Sharp Blue: VBHT2: Coordinate and Proper Time


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Italicised articles are old and unrevised.


Up to now I’ve been explaining the view of space and time that originated with Isaac Newton, and was overturned at the beginning of this century by Albert Einstein. In this part I will explain the difference between proper time and coordinate time, then describe velocity addition in pre-relativistic physics, and then give some hints as to how special relativity differs. This article may seem somewhat pedantic, but later the concepts described here will turn out to be very important.

I’ve been describing the sheets in our model spacetime as being labelled with time values, and these labels are time coordinates. What is more important, however, is the time measured by clocks, which is the proper time; after all, you can’t see the labels on the sheets. The coordinates are just convenient ways to label events, whereas clock ticks are events. Proper time is something that only has meaning along the worldline of the clock, as that is the only place where the clock is ticking. Now suppose that I carry a clock from the event at my front door at 9pm GMT today to the event at the Clifton Suspension Bridge at 11am GMT tomorrow. While I do this the clock will tick a certain number of times. If anybody else in a different frame watches me travelling along this worldline, she may disagree with me about the coordinates of each event along the worldline, but we’ll agree on how many times the clock ticks along my worldline. Proper time is thus invariant. This is true of proper time in a Newtonian world, and will be just as true in special-relativistic or general-relativistic spacetimes. It’s even true if my frame is not inertial (that is, if I accelerate along my path).

Imagine that I’m in an inertial frame. Make the time slices such that my clock ticks once in each slice. I can then label the time slices with coordinate times that coincide with my proper time along my worldline. Now make a Galilean transformation into somebody else’s frame (slide those sheets over). In their frame my clock still appears to tick once in each time slice, and so from their point of view my proper time still coincides with coordinate time. If they have a similar clock to mine then the coordinate time will also coincide with their proper time. Furthermore, if two people travel along different worldlines between two events and they’re carrying similar clocks then the clocks will tick the same number of times between their two meetings, even if their worldlines are very different (that is, even if one or both accelerates along their path). The clocks will just tick once every time the worldline passes through a coordinate time sheet, and the number of ticks of either clock will simply be the number of sheets that lie between the sheet containing the first event and the sheet containing the second event (plus a tick in the final sheet I suppose). Because all of these various times coincide in Newtonian spacetime, it is said that time is absolute in Newtonian physics. However, the world is not Newtonian and when we begin to discuss special relativity, it will turn out that the times do not coincide, and so there is no absolute time.

Another property of Newtonian physics that follows directly from the Galilean transformation is that velocities are additive. An example will make this clear. Suppose I’m travelling on a train at 100km per hour, and I throw a ball in the direction that the train is going at 20km per hour. Now suppose that Eliza is standing on the side of the track watching the train go by. If she looks at the ball, then it will seem to be travelling at 120km per hour. This seems entirely natural and obvious. It too will turn out to be wrong in special relativity; it is only approximately true when all the relevant speeds are small compared to the speed of light.

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