Existential physics: what happens “now” is relative

Excerpted with permission from Existential Physics: A Scientist’s Guide to Life’s Biggest Questions, written by Sabine Hossenfelder and published by Viking.

The fact that the passage of time isn’t universal is mind-blowing enough, but there’s more. Because the speed of light is very fast but finite, it takes time for light to reach us, so strictly speaking we still see things as they were a little earlier. Again, however, we don’t normally notice this in everyday life. Light travels so fast that it doesn’t matter over the short distances we see on Earth. For example, if you look up and observe the clouds, you are actually seeing the clouds as they were a millionth of a second ago. Doesn’t really make a big difference, does it? We see the Sun as it was eight minutes ago, but since the Sun doesn’t normally change much in a few minutes, the travel time of the light doesn’t make a big difference. If you look at the North Star, you see it as it was 434 years ago. But, yes, you will say, so what?

It’s tempting to attribute this time lag between when something happens and our observation as a limitation of perception, but it has far-reaching consequences. Again, the problem is that the passage of time is not universal. If you ask what happened “at the same time” elsewhere – for example, exactly what you were doing when the Sun emitted the light you see now – there is no meaningful answer to the question.

This problem is known as relativity of simultaneity, and it was well illustrated by Einstein himself. To see how this happens, it helps to make some space-time drawings. It’s hard to draw four dimensions, so I hope you’ll excuse me if I only use one dimension of space and one dimension of time. An object that does not move relative to the chosen coordinate system is described by a vertical line in this diagram (figure 1). These coordinates are also called the rest frame of the object. An object moving at constant speed forms a straight line inclined at an angle. By convention, physicists use an angle of 45 degrees for the speed of light. The speed of light is the same for all observers, and because it cannot be exceeded, physical objects must move in lines that are less than 45 degrees inclined.

Einstein now argued as follows. Let’s say you want to build a notion of simultaneity using pulses of laser beams bouncing off mirrors at rest relative to you. You send one impulse to the right and one to the left and move your position between the mirrors until the impulses come back to you at the same time (see figure 2a). You know then that you are exactly in the middle and that the laser beams hit both mirrors at the same time.

Once you’ve done this, you know exactly when in your time the laser pulse will hit both mirrors, even if you can’t see it because the light from those events hasn’t reached you yet. You might look at your clock and say, “Now! In this way, you have constructed a notion of simultaneity which, in principle, could cover the whole universe. In practice, you might not have the patience to wait ten billion years for the laser pulse to return, but that’s theoretical physics for you.

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Now imagine that your friend Sue is moving relative to you and trying to do the same (Figure 2b). Let’s say it moves from left to right. Sue also uses two mirrors, one to her right and one to her left, and the mirrors move with her at the same speed – hence the mirrors are at rest relative to Sue, as your mirrors are relative to you. Like you, she sends laser pulses back and forth and positions herself so that the pulses return to her from both sides at the same time. Like you, it then knows that the pulses hit both mirrors at the same instant, and it can calculate what instant that corresponds to on its own clock.

The problem is that she gets a different result from yours. Two events that Sue says happen at the same time would not happen at the same time to you. This is because from your point of view, she is heading towards one of the mirrors and away from the other. It seems to you that the time it takes for the pulse to reach the mirror on its left is shorter than the time it takes for the other pulse to catch up with the mirror on its right. It’s just that Sue doesn’t notice it, because on the return paths of the mirror pulses, the reverse happens. The pulse from the mirror to Sue’s right takes longer to catch up with her, while the pulse from the mirror to her left comes faster.

You would say that Sue is making a mistake, but according to Sue, you are making a mistake because, for her, you are the one moving. She looks like your laser pulses aren’t actually hitting your mirrors at the same time (Figures 2c and 2d).

Who is right? None of you. This example shows that in special relativity, the statement that two events happened at the same time is meaningless.

It’s worth pointing out that this argument only works because light doesn’t need a medium to travel, and the speed of light (in vacuum) is the same for all observers. This argument doesn’t work with sound waves, for example (or any other signal that isn’t light in a vacuum), because then the speed of the signal really won’t be the same for all observers; rather, it will depend on the medium in which it travels. In this case, one of you would be objectively right and the other wrong. That your notion of now is not the same as mine is an idea we owe to Albert Einstein.

We have just established that two observers moving relative to each other disagree about what it means that two events occur at the same time. This is not only strange, but it entirely erodes our intuitive notion of reality.

To see this, suppose you have two events that are not in causal contact with each other, which means you cannot send a signal from one to the other, not even at speed light. Schematically, “not in causal contact” simply means that if you draw a straight line through the two events, the angle between the line and the horizontal is less than 45 degrees. But look again at figure 2b. For two events that are not in causal contact, one can always imagine an observer for whom everything on this line is simultaneous. You just need to choose the speed of the observer so that the return points of the laser pulses are on the line. But if two points that are not causally related occur at the same time for someone, then each event is “now” for someone.

To illustrate this final step, let’s say one event is your birth and the other event is a supernova explosion (see Figure 3). The explosion is causally disconnected from your birth, which means that its light had not reached Earth at the time of your birth. You can then imagine that your friend Sue, the space traveler, sees these events at the same time, so according to her they happened simultaneously.

Suppose further that by the time you die, the light from the supernova has still not reached Earth. Then your friend Paul might find a way to travel midway between you and the supernova so he sees your death and the supernova at the same time. They both happened simultaneously according to Paul.

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