### October 13, 2011

#### Relativity-4: Measuring time and space

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

To get a better grip on what is involved in the theory of relativity, we need to think in terms of 'events', things that occur instantaneously at a point in space and which every observer will agree happened and is unique. An example of an event might be me clapping my hands once. That occurs at one place in space (where my hands meet) at one moment in time (the instant they make contact) and all observers will agree that I did indeed clap my hands. Of course, actual events will be spread out over a region of space (my hands are quite big objects) and over a small but extended interval of time (the period during which my hands are in contact while clapping) but we can imagine idealized events as things that occur at a single point in space at a single instant in time. Specifying an event also uniquely specifies a location and a time since only one event can occur at any point in space at a particular time.

Suppose we have one event A that takes place at one place at one time (say a neutrino created by a nuclear reaction at CERN) and another event B that takes place at another place at another time (say the detection of the arrival of that same neutrino at the Grand Sasso laboratory). Einstein causality says that since event A caused event B, event A must take place before event B. Even if the neutrino were to travel at a speed greater than the speed of light, all that would do is reduce the time difference between the two events, not reverse their order, as was noted in the example given in the first post in this series. So why is this event seen as such a sensational development?

The answer lies in the fact that Einstein causality is believed to hold true for every observer who sees the same two events, irrespective of the state of motion of the observer. And the existence of faster than light neutrinos means that even though we on Earth will continue to see event A before event B, there are observers who are moving relative to us who will see the neutrino being detected at Gran Sasso before it was created at CERN or, more bizarrely in the case of the shooting example, that the bullet will emerge from person B and seem to travel back into the gun of person A. And unlike in that earlier example, this will not be due to an illusion due to the accident of where the observer happened to be located.

To understand how this can happen, we need to go more deeply into the question of how we measure the location and the time of events and how they differ for observers moving with respect to one another. Location and distance measurements seem pretty straightforward and we do it all the time when we measure the length of something. We simply hold a ruler along the line joining the two events, take the ruler readings at the locations of each of the two events, subtract the smaller reading from the larger, and the resulting number gives us the distance between the two events.

As for the time interval between two events, we can look at our watch when we see event A occurring and note the reading, then look again when we see event B occurring and note the reading, and once again subtract the smaller reading from the larger. The resulting number gives us the time that lapsed between the two events. There is a slight complication here in that it takes time for light to travel from one place to another so the actual time at which event A occurred would be a little earlier than when we see it. But since we know the speed of light, we can take that into account. All we have to do is measure the distance between where we are and the location of event A and divide that by the speed of light to get the time taken for the light to reach us. We then subtract that time from our watch reading to get the 'true' time at which the event A occurs. We can do the same thing for event B.

For example, in the earlier example, if you were standing next to the victim at B, you would have seen the bullet at the 2 meter mark 9 seconds after the gun fires. If you had been standing next to the shooter at A, you would have seen it 3 seconds after the gun fired. If you correct for the time of travel for the light to reach you from the bullet at the 2 meter mark, the bullet would be said to be at that point one second after the gun was fired, irrespective of where you were standing. So the time of an event can be specified uniquely in the case of different observers who are not moving with respect to the events.

What if the observer is moving, though? The question that Einstein pondered is the following. Suppose I, seated in my office, observe two events on the sidewalk outside my window and measure the distance between them and the time interval according to the above methods using my own ruler and watch. Now suppose another person is moving with respect to me (say passing by in a train) and sees the same two events as I do and measures the distance and time interval between them using her ruler and watch. Will that person's measurements of the distance and time intervals agree with mine?

It is the answer to this question that determines whether we live in a world in which Galilean relativity rules or one in which Einsteinian relativity rules.

Next: Galilean and Einsteinian relativity

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Posted by Ben Krill on October 14, 2011 07:25 AM

Mano,
> Einstein causality is believed to hold true for every observer who sees the same two events, irrespective of the state of motion of the observer.

Isn't this believed only because light is the mechanism of observation, and it's believed that lightspeed cannot be exceeded? If either of those two assumptions proved not to be true, Einstein causality would naturally not hold. In other words, isn't Einstein causality an emergent conclusion to be drawn from current physics rather than a driving theory?

And it would still be just an illusion, right? If we could somehow use something faster than light as our mechanism of observation, the proper causality would again be observed. And it wouldn't affect any decision-making. If you planned to shoot someone with a faster-than-light bullet, you would still recognize that "pulling the trigger" is the thing you need to do first, regardless of the conflicting timelines of observers.

Looking forward to your next post in this series!
Robert

Posted by Robert Allen on October 14, 2011 12:34 PM