Time, in physics, is a vague concept. You can look it up here, and you can read Einstein's account of it in Relativity: The Special and General Theory, and you will come away unenlightened.
The difficulty, as I see it, is the circular relationship between time, space (distance), and speed:
t = d/v, where
t = time
d = distance
v = velocity (speed), which is d/t (as in miles per hour).
Substituting for v in t = d/v:
t = d/(d/t)
Transposing:
t(d/t) = d
Combining terms:
td/t = d
Canceling terms:
d = d
By substitution of 1 for d:
1 = 1
The same result obtains for d = vt and v = d/t. Which gets us nowhere, so to speak.
What needs to be done -- and is done -- is to define t arbitrarily as the interval between ticks of a standard clock, define d as the space between two points on a standard rod, and take d/t as the measure of v. That's how the speed of light (c) was established.
Time is therefore an arbitrary measure of the interval between successive events. But the arbitrary measure is applicable only within in the rigid frame of reference (e.g., a building) in which the measurement occurs. Distance also depends on the frame of reference in which it is measured. Speed therefore depends on the frame of reference in which time and distance are measured. Specifically, according to special relativity,
Moving clocks are measured to tick more slowly than an observer's "stationary" clock.
... Objects are measured to be shortened in the direction that they are moving with respect to the observer.
These effects are reciprocal. An observer in the "moving frame" of reference sees no change in time or distance in his frame of reference. But he sees the clocks ticking more slowly and objects shortening in the "stationary" frame of reference. (The "quibble marks" indicate the arbitrariness of designating one frame of reference as stationary and the other one as moving; each is moving relative to the other.)
The reciprocity suggests that time dilation (clocks ticking more slowly) and length contraction (objects shortened in the direction that are moving) may be tricks of perspective. But tests seem to show otherwise.
Nevertheless, I must ask why much made about length contraction but nothing is said of height contraction? Is it because the graphical treatment of special relativity necessarily focuses on length? Or is it because it would be embarrassing to discuss height contraction, which is obviously a phenomenon of distance (perspective), not speed?
There is also an inconsistency in the Lorentz transformations for time and length (which are at the heart of special relativity):
It is said to be impossible for an object to attain the speed of light. When v = c, the equation for time dilation (above left) is indeterminate (the divisor is zero), while the equation for length contraction (above right) yields a value of zero. Those results are contradictory. The first is consistent with the impossibility of light-speed travel; the second implies that it is possible.
Finally, the speed of light in vacuum is said to be a universal constant, and to be the same for all observers regardless of their direction and speed of travel. The idea in the second part of that sentence causes a lot of confusion. If the speed of light is constant, then it must be the same for all observers (that is, when appropriately measured by every observer). But if the speed of light is constant, it cannot be the same relative to the speeds of all observers. (See "Getting Light Right" in "Einstein's Errors".) If it were, the Lorentz transformations would be different -- c would increase as v increases.
Here's another way to look at it. If Mr. Speedy is traveling at half the speed of light -- 0.5c (whew!) -- that's his speed relative to the speed of light. But if he has a flashlight and turns it on, the resulting beam of light will travel away from him at the speed of light -- not at half the speed of light. His forward velocity has no effect on the speed at which the beam of light travels.
This contradicts a principle of Galilean relativity: velocities are additive. Take the case of a pitcher throwing a baseball while standing in an enclosed rail car (with a glass side) traveling at 60 mph. If the speed of the baseball relative to the pitcher is 90 mph, then the speed of the baseball relative to an observer standing on an embankment by the train track is 150 mph.
But there is no contradiction. If the observer on the embankment measures the speed of the baseball relative to speed of the pitcher, he will conclude that the speed of the baseball is 90 mph, not 150 mph. Similarly, an observer of Mr. Speedy will see that the speed of light emitted by Mr. Speedy's flashlight is c, not 1.5c.
Here's the real difference: The speed of the baseball relative to the observer on the embankment is 150 mph (the speed of the train relative to the observer plus the speed at which the baseball is thrown relative to the pitcher). The speed of the beam of light relative to the observer of Mr. Speedy remains c, regardless of Mr. Speedy's velocity when the light is emitted.
How is that possible? Here's my explanation:
The train is a frame of reference, of which the pitcher and baseball are part. Mr. Speedy and his flashlight constitute a frame of reference (you can add a rocket ship for Mr. Speedy, if you like).
The embankment by the train track is a separate frame of reference, of which an observer is part. The rocket ship that zooms along parallel to Mr. Speedy's is a separate frame of reference, of which an observer is a part.
Light is its own frame of reference. It is a massless object with the peculiar characteristic that it moves at the same speed when observed from other frames of reference, regardless of their speeds relative to one another or to light.
You might think of light as analogous to the background for the actors in a play or film. The actors' movements don't affect the background, nor does the background affect the actors' movements. Thus time and space (the actors) play their parts against a fixed background (the speed of light in vacuum).
But ... the speed of light varies according to the medium it traverses. It isn't a fixed background. Therefore, it isn't the measure of all things.
Is there such a measure? (If so, is it the Higgs field?) Or is everything relative?