ts = [t2 + x2 + y2 + z2]1/2 (1)
However, according to currently accepted doctrine in physics, these clocks are said to measure time, and nothing else, but time is considered a relative concept. Time itsself is said to be 'dilated' by motion and gravitation. This conception is due to Einstein [2], who abolished the traditional notion of "absolute time", considering such a quantity as immeasurable and irrelevant to physics. Einstein's conception allowed advances in physics in the absence of any method of measuring common (absolute) time in a physically objective way. The new conception of space-time was subsequently further elaborated by Minkowski [3] and discussed in depth by Reichenbach [4]. However, since the new conception of time is incompatible with the common one, which Galileo and Newton adhered to, relativity theory appears paradoxical to those who have not been adequately indoctrinated. Paradoxes arise when common sense notions are applied instead of the cranky notions required by the theory. Such paradoxes can never be resolved by any physical experiments, but only by an analysis of how people reason. Note 1
Some further confusion has been caused by the widespread practice of referring to the pseudo-space-time of Minkowski just as "space-time". This misleads one to think of it as a four dimensional manifold with three spatial dimensions to wich time is added as a fourth dimension. However, the fourth dimension in Minkowski space-time does not represent any real 'time', but an imaginary one, while the other dimensions are the three real dimensions of space.
Of course, the conceptual revolution initiated by Einstein would need a perestroika if it, unexpectedly, could be shown how time and spatial distances can be measured in agreement with common sense, i.e., without distortion by motion and gravitation. In the following it will be shown that the kind of measurements required for this can be performed by conceptually quite simple means.
For reasons to be motivated below, we are going to refer to the quantity measured by the space-time odometer as the path time ts. For linear motions, this is just the linear distance
ts = [t2 + x2 + y2 + z2]1/2 (1)
in a reference system that is at rest with respect to the background radiation, i.e., in which this radiation appears to be isotropic. From the perspective of this instrument, there is no difference between temporal and spatial distances. Therefore we can do without the velocity of light in this equation. The common absolute unit, the zero-crossing period, corresponds to roughly 1.8 ps and 0.53 mm.
When at rest with respect to the background radiation, a space-time odometer measures just time, and it functions always more like a clock than like a spatial odometer. Only when it reaches the velocity of light, its motion in space contributes to the same extent to the accumulation of zero-crossings as its movement in time. The measured path time can never be shorter than the time interval and never longer than 21/2 times as much. This is the limiting value that would be obtained if the instrument was moving at the velocity of light.
Suppose that the light-clock is moving with constant velocity in the direction of the coordinate x. We orient the ray in a direction transversal to that of the motion as seen by a co-moving observer. It is easy to see that the length L that the light pulse travels between the counting events increases with the velocity of the clock. Since
y)2 + x2, (2)
where
x = L v/c, (3)
we obtain
y / (1 - v2/c2)1/2. (4)
The unit of measurement increases as a consequence of this increase in path length. In Einstein's theories, this phenomenon is called time dilation. It is known that the pace of an atomic clock depends on motion and gravitation in the same way as that of a light-clock, and experiments such as those by Hafele and Keating [1] have shown that the behaviour of such clocks is adequately described by the theory of relativity. Nevertheless, if we want to avoid a conflict with the common people's concept of time, we have to say that these clocks do not measure time t, but a related quantity that we shall refer to as the proper time to. This expression has some tradition within the frame of relativity theory, but in the present frame it has a wider range of application. (The expression "path time" is analogous to this - and it is even more adequate to speak of a 'time' here, since path time can never deviate so much from common time as proper time can.) For linear motions, it holds in general that
to = [t2 - x2 - y2 - z2]1/2, (5)
which reminds one of equation (1). Comparing these equations, we see that the passage of time affects the readings of the two instruments in the same way, while motion in space affects them in opposite senses. This holds provided that the behavior of the light-clock is analyzed in a reference system in which the background radiation appears isotropic. The space-time odometer does not give us any choice in this matter.
