Question

Consider motion in one spatial dimension. For any velocity \(v,\) define parameter \(\theta\) via the relation \(v=c \tanh \theta\) where \(c\) is the vacuum speed of light. This quantity is variously called the velocity parameter or the rapidity corresponding to velocity \(v\). a) Prove that for two velocities, which add according to the Lorentzian rule, the corresponding velocity parameters simply add algebraically, that is, like Galilean velocities. b) Consider two reference frames in motion at speed \(v\) in the \(x\) -direction relative to one another, with axes parallel and origins coinciding when clocks at the origin in both frames read zero. Write the Lorentz transformation between the two coordinate systems entirely in terms of the velocity parameter corresponding to \(v\), and the coordinates.

Short answer

In this exercise, we have a) proven that for two velocities (which add according to the Lorentzian rule), their corresponding rapidity parameters simply add algebraically, and b) derived the Lorentz transformation equations between two coordinate systems moving at speed v in the x-direction relative to one another in terms of the rapidity parameter θ. The final Lorentz transformation equations are: \(x' = \cosh \theta (x - ct\tanh \theta)\) \(t' = \cosh \theta (t - \frac{x\tanh \theta}{c})\)

Step-by-step solution

a) Prove algebraic addition of velocity parameters

Consider two velocities \(v_1\) and \(v_2\) which add according to Lorentzian rule: \(v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}\) Let's find the rapidity parameters \(\theta_1\) and \(\theta_2\) corresponding to \(v_1\) and \(v_2\), by using the given relation: \(v_1 = c \tanh \theta_1\) and \(v_2 = c \tanh \theta_2\) Our goal is to prove that \(\theta_1 + \theta_2 = \theta\), where \(\theta\) is the rapidity corresponding to the velocity \(v\), which is the sum of \(v_1\) and \(v_2\). Let's substitute \(v_1\) and \(v_2\) in the Lorentzian rule equation using the given relations: \(v = \frac{c\tanh \theta_1 + c\tanh \theta_2}{1 + \frac{(c\tanh \theta_1)(c\tanh \theta_2)}{c^2}}\) Now, we simplify the equation to get \(v\) in terms of \(\theta_1\) and \(\theta_2\): \(v = c\frac{\tanh \theta_1 + \tanh \theta_2}{1 + \tanh \theta_1 \tanh \theta_2}\) Remembering the hyperbolic trigonometric identity \(\tanh (a+b) = \frac{\tanh a + \tanh b}{1 + \tanh a \tanh b}\), we can write: \(v = c \tanh(\theta_1 + \theta_2)\) This proves that the velocity parameters simply add algebraically, following the Galilean velocities.

b) Lorentz transformation in terms of velocity parameter

The standard Lorentz transformations between two coordinate systems \(K\) and \(K'\) moving at a velocity \(v\) relative to one another in the \(x\)-direction are given by: \(x' = \gamma (x - vt)\) \(t' = \gamma (t - \frac{vx}{c^2})\) Where \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) We have to rewrite these equations in terms of the rapidity parameter \(\theta\). Recall the relation \(v = c \tanh \theta\). From this, we get: \(\frac{v}{c} = \tanh \theta\) Now, let's find \(\gamma\) in terms of \(\theta\). We know \(\gamma\) can be expressed as: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1 - \tanh^2 \theta}}\) Using the identity \(1 - \tanh^2 \theta = \text{sech}^2\theta\), we have: \(\gamma = \cosh \theta\) Substitute \(\frac{v}{c}\) with \(\tanh \theta\) and \(\gamma\) with \(\cosh \theta\) in the Lorentz transformation equations: \(x' = \cosh \theta (x - ct\tanh \theta)\) \(t' = \cosh \theta (t - \frac{x\tanh \theta}{c})\) These are the Lorentz transformation equations in terms of the velocity parameter \(\theta\) and the coordinates.

