Vinaire's Blog

Reference: Einstein’s 1920 Book

Section XVII (Part 1)
Minkowski’s Four-Dimensional Space

Please see Section XVII at the link above.

.

Summary

The world in which we live is a four-dimensional continuum. It is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates x, y, z and a time co-ordinate t.In classical mechanics, t is considered to be independent and absolute; but in the theory of relativity, t is not independent of x, y, and z.

Pure “space-distance” of two events with respect to K results in “time-distance” of the same events with respect to K’. Therefore, space and time are related on a larger scale, though this is not so apparent in the material domain. Minkowski made this very clear by presenting the time coordinate as “√(-1) ct“ so that the time co-ordinate plays exactly the same rôle as the three space coordinates.

Minkowski’s idea made the mathematical foundation of the general theory of relativity possible.

.

Final Comments

The four dimensions are length (x), width (y), height (z) and time (t). The dimensions are like scales and x, y, z, and t, called co-ordinates, represent specific values on those scales. These dimensions are attributes of substance. For example, the three space coordinates x, y, and z represent “a location in the space occupied by substance”; and the time coordinate t represents “the duration of substance at that location.”

The important point is that these dimensions do not exist in the absence of substance. A point  in a space occupied by substance is not dimensionless. It has four dimensions, and the fourth dimension represented by t is the inertia (rigidity or flexibility), associated with that point. Minkowski’s contribution made it possible to model this property of inertia, or consistency, mathematically.

Time (duration) is the inertial dimension. This inertia is very large throughout the material domain, and the changes in it are imperceptible; so the matter is very rigid. But this inertia is negligible throughout the energy domain, and it is hardly perceptible; so the energy is very flexible.

The four dimensions together represent the continuum of substance. There is no empty space; thus, we have matter floating in energy. There is no clear separation at the boundary between matter and its surrounding energy.

The consistency decreases from matter to energy. As a result, the flexibility increases, which manifests itself as motion. So, there is a natural relationship between consistency (inertia) and motion (velocity). This is the meaning of the contribution of Minkowski to four-dimensional space.

.

Reference: Einstein’s 1920 Book

Section XV (Part 1)
General Results of the Theory

Please see Section XV at the link above.

.

ith the velocity of light always takes the place of instantaneous action at a distance or of action at a distance with an infinite velocity of transmission.

Summary

The origins of the special theory of relativity lie in electrodynamics and optics. In its turn, the special theory of relativity has considerably simplified the theoretical structure of electrodynamics and optics.

Classical mechanics required to be modified, mainly in regards to rapid motions, before it could come into line with the demands of the special theory of relativity.

The special theory of relativity points to the equivalence of matter and energy by conserving them together. In other words, it looks at energy as a form of substance.

The conclusion that matter and energy are equivalent is an important one. [NOTE: This prediction of energy converting into inertial mass, and vice versa, was prophetic. It led to the development of atomic bomb and nuclear reactor.]

According to the theory of relativity, action at a distance with the velocity of light always takes the place of instantaneous action at a distance or of action at a distance with an infinite velocity of transmission.

.

Comments

Electrodynamics and optics basically introduced the idea of lines of force replacing the idea of empty space. These lines of force form the basis of light in the special theory of relativity. According to Faraday, these lines of force flow in space as light, and they become concentrated in atoms to form mass. This provides the basis for the equivalence of energy and matter. This also provides a bridge between classical mechanics, electrodynamics and optics.

The degree of concentration of the lines of force may be called the “consistency of substance.” Light has very low consistency but very high velocity. On the other hand, matter has very high consistency but very low velocity. Thus, consistency seems to have an inverse relationship to motion. It is the change in consistency that manifests as the change in the velocity of a material body. However, this change denotes the mass that is being converted into energy (and vice versa), which is so small that it is imperceptible. The magnitude of this change in consistency may be assessed from the third term in the mathematical series above. It is of the order of 10^-8, which is obtained by substituting v by  the velocity of the earth. 

The Lorentz transformation uses the extrapolation of consistency between light and matter to determine the relativity of motion of material bodies. This extrapolation works in the range of matter only. It takes into account the imperceptible change in consistency that contributes to motion of material bodies. This is how the special theory of relativity modifies classical mechanics. 

The lines of force provide substantiality to space in the form of a “fabric.” The word “space” then defines the extents of this fabric. The word “time” then defines the durability of this fabric at any location in space. Both space and time shall vary in their characteristics as the “fabric” varies in its consistency. 

A totally rigid coordinate system represents a “fabric” of space that is rigid like matter. Any body attached to this coordinate system shall exhibit infinite inertia and a state of no motion. As the coordinate system becomes less rigid, the body attached to it shall display a decrease in inertia and an increase in motion. The amount of motion gained shall depend on the reduction in inertia.

The law of addition of velocities applies when the reduction in inertia is infinitesimal and can be ignored. When the change in inertia cannot be ignored the Lorentz transformation replaces the Galilean transformation. However, when the change in inertia becomes large even the Lorentz transformation becomes inapplicable because it is also an approximation. 

The electromagnetic spectrum may be now expanded into a spectrum of substance. This spectrum shall extend from energy of lowest frequency to matter of highest density. It will show decreasing motion with increasing consistency.

This means that the orbits of planets in the solar system purely are the result of a balance among their inherent consistencies or motion. We see this balance as gravity. It defines gravity as being more than merely a force.

.

Reference: Einstein’s 1920 Book

Section XIII (Part 1)
Theorem of the Addition of Velocities. The Experiment of Fizeau

Please see Section XIII at the link above.

.

Summary

To verify the results from the special theory of relativity, an experiment needs to be devised in which the v/c ratio is small but not insignificant, so one can see that more than simple addition of velocities (per Galelei transformations) is required to match the reality.

An experiment, which was performed by Fizeau, meets this requirement. When the experimental results are compared with theoretical calculations, they are verified by the transformation of the special theory of relativity, rather than the simple addition of the two velocities. The experimental result thus validates the special theory of relativity.

Fizeau’s experiment presents the circumstance in which the theorem of the addition of velocities employed in classical mechanics does not work. Maxwell-Lorentz theory explained the difference by the use of particular hypotheses as to the electromagnetic structure of matter. But the special theory of relativity uses more general and a much simpler hypothesis to come up with the same conclusion.

.

Comments

Addition of velocities is valid when the velocities are of about the same order of magnitude. Lorentz transformation becomes crucial when the velocities to be added are very different in their orders of magnitude.

.