Schrödinger's Cormorant

A2.2 Changing Perspective

The Lorentz transformations enable you to convert the time and spatial position measured in one frame of reference to another one.

To begin with, we need to learn how to transform space-time coordinates from one reference frame to another: that will provide us with the mathematical hardware needed to get stuck into some pretty interesting questions.

Here’s the picture: Nefertiti is standing on the ground, while Cormorant is flying overhead. Nefertiti is annoying him by flying a drone in pursuit of him. They are flying in a straight line, at a constant velocity (so the distance between drone and seabird remains constant). Nefertiti and Cormorant are each at the centre of their individual frames of reference – Hers is frame N and his is frame C. We’ll define Cormorant’s line of flight as the x axis; we don’t care about the other spatial coordinates.

In Cormorant’s frame of reference, the drone is stationary, so it will have a fixed x-coordinate, xC. In Nefertiti’s frame of reference, the drone is moving, so its x-coordinate, xN, will be changing with time. Our mission is to be able to relate xN to xC at any given point in time. We therefore need to introduce a time coordinate too. In Galileo’s relativity this would be easy because the time would the same everywhere at any instant. From what we learned in Chapter 2, however, we know that the reality is actually more complicated: in Uncle Albert’s relativity, the time will in general be different at different places when observed from a moving reference frame (think back to Margaret and Vlad in the railway carriage). So we’re going to look for two transformations: to convert the drone’s x-coordinate as Nefertiti sees it to the coordinate as Cormorant sees it, and to relate the time at that coordinate as Nefertiti measures it to the time as Cormorant measures it.

To begin, let’s focus on the moment when Cormorant is exactly overhead, so that the x-origins in the two reference frames (Nefertiti and Cormorant) are momentarily coincident. We’ll define this moment as time zero in both frames of reference, as shown in the picture above.

Since this is time zero, we’ll call the drone’s coordinate at this moment, in Nefertiti’s frame of reference, xNo. We know however that the distance between Cormorant and the drone, as it is measured by Nefertiti on the ground, has been shrunk because of their movement relative to her, so xCo will be different.

Question A2-1

Now, some time later, Cormorant (and the pesky drone) will have moved on, travelling towards the right with a constant velocity, u, relative to Nefertiti. See the diagram above to visualise the situation after time tN has elapsed (as measured by Nefertiti):

Question A2-2

Now we come to the crunch: in Cormorant’s reference frame, the drone’s x-coordinate is still xC, because it is not moving relative to him. But how is xC related to xN?

Question A2-3

If you've worked through these questions, you have, hopefully, arrived at is this:

This would be valid whatever position the drone maintains relative to Cormorant, so what it means, more generally, is that that if you measure the time and the position (x-coordinate) of something in one frame of reference, you can now find the x-coordinate of this thing, at this time, in any other inertial frame of reference. This is an awesome power to possess, so use it wisely.

As we have learned in Chapter 2, the relativistic length contraction works reciprocally, so just as the distance between drone and Cormorant is contracted when observed by Nefertiti, so the distance between Nefertiti and a gooseberry bush located somewhere along the x-axis from her would be shortened in Cormorant’s reference frame. So there will be a very similar transformation equation that allows you to convert an x-coordinate measured by Cormorant at a specific time in his frame of reference and convert it to Nefertiti’s reference frame.

Question A2-4

Now we have two coordinate transformations, from xN to xc and the reverse:

The complication, however, is that they use time measurements made in different frames of reference. So, to be fully in charge of things, we need to be able to interconvert not just the spatial coordinate but the time coordinate too.

The key to this is to recognise that if we focus a specific point in space and time, both of these transformations must simultaneously be true. We can therefore legitimately combine the two expressions and, we’ll find that this lets us relate the time coordinates tN and tC. So let’s do that.

Question A2-5

The answer you should eventually reach in (A2-5) is:

This is a transformation equation for relating time at the same point as observed from two different reference frames. If you follow the same approach to find tC in terms of tN, you should expect to find a very similar formula because, as we know, time dilation works the same in both directions. The only difference stems from the fact that the velocity of Nefertiti’s frame of reference relative to Cormorant’s is -u, rather than u. This leads to:

What you have just succeeded in deriving are called Lorentz Transformations, derived by the great Dutch physicist Hendrik Lorentz who achieved them – and thereby laid the foundations for Uncle Albert’s special relativity theory – without reference to seabirds of any kind. This achievement has turned you into a very powerful being, with the ability to convert both space and time coordinates between different frames of reference. And if you are master of both space and time, you are now ready for the next challenge, which is velocity.

We can generalise these results to provide seabird-free expressions which can be used to transform coordinates between any inertial frames of reference: where frame 1 is moving at velocity u, relative to frame 0:

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