Key concepts

Velocity Addition

In the realm of physics, especially when dealing with high-speed particles close to the speed of light, velocities don't simply add together as they would at everyday speeds. This is where the concept of velocity addition under the Lorentz transformation comes into play.

• Unlike classical physics, where you would just add two velocities to get a resultant velocity, in relativity, the formula to combine two velocities \(v_1\) and \(v_2\) is given by: \[ v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \]
• This equation makes sure that the resultant velocity \(v\) never exceeds the speed of light, maintaining the universality of light speed in all frames of reference.
• It's a crucial adaptation that accounts for the relativistic effects that become prominent at speeds approaching the speed of light.

Understanding this concept reinforces why classical ideas of motion aren't sufficient for explaining the behavior of objects at those extreme velocities.

Rapidity

Rapidity is a concept that helps us describe velocities in a more manageable form, especially useful in relativistic contexts. It’s defined via the expression \(v = c \tanh \theta\), where \(\theta\) is the rapidity.

• One key advantage of using rapidity is that, unlike velocities, rapidities simply add up directly when combining two motions, without need for the complicated Lorentz addition formula.
• This makes equations simpler because you can treat additions of rapidities as if they were additions of velocities under classical mechanics.
• Moreover, rapidity can take on values from negative to positive infinity, offering a consistent measure of motion that's not bound by the speed of light.
• In particle physics and accelerators, rapidity is often used instead of velocity to simplify calculations and theoretical models.

Embracing rapidity as a term helps us delve deeper into the study of high-speed motion in a more intuitive and efficient manner.

Hyperbolic Functions

Hyperbolic functions, like \(\sinh\), \(\cosh\), and \(\tanh\), are instrumental in understanding relativistic physics. They are counterparts to the trigonometric functions, but serve a special role in relativity.

• The function \(\tanh \theta\) inherently appears in the relationship between velocity and rapidity: \(v = c \tanh \theta\).
• The addition formula for hyperbolic functions, \(\tanh(a + b) = \frac{\tanh a + \tanh b}{1 + \tanh a \tanh b}\), mirrors the addition of velocities in relativity, simplifying such expressions significantly.
• They help us describe quantities in a more linear way at relativistic speeds, unlike the circular trigonometric functions.
• Hyperbolic functions are used extensively in special relativity notation and calculation because they handle transformations and rapidity so elegantly.

Recognizing these functions and distinctions aids in aligning our understanding with the mathematical tools that best describe relativistic phenomena.

Relativity

The theory of relativity, introduced by Albert Einstein, reshaped our understanding of how objects behave at high speeds and in gravitational fields. There are two main principles to note:

• Special relativity deals with the physics of objects moving at constant speeds close to that of light. It incorporates time dilation, length contraction, and mass-energy equivalence.
• An essential feature is the constancy of the speed of light in all inertial frames, leading to the need for the Lorentz transformation.
• General relativity expands the theory to include acceleration and gravity, fundamentally redefining our understanding of gravitational fields as warps in space-time.
• At its core, relativity tells us that the laws of physics are the same for all observers, regardless of their relative motion.

Relativity provides the framework that helps explain the strange yet fundamental behaviors of the universe at large scales and high velocities.

Reference Frames

Reference frames are the coordinate systems or perspectives used for observing physical phenomena. In relativity, understanding reference frames is crucial for analyzing motion:

• A reference frame is essentially a point of view or set of coordinates from which an observer measures various quantities, like time, distance, and velocity.
• Inertial reference frames move at constant velocity and are where Newton's laws hold without modifications.
• Under Lorentz transformations, one can convert measurements from one frame to another that has a relative velocity, ensuring the consistency of physical laws such as the speed of light.
• The ability to switch between reference frames using transformations helps us see how events that occur simultaneously in one frame might not appear so in another, illustrating relativity’s core insights.

Recognizing the role of reference frames underpins the broader understanding of how relative motion affects observations and calculations in physics.