[Physics FAQ] - [Copyright]
Don Koks, 2018.
Yes, they most certainly are. They are both a result of what we measure—and what we measure is as real as things get.
Time dilation is easier to understand: we can easily measure if, say, a fast-moving particle is decaying after a longer time than it would if it were at rest. But Lorentz contraction still confuses people. Consider that simple discussions in relativity can go awry when the distinction between "seeing" and "observing" (or "measuring") is not made clear. What we see is what we really see with our eyes or cameras: this has visual perspective, and can contain optical illusions due to the distances of objects, such as the apparently faster-than-light motion seen of matter that is being ejected from some galaxies. "Seeing" can contain the Doppler effect, which is due to changing arrival times of signals at our eyes or measuring gear. Contrast this seeing with what we observe or measure, which you can envisage as the end result of seeing after we have subtracted all the complications caused by signal travel times and distances of objects. For example, even outside the realm of relativity, if you hear two explosions at different times, this does not mean they really happened at different times. Once you have corrected for the travel times of the sound from possibly different distances, you might conclude that they really occurred at the same time. So, you heard them happening at different times, but you observed that they happened at the same time.
In fact, in relativity we seldom analyse a situation on paper by asking what is seen and then accounting for signal travel times. Instead, we invoke the properties of a frame. A frame is a collection of observers, each equipped with all manner of measuring devices, and who each measure only what is happening in their own immediate vicinity. What makes them a frame are two properties:
(I presume that those who make the above incorrect statement about a receding clock have never thought about the case when the clock is moving toward you: when it's moving toward you, the Doppler effect means that you will see it ticking quickly, even though relativity says it is actually ticking slowly.)
Similarly, this kind of analysis makes it clear that when we observe a moving object, its length is contracted. What we see is more complicated: see the FAQ entry on Penrose–Terrell Rotation for a discussion of that. The fact that this contraction is frame dependent is neither here nor there. As an analogy, consider the air flow over an aircraft's wings that makes it fly: this is frame dependent too—it changes as the plane flies faster—but if you insist that this air flow is not real in the aircraft's frame, then while you might still be able to crunch numbers in an aircraft-engineer exam, you will forever be wondering how aircraft can fly. So, don't think that the Lorentz contraction is not real. It's as real as anything else.
The fact that the air flow over an aircraft's wings does change in a particular well-defined way when we change frame is a reflection of the fact that the whole scenario of the aircraft flying through air is "real", and this can be described using the elegant maths of vectors. Similarly, when certain measurements of an object change in a particular well-defined way when we change frame in relativity, this is also an indicator that what is going on is real. And that is the idea behind using tensor notation in the maths of advanced relativity.