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### October 17, 2011

#### Relativity-5: Galilean and Einsteinian relativity

(For previous posts in this series, see here.)

In the previous post in this series, I posed the situation where, seated in my office, I observe two events on the sidewalk outside my window and measure the locations and time of two events and deduce the distance between them and the time interval according to the rules for using my own ruler and watch. Now suppose another person is moving with respect to me (say in a train that passes right by where the two events occur) and sees the same two events as I do and measures the locations and times of the two events and deduces the distance and time interval between them using her ruler and watch. Will her measurements agree with mine?

When it comes to location and distance measurements, it is not hard to see that the two results will be different. When I take ruler readings of the two events, the ruler is not moving compared to the two events. But because the person in the moving train's ruler will be moving along with her in the train, the ruler readings of where the two events occurred will be affected by her motion. After the person in the train takes the reading on her ruler at the location where event A occurred, by the time the later event B occurs, she and her ruler would have moved along with her train and so the ruler reading for event B would be different from what would have been obtained if the ruler had been stationary. So the locations and the measured distance between the two events based on her two ruler readings will be different from those based on my two ruler readings.

What about the time interval between events A and B? It used to be thought that even though the two observers used different clocks and they were moving relative to each other, as long as the clocks were identical and synchronized properly, the two observers would at least agree on this because it seemed so commonsensical that time was some sort of universal property, independent of the observer measuring them or her state of motion. Time measurements were said to be *invariants*.

These relationships between the location and time measurements made by observers moving with respect to one another were first postulated by Galileo. It is now known as 'Galilean relativity'. Galileo used these relations to show why, even though the Earth was moving quite fast through space (a seemingly absurd idea at that time), a ball thrown vertically upwards would fall back down to the same point from where it was thrown, and not be displaced because the Earth had moved during the time that elapsed. This everyday observation had previously been used to argue that the Earth must be stationary but Galileo turned it around to show that it was consistent with the Earth moving.

But one consequence of the assumption that time is an invariant is that if you measure the speed of light (by taking two events, one consisting of light being emitted at one point and the other of it being detected at another point and dividing the difference in ruler readings between the two events by the time interval between the events), you would get different values for two observers in relative motion to each other, since the distances traveled (i.e., the differences in the ruler readings) would be different for the two observers but the time interval would be the same. In other words, the measured speed of light was not an invariant but depended on the speed with which the observer was moving.

What Einstein postulated (based on several reasons that I will not get into here) was that the *speed of light was the same for all observers*. In other words, it is the measured speed of light that is an invariant, the same for all observers irrespective of how they are moving. One important consequence of this is that the elapsed time between two events is no longer an invariant, and depends on the observer. Time is no longer a universal property but depends on who is measuring it. The difference in measured times is tiny for the normal speeds we encounter in everyday life, which is why we don't perceive it. But it does leads to things like the celebrated 'twin paradox' where if you have a pair of identical twins, one remaining on Earth and the other going in a rocket at high speed to a distant star and returning, the traveling twin would have aged much less than the one who stayed home.

Needless to say, this caused some consternation and it took some time for people to be persuaded that this seemingly bizarre result was correct. What Einstein did was force us to be more precise about how we measure the location and time at which events occur, so that we can meaningfully compare the results of different observers viewing the same events.

Next: Measuring time and space more precisely.

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