Question

A particle ts m hyperbelhe motion along a world-line whose equaton ts given by $$ x^{2}-c^{2} t^{2}=a^{2}, \quad y=z=0 $$ Show that $$ \gamma=\frac{\sqrt{a^{2}+c^{2} t^{2}}}{a} $$ and that the proper tme starthng fronn \(t=0\) along the path is grven by $$ \tau=\frac{a}{c} \cosh ^{-1} \frac{c t}{a} $$

Short answer

The Lorentz factor, \(\gamma\), is given by \(\gamma = \frac{\sqrt{a^2 + c^2 t^2}}{a}\). The proper time, \(\tau\), along the path starting from \(t=0\) is given by \(\tau = \frac{a}{c} \cosh^{-1} \frac{ct}{a}\)

Step-by-step solution

Calculating the Lorentz factor (\gamma)

First, write down the definition of the Lorentz factor \(\gamma\), which is given by \(\gamma = 1/\sqrt{1 - v^2/c^2}\) where \(v\) is the velocity of the particle. Solve this equation for \(v\), we get \(v = c \sqrt{1 - 1/\gamma^2}\). Differentiate the given equation of the world-line, we get \(2x dx/dt - 2 c^2 t = 0\). Solve this for \(dx/dt\). Insert \(dx/dt\) into the equation for \(v\) to get the first relation.

Calculating the Proper Time (\tau)

To compute the proper time, we use the definition of proper time which integrates over dt, i.e., \(\frac{d\tau}{dt} = \gamma\). We have already calculated \(\gamma\) from step 1. Insert it into this equation and integrate from \(t = 0\) to \(t = t\) to obtain the expression for proper time.

Key concepts

Lorentz Factor

The Lorentz factor, denoted as \( \gamma \), is a fundamental concept in special relativity. It describes how time, length, and relativistic mass change for an object moving at a significant fraction of the speed of light. The Lorentz factor is defined as\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]where \(v\) is the velocity of the object and \(c\) is the speed of light. In this particular problem, we are given a specific world-line equation that allows us to compute a different form of the Lorentz factor based on the hyperbolic path of the particle.The given equation \(x^2 - c^2 t^2 = a^2\) implies a hyperbolic trajectory, which helps determine the relationship between space and time for the particle. By differentiating this equation with respect to time "t", and manipulating the result, we find\(\gamma = \frac{\sqrt{a^2 + c^2 t^2}}{a}\).

• The numerator \(\sqrt{a^2 + c^2 t^2}\) stems from including relativistic effects on space and time.
• This expression illustrates how the particle's relativistic effects depend on its position along the hyperbolic path.

Understanding the Lorentz factor is crucial, as it reveals how different observers perceive time and space differently depending on their relative speed.

Proper Time

Proper time, denoted by \( \tau \), is a concept in special relativity that refers to the time interval between two events as measured in the rest frame of the observer experiencing the events. It is an observer-independent interval that is equivalent to the time measured when the observer is at rest relative to the events.In this exercise, we are asked to compute the proper time for the particle starting from \( t = 0 \). The calculation involves the relation \[\frac{d\tau}{dt} = \gamma\]where \( \gamma \) is the Lorentz factor we derived before.Integrating this relation from time \(t = 0\) to some time \(t\), and replacing \(\gamma\) with its derived form, reveals the expression for proper time:\[\tau = \frac{a}{c} \cosh^{-1}\left(\frac{c t}{a}\right).\]

• The factor \(\frac{a}{c}\) represents the initial conditions and scale related to the hyperbolic path of the particle.
• \( \cosh^{-1}\) denotes the inverse hyperbolic cosine function, which is mathematically associated with hyperbolic motion.

Therefore, proper time gives us an intrinsic way of understanding the motion experienced by the particle without reference to external frames.

World Line

A world line is a concept in relativity that maps the entire history of an object through spacetime. It represents the path that an object takes as it moves, detailing how it progresses through both space and time.In this exercise, the particle's world line is given by the hyperbolic equation:\[x^{2} - c^{2} t^{2} = a^{2}, \quad y = z = 0\]

• This shows motion confined to the x-axis, with no change in y or z coordinates.
• The mathematical form resembles a hyperbola which is a classic signature of relativistic motion.

This world line provides a visual representation of the particle's motion under the influence of special relativity. It shows how the particle travels through space and time, depicted as an unbroken path. World lines are essential in physics because they help visualize the effects of relativity by mapping trajectories in spacetime. This representation aids in understanding how different observers perceive events depending on their relative velocities.