Abstract
Time has been discussed in philosophy, metaphysics, and physics. It continues to be a confusing notion to many people, especially because of a lack of understanding of Einstein’s relativity theories, which are often misinterpreted and confused with the subjectivity of observers or even their psychological states. It has even been questioned by some whether “time” exists, that it may be only an illusion created by our minds.
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Notes
- 1.
The correct relativistic rule is as follows. If the moving observer has speed v relative to a static observer, and if the moving one throws a particle with speed V in the same direction of motion, then the static observer will see the particle moving at a total speed \(V_{{\rm{total}}} = \frac{{v + V}}{{1 + \frac{{vV}}{{c^2 }}}}\). If the direction of motion of the particle is in the opposite direction, then V is replaced by −V. In this formula, if you replace V by c, then you will find V total = c, and if you replace V by −c, then you will find V total = −c. This formula illustrates how all inertial observers moving at any v see the same speed of light c. Furthermore, for any particle velocity V, if the moving observer himself has the speed of light, v = c, then again we find V total = c, indicating that the static observer sees the other observer as well as any particle in his frame moving at the speed of light. Note that for speeds much smaller compared to c, we can neglect \(\frac{vV}{c^{2}}\) compared to 1. Then the formula above reduces approximately to V total ≈ υ + V, which agrees with experience in everyday life.
- 2.
According to relativistic rules, the relation between the momentum, the rest mass, and the energy of a freely moving particle is determined, as shown in Fig. 3.4, by a right triangle. The perpendicular sides are proportional to the momentum pc and rest mass mc 2, while the hypotenuse represents the energy \(E = \sqrt{\left( pc \right)^{2} + \left( mc^{2} \right)^{2}}\). For any momentum p, the speed of the particle is determined by the ratio of the vertical side to the hypotenuse, \(\frac{v}{c} = \frac{pc}{E} = pc/\sqrt{\left( pc \right)^{2} + \left( mc^{2} \right)^{2}}\). This gives the relativistic relation between rest mass, momentum, and velocity. From the figure one can see that the ratio \(\frac{v}{c} = \frac{pc}{E}\) is always less than 1 as long as the horizontal side is not zero. Therefore, a massive particle m ≠ 0 must always travel at speeds v less than c. Two limits of the triangle in Fig. 3.4 are of particular interest. Consider at first a very short horizontal side, when the hypotenuse approaches in length the vertical side. This represents a fast moving particle since v/c approaches 1, the maximum possible. For massless particles m = 0, such as the photon or light, the horizontal side collapses to zero. Then the hypotenuse has exactly the same length as the vertical side, and therefore v = c. So massless particles must always move at the speed of light v = c. Massless particles cannot stop no matter how small their momentum or energy is. Next, consider the opposite limit of a very short vertical side, when the hypotenuse approaches the length of the horizontal side. This represents a slow moving particle since v/c gets small. In the zero limit, the momentum is zero p = 0, so the particle is at rest v = 0. Then the energy E reduces to its rest energy E = mc 2, same as the horizontal side. When the horizontal side is much larger than the vertical side, which happens either when the rest energy mc 2 is very large, or the momentum pc is very small, then the formula for the energy E can be approximated by \(E = mc^{2} + \frac{p^{2}}{2m} + {\rm smaller\,terms}\), where mc 2 is dominant by far. The small part \(\frac{p^{2}}{2m}\) is the kinetic energy (due to motion) of a slow particle, and it agrees with Newton’s laws for kinetic energy. Similarly, in the same approximation, using E ≈ mc 2 we obtain \(v \approx \frac{p}{m}\) from the speed formula, which also agrees with Newton’s relations p = mv among momentum, mass, and velocity. So, Newton’s formulas for energy and speed are valid only for the case of sluggish particles since they are only approximations to Einstein’s relativistic formulas.
- 3.
In particular, one can define the concept of the invariant relativistic distance between any two events that any inertial observer would obtain when moving at any constant speed. This quantity is given by the following simple formula:
$$({\rm invariant\,distance})^2 = ({\rm space\,interval})^2 - c^2 ({\rm time\,interval})^2.$$We have explained before that the space or time intervals between any two events are different when measured by a static observer versus a moving one. Despite the fact that these intervals are different, for different inertial observers, the combination above gives the same numerical value for the invariant distance, for either the static or the moving inertial observer. For example, if a particle of light travels from one point to another, we know that every observer must see the light signal moving at the speed of light. So for every inertial observer, the space interval, the time interval, and the speed must satisfy the standard formula (space interval) = c × (time interval). In that case, the formula above would say that, for any two events connected by a light signal, the invariant distance must be zero, according to the measurements of every inertial observer. Such events are said to be separated by a light-like invariant distance (i.e., 0). Similarly, when two space–time events yield a positive value in the formula above, they are said to be separated by a space-like distance, and when they yield a negative value they are said to be separated by a time-like distance. For example, consider the two space–time events defined by you sitting in your chair looking at your watch at one instant, waiting a minute, and looking at it again. The space interval between the two events is zero since you did not move at all, while the time interval is 60 s. The formula above gives a negative result; this is consistent with the fact that the second event is separated from the first by a time-like invariant distance. Matter, energy, or people are able to travel between events separated by time-like or even light-like distances, but not between events separated by space-like distances. So, the universe we see today is only that part of the early universe that is connected to our present space–time by time-like or light-like signals. We cannot see those regions of the early universe that became space-like separated relative to our visible universe by moving faster than light during inflation.
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Bars, I., Terning, J. (2010). What Is Space–Time?. In: Nekoogar, F. (eds) Extra Dimensions in Space and Time. Multiversal Journeys. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77638-5_3
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