help us, though, as we wish to generalize this to spacetime, and there are no spacetime rulers that you can hold in your hand. We can, however, express this distance in terms of the coordinates of the point. The distance is given by the formula:
distance = square root of (x coordinate times x coordinate plus y coordinate times y coordinate)
Yes, this is mathematics, but we will soon get used to it. First let's write it in a more convenient way. I want to write x for x coordinate and y for y coordinate. Also I will write x2 for x times x and y2 for y times y Then the formula looks like:
distance = square root of (x2+y2)
Finally, I will write d for distance, getting d = square root of (x2+y2). Now it is clumsy to carry the phrase "square root of" all the time, so I will simply square both sides, in order to write the formula in the equivalent form:
d2 = x2+y2
This is the formula for the distance of a point from the center of the coordinate axes in terms of the coordinates of that point. It will give the same number d2 no matter how you rotate the coordinate axes. This is a remarkable fact, for the values of x and y will be different if you rotate your coordinates. This formula is special in that the differences exactly cancel out--the x gets a little bigger perhaps and the y gets a little smaller, so that if you add them together the differences cancel.
Now we are viewing the Lorentz transformation as a rotation in spacetime. Is there a spacetime distance formula that takes x and ct (the new time coordinate) and produces a distance that does not change under Lorentz rotations? Let us call the spacetime distance s. It turns out that the formula s2 = c2t2+x2 changes under Lorentz rotations. Therefore it cannot be the correct formula for the spacetime distance of a point (event) from the origin.
There is, however, a formula that does work. Here it is:
s2 = c2t2-x2
It is the same as the other distance formula, except that instead of adding the squares of the coordinates, you subtract them. The reason for this is that a rotation in spacetime is different from a spatial rotation. This difference is completely described by this distance element. All of this will be covered in a soon to arrive technical handout.
This distance formula is indeed the same no matter how I rotate my axes in spacetime, using Lorentz rotations. Let me say this in a technical way. Let x and ct denote my unrotated coordinates, and x' and ct' denote your rotated coordinates. Then:
c2t2-x2 = c2t'2-x'2
Thus though you and I will measure the time and spatial coordinates of an event differently if you are moving relative to me, there is something we can agree upon, the spacetime distance.
Using this distance formula, we can derive formulas for the quantitative difference between our time and space measurements. We find that if you are moving relative to me and you say that an event has coordinates x' and ct' then I will see that the same event has my coordinates x and ct given by
This will be derived in the technical handout.
This completes our visitation with the foundations of the special theory of relativity. Next time we will look at all the implications of the Lorentz transformation.