If the distance between the mirrors is constant, the path length L of the light pulse increases when the clock is oriented differently, e.g., with the path in parallel with the coordinate x. In this case, the pace of the clock would slow down, since
x [1/(1 - v/c) + 1/(1 + v/c)], (6)
i.e.,
x/(1 - v2/c2). (7)
However, the distance between the mirrors does not remain constant when the clock is rotated. The length of a rigid body is given by the interactions between the particles of which it consists, mainly by the electrical interactions among electrons and protons, and it has been shown that the distances in such a system of particles decrease in the direction of motion when it moves through a stationary 'aether' [7] so that
x' = x (1 - v2/c2)1/2. (8)
Due to this phenomenon, known as the FitzGerald-Lorentz contraction, the pace of the light-clock remains unaffected by rotation. Note 2
However, true measurements of time and spatial distances can also be made very simply with a space-time odometer and a light-clock that is closely linked to it. For linear motion and for sufficiently linear pieces of more complex motions, temporal (t) as well as spatial distances (s) can be calculated from the path time interval
ts = [t2 + s2]1/2 (9)
measured with the space-time odometer, and the proper time interval
to = [t2 - s2]1/2 (10)
measured with the light-clock linked to it, as
t = [ts2/2 + to2/2]1/2, (11)
and
s = [ts2/2 - to2/2]1/2. (12)
Thus, for cases in which the effects of gravitation can be neglected, it is evident that time and spatial distances can be measured in agreement with common sense. A good clock, that gives us a valid measurement of t can be realized by implementing equ. (11). This implies also that good clocks can be synchronized with any number of other good clocks, irrespective of their mutual distances and velocities, by comparison with a good clock that is moved around to each place.
Therefore, a space-time odometer that remains at a constant distance from a local center of gravity will be affected as if it was moving at vesc, and its pace will increase so that
ts = t(1 + vesc2/c2)1/2. (14)
A light-clock in the same place will also be affected as if it was moving at vesc, and its pace will decrease accordingly. The detailed analysis of this involves also length effects, as in the case of a moving light-clock. The general theory of relativity which, as far as we know, correctly describes the behaviour of the light-clock, predicts according to Schwarzschild [9]
to = t(1 - 2GM/rc2)1/2. (15)
Since we can substitute vesc2 for 2GM/r, this gives us
to = t(1 - vesc2/c2)1/2. (16)
The equations (14) and (16) can be seen to be equivalent with (9) and (10). Due to this equivalence of the effects of gravitation and velocity of motion, the equations (11) and (12) remain valid also in this case. If this holds in general, space-time can be surveyed in agreement with common sense without any need of knowing how much of an observed discrepancy in pace between the two instruments should be ascribed to gravitation. Good clocks, as defined at the end of section 1.3 remain good clocks even if brought into and/or out of local gravitational fields. However, initially we have to calibrate each space-time odometer and light-clock so that they run at the same pace when they are outside any strong local gravitational fields and at rest with respect to the background radiation.
In the general theory of relativity, the notion of gravitation has been re-interpreted in addition to that of time. Unlike the other types of interaction, it is treated as a property of space-time. Due to the universality of gravitation, it may be possible to describe the world in this fashion, but if we survey space-time in the way described here, we obtain a more readily intelligible Euclidean description that calls for treating gravitation more like the other fundamental interactions.
"In the last lecture, we described a number of relativistic effects, including the notion that "moving clocks run slow." Today, we're going to delve a little deeper into that idea, to give you a little more idea of how this works. ¶ Abandon All Preconceived Notions, Ye Who Enter Here. You already have intuitive ideas about how time works. Unfortunately, they're probably wrong. It will take a while to get used to some of the more counterintuitive ideas contained in Relativity. In the meanwhile, the best thing to do is forget everything you think you know about time, and work on figuring out the ideas of Relativity. ¶ What is Time? What is a Clock? ¶ If we're going to talk about time, we need to first decide what time is. Even though we all have intuitive ideas about what time is, and how time passes, if you try to nail down your intuitive ideas, you'll find that they're not as concrete as you thought. ¶ We can get around this problem by defining time operationally. This means that we're not going to worry about what time "really" is, we're just going to figure out how to measure it, and call whatever we're measuring "time." So, the operational definition of time (and the definition we're going to use) is: "Time is what you measure with a clock." ¶ So what is a clock? ¶ If time is something you measure on a clock, then what is a clock? One good definition of a clock is something that measures regular intervals of time. Is this circular? You bet....but since we don't "really" know what time is, it is very difficult to avoid circular definitions."
When you have read the present paper, you will understand that there is nothing wrong with the preconceived notion of time. It can be defined operationally and without circularity. An interval of time is the non-spatial distance between two events, and it can be measured using the method described. The clocks referred to in the quotation do not produce a valid measurement of time. They measure proper time, which is a different concept that is to be given a different definition.
(2) In contemporary teaching of Relativity, the FitzGerald-Lorentz contraction is often downplayed as an "ad hoc hypothesis", as described, e.g., in Zur speziellen Relativitätstheorie, and it is left unclear whether this phenomenon is "real" or just "apparent". In his enjoyable paper, How to teach Special Relativity, J. S. Bell makes it clear that honoring the Lorentzian view does indeed have certain pedagogical advantages.
(3) To say it more clearly: It would be philosophically, pedagogically, and ethically wrong to continue teaching Relativity in the same mentally violent way as before (cf. Note 1).