To describe processes in nature we need a *system of reference* (sor).
A system of reference gives *coordinates* that indicate the position of a particle in space and also gives *clocks* fixed in each system that indicate time. We can imagine a clock pinned to each position.
A freely moving body is that in which no external forces act upon it. For such bodies, there exist sor in which these bodies move with constant velocity. These sor are called *inertial* or isor.
If two sor move uniformly relative to each other, if one is inertial, so is the other.
Principle of relativity: all laws of nature are identical in all isor. We know this because it is what *experiments* show. So, if we transform from one isor to another, laws must show themselves as invariant under such transformations. From now on, we cal transformations as transf. Transfs involve changes both in space and time coordinates.
In mechanics, we usually consider material particles: objects for which we consider that all its mass is focused in a single point, its centre of mass (CM). This usually means that we don't consider its rotation features.
These particles interact through a potential energy of interaction, which is described as a function of the (space) coordinates of these interacting particles. This dependence on only space coordinates means that the interaction is assumed to travel at infinite speed. Since no time appears in the functionality, for a given time, one particle feels the position of the others, no matter how far are they. A change in position of one of the particles would instantaneously perceived by our particle.
From experiments we know this is not what happens. Instantaneous interaction is just an approximated description of reality. When a particle changes its position, the potential must change accordingly, and this change propagates at finite speed until, after some time, other particles feel this change. So there is a velocity of propagation of interactions.
The speed of propagation of interaction cannot be exceeded in nature, since that would mean that we could measure such a speed, which would mean we could notice such speed, which would mean we would be affected by such a speed, and then there would be an interaction propagating faster than the propagation of interaction, which is absurd.
We can call this maximum speed of propagation of interaction as signal speed (or velocity if we include direction).
Now, an extremely counter-intuitive conclusion: the principle of relativity suggests that the signal speed must be the same in all isor! So it is a universal constant, which happens to equal the speed of light in empty space = ๐. The unicode symbol ๐, an italic c, is what we have used here. However, most of the times I am going to use a regular c. Maybe, if the letter appears within text and alone, it is better to write ๐, but if are dealing with equations, especially in relation with v as in v/c, I am definitely writing c.
The speed of light is ๐=299792458 m/s. In the book, we are given ๐= 2.998ยท10ยนโฐ cm/s. What is important here is that this speed is so big that the assumption of instantaneous interaction is a good approximations in everyday distances. For our daily experiences we can consider this number as infinite.
The principle of relativity added to an infinite signal speed constitutes the relativity of Galileo. By contrast, the principle of relativity added to the finite signal speed constitutes the relativity of Einstein. From now on, by relativistic we will mean belonging to Einstein's relativity.
Einstein's mechanics (relativistic mechanics) must be able to recover classical mechanics (non-relativistic mechanics or Newtonian mechanics) for speeds small with respect to (wrt) ๐. We usually write this limit as v/c << 1.
In Newtonian mechanics, if an observer measures its distance to an object and another observer from a different sor measures its distance to the same object, both observers will measure different distances, so distance is something that is already relative here.
However, time is absolute in Newtonian mechanics. All observers will agree on a time interval measurement, regardless of the sor in which they measure it. In particular, if two events are simultaneous for one observer, they will be simultaneous as well for every other observer.
This universal notion of simultaneity fails in Einstein's relativity.
Another thing that breaks in Einstein's relativity is the simple addition of velocities as a vector sum. This becomes clear from having a universal signal speed. If you observe a signal travelling along a direction, you must measure its speed as ๐, but if you increase your velocity in the same direction, the signal's speed cannot be reduced. It must still be ๐! If you see someone pursuing a photon at c/2, you cannot conclude that the pursuer sees the photon with c/2. Instead, the pursuer will observe the photon with c, as we do. This is one of the most counter intuitive aspects of relativistic mechanics. However, results of experiments like those of Michelson are unambiguous: the Einstein relativity principle is what agrees with empirical evidence.
Consequence: time is not absolute. Intervals of time (the equivalent of distance in space) are not universal quantities for observers. They depend on the sor. Also, two events that are simultaneous for one observer will not be for another observer.
y
| v
| y' -->
| |
| |
| |
| |
| |
|______|_____________ x v
+-----|--|--|---------x' -->
B<-A->C
Consider two frames, the K system (x,y) which we see at rest and the frame K' (x',y') which we see moving along the x axis. Both sor are inertial.
Suppose a signal is emitted by A in two directions, towards B and towards C. B and C are equidistant from A. In the K' system, the observer will see how the signals arrive to B and C simultaneously. However, for the observer in K, the signal from A to C still travels with speed c, but the point C also travels with speed v. On the other hand, the signal from A to B travels at -c but the point B travels with v. This means that for K', the time for A to C is greater than the time from A to B. So the events that were simultaneous to K' are not simultaneous to K.
Our notions of absolute time, infinite speed of signals, absolute simultaneity, etc are just approximations justified by the small speeds of daily events wrt to the speed of light.