World-Line

in relativity theory, a geometrical representation of the four-dimensional “trajectory” of a mass point (particle) in space-time or in its equivalent, Minkowski space, independent of the frame of reference. Each point on the world-line is a “world-point” or “event” that indicates the position of the particle—in space coordinates x₁ = x, x₂ = y, and x₃ = z—and the instant of time t corresponding to this position; the time t is related to the time coordinate x₀ of four-dimensional space-time by the equation x₀ = ct, where c is the velocity of light. A world-line parallel to the x₀-axis represents a particle at rest.

In the special theory of relativity, the world-line of a particle moving uniformly and linearly is given by a straight line inclined at a certain angle θ (<45°) relative to the x_o-axis. This angle depends on the velocity v (tan θ = v/c); an angle of 45° corresponds to the world-line of light. The world-line of a nonuniformly moving particle is a curve. In the presence of a gravitational field (in the general theory of relativity), the world-lines of light and of a freely moving particle are curved.

G. A. ZISMAN

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7 Expressing Synge-Weber equation (the world-lines deviation equation) in the terms of physical observable quantities, and its exact solutions

These systems are described by the world-lines deviation equation--the Synge equation of geodesic deviation (2.

The greater their velocity with respect to the space reference and the observer, the greater the deviation between the time flow on both world-lines.

But if we wish to obtain solutions to the world-lines deviation equation, we need to express the quantity [D.

We merely need to obtain exact solutions to the world-lines deviation equation, applied to detectors of that kind which this experiment uses.

Therefore, using the world-lines deviation theory developed here in the terms of physically observable quantities, we are going to:

The behaviour of two neighbouring particles in their motion along their neighbouring world-lines is described by the world-lines deviation equation.

and substituting this into the world-lines deviation equation in its initial form (2.

This is the final form of the world-lines deviation equation for two test-particles connected by a spring.

In other words, experimental physicists expect that oscillations of the acting gravitational wave field give rise to a force in the world-lines deviation equation (the Synge-Weber equation), thereby displaceing the test-particles which were at rest at the initial moment of time.

Before ratifying the aforesaid conclusions it would be reasonable to study the world-lines deviation equation for two interacting test-particles that model a Weber pig, because this equation is the theoretical basis of all experimental attempts to register gravitational waves made by Weber and his followers during more than 30 years.

Gravitational waves and gravitational inertial waves in the general theory of relativity: a theory and experiments

Let us consider the entanglement condition d[tao] = 0 in connection with the world-lines deviation equations.

Entangled states and quantum causality threshold in the general theory of relativity