In the common narrative of the history of science, one finds the story that classical mechanics was replaced by Einstein's theory of relativity and is contained in the theory of relativity as a special case for small velocities. On closer inspection, however, this narrative is not quite right.
As originally formulated by Einstein in 1905, special relativity describes the transformation between reference systems that move uniformly in a straight line relative to each other, only. Accelerated motion or motion with constant velocity on curved paths cannot be described in this way. General relativity deals with accelerated motion due to gravity, but not accelerations due to other forces. Newtonian mechanics, on the other hand, can describe both uniform linear motion and accelerated motion - regardless of the nature of the force - within the same theoretical framework. It is based on Newton's three axioms:
Law of inertia: A body on which no forces act remains at rest or moves in a straight line at a constant speed.
Law of force: Force is mass times acceleration.
Actio equals Reactio: If the body A acts on the body B with the force F, then B acts on A with an opposing force of the same magnitude.
In the further development of relativistic mechanics, attempts were made to find a generalized equivalent for Newton's law of force, which was successful. Today's textbooks use the four-vectors, which go back to the Göttingen mathematician Hermann Minkowski (1864-1909). However, the description using four-vectors often involves mathematical expressions that are not measurable quantities and therefore have no direct physical meaning. For example, the four-vector velocity is defined as uµ = (γ · c, γ · v) with the Lorentz-Ffctor γ = (1 - (v/c)2)-1/2 . The spatial components of the speed of four therefore do not correspond to the measurable speed v, but are greater by the Lorentz factor γ, which leads to faster-than-light speeds at high values of γ; in addition, the zeroth component γ . c is generally greater than the speed of light c for every moving object. The occurrence of faster-than-light speeds and not directly measurable descriptive variables is questionable from the perspective of scientific theory. An alternative formulation of relativistic mechanics, whose theoretical terms are closer to the actually observable quantities, would be desirable
Another headache is the twin paradox. In this thought experiment, one twin travels in a rocket from Earth to a distant star and then returns to Earth, where the other twin is waiting for him. According to relativity, all inertial systems are physically equivalent. If the twin in the rocket travels at a constant speed, each twin can assume that he is at rest and the other twin is in motion. Due to the relativistic effect of time dilation, each twin will have the impression that the other twin's clock is slowing down during the trip. However, when they meet again on Earth, it turns out that less time has actually passed for the twin who was in the rocket, and his clock shows a shorter period of time since departure than the clock of the twin who remained on Earth. This paradox is explained by the fact that only the twin in the rocket was exposed to acceleration, which he could objectively determine at the moment of acceleration during the take-off from Earth and during the turn at the distant star.
However, for this explanation to work, the twin's clock in the rocket would have to have a "memory" of the fact that it was accelerated and how long it traveled at what speed. In the formalism of relativity, however, there is no quantity in which this memory could be stored. Physical quantities such as position, velocity, and momentum are measured relative to their respective reference systems - according to Einstein's theory, there is no good reference system against which an acceleration could be measured in absolute terms. In both classical mechanics and relativity, the physical behavior of an object depends solely on its state of motion at the start of the observation and any forces acting during the observation period, but not on events in the object's past before the observation began. However, the twin paradox can only be resolved and reconciled with the facts of observation if it is conceded that the theoretical description lacks a quantity that can be used to describe the sum of the accelerations that an object has experienced in its past.
In classical mechanics, several mathematical formulations are known that are based on differential equations. In addition to Newton's formulation, Hamilton's phase space representation is also commonly used. Both representations are equivalent to each other and contain all physically relevant aspects equally. The two theories of relativity, on the other hand, each describe only selected types of motion (uniform rectilinear motion or accelerated motion due to gravity) and also differ fundamentally in their mathematical structure.
Special relativity is based on transformations between coordinate systems, called Lorentz transformations. General relativity is completely unusual - even compared to all other physical theories. It is the only theory that describes a force effect by changing the space-time metric. In his later years, Einstein tried to apply this conceptual approach to electromagnetism, but ultimately failed. As much as general relativity is admired, one should not rule out the possibility that the geometric formulation via a curvature tensor may have led to a dead end. If we want to make the description of gravity compatible with other physical theories, we will probably have no choice but to reformulate general relativity on a different mathematical basis.
Against this background, we would therefore like to pose the question: Can relativistic mechanics be formulated in a unified mathematical corset for arbitrary forces?
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