A PHILOSOPHY OF THE FOUR-DIMENSIONAL SPACE-TIME
The Worldview of Relative Simultaneity (MURAYAMA Akira)
CHAPTER I
Relativity and Four-dimensional spacetime
[The First half] ---- Outlining the Theory of Relativity ----3. The Relativity of Simultaneity, Time Dilation, and the Length Contraction
(4) The Lorentz Transformation and the Special Theory of Relativity
When specifying a particular place and time, we must identify the distance and direction from a landmark, as well as the time elapsed from the benchmark. However, this makes no sense in different vehicles. It is necessary to specify the vehicle you are boarding, with one being considered the static object. The necessity of specifying this vehicle has widely been recognized with regard to a specific place, but the theory of relativity introduced the necessity of specifying a particular vehicle with respect to time as well.
The conceptual framework of a vehicle, on which time and places can be specified, is called a system of coordinates, and a group of numerical values of distance, angles, and time indicating specified places and time schedules is called a coordinate. In addition, coordinate transformation refers to the conversion of coordinates expressed in one system of coordinates (the vehicle), i.e., a particular time and specific location (the space-time point), becoming represented in another system of coordinates (the vehicle).
If the speed between these systems of coordinates is consistent, transformation can be conducted by adding or subtracting those values obtained by multiplying elapsed time by speed to or from the benchmark's coordinate. This methodology represented a transformation in the mechanics that were used before the advent of the theory of relativity, and this became known as the Galileo transformation. In this model, only the space coordinate, that is, those values showing places, was transformed. Time, however, was not transformed, because the model was based on the assumption that time elapses in a common manner.
In contrast, the theory of relativity holds that time elapses differently in varying systems of coordinates (vehicles) and that both the space and time coordinates need to be transformed. This type of transformation is referred to as the Lorentz transformation. The Lorentz transformation of the space coordinate is conducted, similar to the Galileo transformation, by adding or subtracting those values obtained by multiplying elapsed time by speed to or from the benchmark's coordinate, and the figure is also multiplied by the above-mentioned time dilation γ as a coefficient. Additionally, a similar method is applied to the Lorentz transformation of the time coordinate. The transformation is conducted by adding or subtracting values obtained by multiplying distance gaps by speed to or from the benchmark time, and that figure is multiplied by the time dilation γ as a coefficient. This term of addition and subtraction represents the relativity of simultaneity. Time dilation and the length contraction are affected by this coefficient.
Many conceivable methods can derive the Lorentz transformation, and most textbooks describe Einstein's method of establishing an equation for the spherical wave of a certain light in a different system of coordinates and then establishing the coefficient of a linear transformation by solving the equation. However, this study does not explain Einstein's method in detail. Instead, the following section will describe a geometric method of using the space-time diagram and explain the Lorentz transformations.
The Lorentz transformations facilitate these innovative mechanics, as well as the derivation of the relativity of simultaneity, time dilation, and scale contraction. These factors constitute the core of the theory of relativity.
One rudimentary derivation method is the velocity-addition formula. In the Galileo transformation model, the formula was based on simple addition and subtraction. However, the velocity-addition formula in the theory of relativity based on the Lorentz transformation is more complex. More specifically, the calculation method demonstrates that if you throw an object moving at a rapid speed in close proximity to light velocity from another object moving at a similar speed, the first object will simply continue traveling at a speed close to that of light velocity instead of doubling speed. The speed of light represents the very limit of speed and no acceleration can break that limit. The closer an object gets to the speed of light, the more difficult acceleration becomes.
This can be interpreted as an indication showing that inertial mass, which represents the extent of acceleration difficulty, becomes infinitely larger as it approaches the speed of light. That is, an increase in kinetic energy means an increase in mass. The equivalence principle of mass and energy, which is represented by the well-known equation E = mc2, can be derived on the basis of the special theory of relativity. This equation became particularly famous as the basis of energy in the atomic bomb, but the equation is actually a general law encompassing the entire scope of mechanics and all energy issues.
Newtonian dynamics were hugely modified, but electromagnetic laws did not undergo any major modifications because electromagnetism intrinsically computes the Lorentz transformation. However, a minor modification was applied to electromagnetic theories on the basis of the theory of relativity, and it included the theory that magnetic power is affected by the Lorentz contraction of electric power.
Basic achievements using the special theory of relativity as a foundation were the modification of Newtonian dynamics and the theory's integration with electromagnetism through recognition that Lorentz coordinate transformation should be a fundamental operation. However, at the same time, the special theory of relativity left other problems unsolved. For example, what about gravity? The special theory of relativity could not provide any substantial clues regarding an examination of Newton's law of gravitation. However, Einstein had serious misgivings about accepting the law of gravitation as it was then understood. In fact, Einstein's strong will to solve that question bore fruit ten years later when the general theory of relativity was established. The process whereby the theory of relativity came to address gravitational fields involved an important representational development, and that was the concept of four-dimensional space-time. Let us in the following section focus on the subject of four-dimensional space-time, the major focal point of this paper.