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by Bruce Linnell |
Let's deal with a simple seeming question. Let's say you have a trolley that goes by in a straight line traveling at 5 miles per hour, right outside your house. You get up in the morning, and three hours later you get on the trolley.
The answers to both of these questions is probably opposite of what you are thinking. In all likelihood, for 1, you answered (a), when the actual answer should be (b) Why? Because your "here" is no longer the "here" of your home, but the "here" of the trolley. You got up 15 miles from the trolley, and once you are on the trolley, you are sharing the trolley's reference frame, so essentially, the event happened 15 miles ahead of you, even though all you did was change your velocity.
For question 2, you probably answered (b) while the actual answer should be (a) Why? Because when you get off the trolley, you will be AT the trolley. It is not going to happen 15 miles from where you are now, because where you are now is IN THE TROLLEY. The event will happen at your current position, but three hours in the future.
The diagram below shows the spacetime diagram from the perspective of home. The event at (0 miles,-3 hours) is where and when you woke up in the morning. The event at (0 miles,0 hours) is the event where and when you get on the trolley. The event at (5 miles,1 hour) is the event where and when you get off the trolley.
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Once you get on the trolley, you are in a new reference frame:
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Now the event where you woke up this morning is located at (15 miles, -3 hours), the event where you get off at the store is located at (0 miles, 1 hour), and the event where you get on the trolley is still at (0 hours, 0 miles).
What has happened is that every event in the future has moved to the left (toward the back of the trolley) and every event in the past has moved to the right (toward the front of the trolley). What is amazing and at first counter-intuitive, is that the event of you getting up this morning has suddenly moved 15 miles!
You would think that since you have not yet moved, you should still be "at the same location where you got up." The reason that we have this discrepancy is that in our minds, we never fully adopt the trolley reference frame. If we ride around on trolleys, our "common sense" is that we are riding around on trolleys; not that we are standing still on a stationary trolley while the ground moves underneath us.
We are perfectly happy to say "the trolley is moving" and "the store and home are stationary." As long as we continue to use the store-and-home reference frame, then the events don't move around. But if we fully make the switch, conceptually, to the trolley reference frame, then the events, both in the past, and the future must move, to adapt to the new reference frame.
It is, of course, possible to adopt the home-and-store reference frame, the trolley reference frame, or any frame in between.
This transformation is only a 5 mile per hour difference, and the motion of the events an hour or two in the past or future is quite dramatic. The Galilean transformation can continue to transform to 10, 15, 20, 25, 30 miles per hour, and keeps going forever.
The equations to use here are simply:
The point in space-time where we're gettting off the train is ( x,t ) = (5 miles, 1 hour). After the transformation,
The point in space-time where we got up this morning is ( x,t ) = (0 miles, -3 hours). After the transformation
For this little example, we only had three events; waking up this morning, getting on the trolley, and getting off the trolley. We could add other events, of course. Where we brushed our teeth, drank our coffee, checked our email, etc. If you include every event that happened to you all day, (Consider every photon that bounced off your skin today an event.) you get a continuous sequence of events, forming a curve through space-time.
To see what that curve looks like in another reference frame, you simply do the same transformation for every point on that curve.
Another way to write this transformation is in the form of a matrix
If you are completely unfamiliar with the basics of matrix multiplication, you can google any number of basic examples. Here, for instance is a ten-minute video tutorial.
In the example above, with our three relevant events (t,x) = {(-3,0),(0,0), (1,5)}, we can set up the list of events as a 2 by 3 matrix, and multiply the transformation matrix by the whole list, as follows:
To get the space-time diagram in both frames, it is just a matter of plotting the new coordinates, and then playing connect-the-dots between the relevant events. In our original "home" frame, we got up at position 0, got on board the trolley at position 0, and get off at the store at position 5. In the "trolley frame" we got up at position 15, 15 miles in front of the trolley, got on board the trolley at position 0, and got off the trolley at position 0.
The matrix multiplication helps to organize all of the relevant events into a single input, perform the transformation, and get the new coordinates of all of the events out as a single ouput.
When dealing with relative velocities such as 5 miles per hour, 500 miles per hour, or even 50 000 miles per hour, the Galilean transformation is sufficient. Why? Because the speed of light is around 671 000 000 miles per hour.
(Conversions: 1 mi/h = 0.447 m/s 300 000 000 m/s = speed of light)
As long as you are dealing with speeds significantly less than the speed of light, the Galilean transformation will give answers almost identical to the Lorentz Transformation.
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by Bruce Linnell |