Introduction
This property appears in the theory of special relativity. Of course, it is also included in general relativity since locally the laws of special relativity apply. is preserved
A velocity invariant in the theory of special relativity
1-The principle of relativity
Einstein had made two assumptions when he finalized the theory of special relativity (1905).
- The laws of mechanics and electromagnetism are the same in all inertial frames (Galilean frames). In these inertial frames one does not « feel any force »: we float, like astronauts in the International Space Station where in fact it is not totally an inertial-frame, just a micro-gravity frame, but it is for illustrating the phenomenon.
- These inertial frames are deemed « indistinguishable » (no absolute or preferential frame): observers in a closed and opaque cockpit (without a view to the outside), could not determine, by local experiments of physics or electromagnetism, whether they are at rest or in uniform motion in relation to a supposed inertial absolute “rest” frame (the absolute space of Newtonian mechanics).
- This is the principle of relativity: no privileged inertial frame, they are all equivalent.
- To satisfy it, it is necessary to establish a certain number of equations describing the laws of transformation of coordinates, (it was the Lorentz transformations that he established empirically, without any epistemological base for explaining the negative result of the Michelson-Morley experiment), between several inertial frames of reference.
- Einstein’s derive these transformations from epistemological principles.
- This is in total rupture with Newtonian mechanics and has consequences that go far beyond what can be imagined by reasoning in a context of space and time such as that used in Newtonian mechanics, where we can have the impression that imposing a velocity invariant is only an accessory constraint.
- This will involve to get rid our fundamental and independent concepts of time and space.
- Emergence of spacetime.
- Indeed, as Minkowski would declare shortly afterwards: « The only physical reality is spacetime (a new concept) whose space and time are no more than shadows.
- This spacetime, defined locally by the spacetime interval generally denoted « ds2« , a bilinear form including space and time, (in fact it is a « tensor », the bilinear form being its representation in analytic geometry) is an invariant.
- All inertial observers, regardless of their relative uniform velocities, for a given phenomenon, agree on the value of this ds2 (and that of s2 which results, macroscopically, by integration), while they will disagree for the measurements of space and time separately.
- This is called loss of universal synchronization. Two simultaneous events for an inertial observer will not be simultaneous for all inertial observers. This results from the fact that there is no absolute space and time, as in Newtonian mechanics that served as an absolute reference frame, in relativity.
2-The velocity of light is the same in all inertial frames of reference [1]
A velocity invariant already predicted by the principle of relativity
This second postulate, which seemed as fundamental as the first, in fact, specifies an experimental data on something that was already present in the first postulate, namely a velocity invariant. Indeed, the equations to establish the transformations between inertial frames shows the existence of a velocity invariant, but does not specify its value.
If this constant is infinite (instantaneous propagation) we find Newtonian mechanics, if, for satisfying the observations, we set it equal to the velocity of light we obtain special relativity.
Note that in fact light is only a marker of a property of spacetime of which this limit velocity is a structural property. This is developed in other pages of this site.
An other physics
This property, which may seem innocuous in a Newtonian context, will totally upset physics because it gives to the spacetime a « hyperbolic » structure with folios that, if properly described by equations, is very difficult or impossible to conceive by our mind forged in space and time.
It is then necessary to trust mathematics, renouncing (at least temporarily?) any desire for conceptualization, to admit phenomena that defy our mind (for example Langevin’s paradox of travelers).
Experimental confirmations of the strangeness of relativity
We still have the experience to verify that mathematics does not lead us astray.
Although the effects of spacetime are very small in our physical world, the accurate observations we have been able to make, subject to misinterpretation, confirm what mathematics predicts.
A paradoxical situation for our mind
As mathematics is a product of the human mind, we are in a paradoxical situation where our mind produces concepts that it is unable to understand.
This has been discussed in more detail in other pages on the site, it results from the mechanism of functioning of our brain that can only conceive of what it has been confronted with.
The heuristic power of mathematics
Obviously, mathematical language is more powerful in describing nature than are our conceptions from our usual experience.
Indeed, the structure of the formalism that we have been led to develop to treat a problem, when it turns out that it is particularly adapted to a phenomenon to be treated, which is manifested by its simplest and most synthetic representation, informs us about the structure of the phenomenon itself, since this adaptation can only result from a morphism between the two entities: method of best fit.
The adapted tool informs us about the object to which it is adapted. This reveals the exceptional heuristic power of mathematics.
Let us recall a few examples.
Penrose, associated with Newmann, on his deep conviction that null geodesics play a fundamental role in general relativity (especially in causality) develops in the 60s a strange formalism, in total rupture with uses, totally counterintuitive, but which accounts in the simplest way of the phenomenology of Kerr and Kerr-Newmann black holes.
The study of spatial rotations in Euclidean space results in a group SO(3), whose associated Lie algebra shows that there is another more fundamental group, the SU(2) group, which shows that it takes a rotation of 720° to return to the initial state and not 360° as might have been assumed and as the SO(3) group seemed to indicate.
Again, it was mathematics that revealed the solution.
One might object that mathematics only provides formal information, whereas we consider physical problems. But, in fact, what matters are the laws of nature, which are formal relationship, and when we analyze knowledge more closely we realize that all science is described in terms of relations governing actions between objects, of the same nature between them and with others. It is the action that counts, an object that does not act does not exist (Leibnitz).
Additional information can be found on other pages of this site.
A constructive approach
In conclusion, it is up to us to work to reconcile these two parts of ourselves. We see that for this we have a powerful tool. It is mathematics that should serve as a guide to achieve this.
This may lead us to an improvement in our way of thinking. We can only hope for it and still have a certain confidence, because the situation although paradoxical is not without interest because it allows us to make progress in science even if it is in our defending mind!
To conclude, we add that in quantum mechanics we are confronted with this same type of problem, see other pages of this site.
[1] The velocity of light, relative to all inertial observers, is the same but not their frequency. Let us recall that for light, the affine parameter is the momentum bound to the frequency. In other words, null geodesics of different frequencies are different null geodesics. The null geodesics may provides a system of reference. See the page:
Spacetime, space, time. Spacelike, timelike and null coordinates. What is physical reality? (upated 03-11-21), on this site and related links:
http://astromontgeron.fr/s%C3%A9minaire-maths-philo_2019.pdf http://astromontgeron.fr/SR-Penrose.pdf