Geometric Representation

Spacetime:

 Spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum (a continuous sequence in which adjacent elements are not perceptibly different from each other). Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive where and when events occur.

Minkowski Space: Minkowski space (or Minkowski spacetime) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.

Minkowski Space:

Minkowski space is closely associated with Einstein's theory of special relativity, and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frame of references will agree on the total distance in spacetime between events. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.

Space-Time Diagrams:

According to classical physics, the time coordinate is unaffected by a transformation from one inertial frame to another i.e. the time coordinate, t', of one inertial system does not depend on the space coordinates, x, y, z of another inertial system, the transformation equation being t' = t.

In relativity, however, space and time are interdependent. The time coordinate of one inertial system depends on both the time and the space coordinates of another inertial system, the transformation equation being.

t′= {𝑡−(𝑣𝑥/𝑐2)}/ √(1−𝑣2𝑐2)

Hence, instead of treating space and time separately, as is quite properly done in classical theory, it is natural in relativity to treat them together. H. Minkowski was first to show clearly how this could be done. In what follows, we shall consider only one space axis, the x-axis, and shall ignore the y and z axes. We lose no generality by this algebraic simplification and this procedure will enable us to focus more clearly on the interdependence of space and time and its geometric representation. The coordinates of an event are given then by x and t. All possible space-time coordinates can be represented on a space-time diagram in which the space axis is horizontal and the time axis is vertical. It is convenient to keep the dimensions of the coordinates the same; this is easily done by multiplying the time t by the universal constant c, the velocity of light. Let ct be represented by the symbol w. Then, the Lorentz transformation equations can be written as follows.

Notice the symmetry in this form of the equations. To represent the situation geometrically, we begin by drawing the x and w axes of frame S orthogonal (perpendicular) to one another

Fig-1

If we wanted to represent the motion of a particle in this frame, we would draw a curve, called a world line, which gives the loci of space-time points corresponding to the motion. Minkowski referred to space-time as "the world." Hence, events are world points and a collection of events giving the history of a particle is a world line. Physical laws on the interaction of particles can be thought of as the geometric relations between their world lines. In this sense, Minkowski may be said to have geometrized physics.

The tangent to the world line at any point, being 𝑑𝑥𝑑𝑤=1𝑐𝑑𝑥𝑑𝑡, is always inclined at an angle less than 45° with the time axis. For this angle (see Fig. 1) is given by tan𝜃 = 𝑑𝑥𝑑𝑤 = 𝑢𝑐 and we must have u < c for a material particle. The world line of a light wave, for which u = c, is a straight line making a 45° angle with the axes