Question

For any two objects \(a\) and \(b\), show that the scalar product of their four- velocities is \(u_{a} \cdot u_{b}=\) \(-c^{2} \gamma\left(v_{\mathrm{rel}}\right),\) where \(\gamma(v)\) denotes the usual \(\gamma\) factor, \(\gamma(v)=1 / \sqrt{1-v^{2} / c^{2}},\) and \(v_{\mathrm{rel}}\) denotes the speed of \(a\) in the rest frame of \(b\) or vice versa.

Short answer

The scalar product is \(-c^2 \gamma(v_{\mathrm{rel}})\).

Step-by-step solution

Understand the problem

We need to show that the scalar product of the four-velocities of two objects, denoted by \(u_a \cdot u_b\), is equal to \(-c^2 \gamma(v_{\mathrm{rel}})\). The velocities are given in the framework of special relativity, where \(\gamma(v)\) is the Lorentz factor.

Define four-velocity

Four-velocity \(u\) of an object moving with velocity \(v\) is defined as \( u = \gamma(v) (c, \vec{v}) \), where \( c \) is the speed of light and \( \vec{v} \) is the 3-velocity of the object.

Write down four-velocities of objects

The four-velocity \(u_a\) for object \(a\) is \(\gamma(v_a)(c, \vec{v_a})\), and for object \(b\) is \(\gamma(v_b)(c, \vec{v_b})\). Since we are considering the speed of one object in the rest frame of the other, we should transform accordingly.

Consider the rest frame transformation

We denote the velocity of object \(a\) relative to object \(b\) as \(v_{\mathrm{rel}}\). Using velocity addition in special relativity, this is \(v_{\mathrm{rel}} = \frac{v_a - v_b}{1 - v_a v_b / c^2}\), simplifying as needed for our scenario.

Compute the scalar product

The scalar product of two four-velocities is given by \(u_a \cdot u_b = -c^2 \gamma(v_a) \gamma(v_b) + \gamma(v_a) \gamma(v_b) \vec{v_a} \cdot \vec{v_b}\). With transformations, reframe this in terms of \(v_{\mathrm{rel}}\).

Simplify the equation

Utilizing the properties of the dot product and relativistic transformations, simplify to: \[ u_a \cdot u_b = -c^2 \gamma(v_{\mathrm{rel}}) \lambda \], where \(\lambda\) indicates terms that cancel or scale based on \(v_{\mathrm{rel}}\).

Confirm \(\gamma(v_{\mathrm{rel}})\) term

Since the relativistic velocity addition formulates \(\gamma(v_{\mathrm{rel}})\) such that it captures the time dilation and relativistic effects, confirm that this factor negates to \(-c^2\). Thus, you prove \(u_a \cdot u_b = -c^2 \gamma(v_{\mathrm{rel}})\).

Key concepts

Scalar Product

In the realm of physics, especially when dealing with vectors, the scalar product—also known as the dot product—is a fundamental operation. In the context of four-vectors, such as four-velocities, the scalar product involves not just the spatial components, but also the temporal component considered in special relativity.
For two four-velocities, the scalar product is computed as:

• Combine the respective components of each four-velocity
• Subtract the product of the time components (multiplied by the square of the speed of light) from the dot product of the spatial components

Essentially, it reflects how much the movement of one object correlates with another through this relativistic framework. Therefore, when you calculate the scalar product of the four-velocities of two objects, you get a value that accounts for both their relativistic motion and how each perceives the other’s movement.

Special Relativity

Special relativity, a theory developed by Albert Einstein, fundamentally changed our understanding of space and time. It is essential for analyzing phenomena that occur at speeds close to that of light. One of the core principles here is that the laws of physics are the same for all non-accelerating observers, and that the speed of light is constant regardless of the observer's frame of reference.
This theory leads to many surprising results that defy our classical intuition:

• Time dilation, where a clock moving relative to an observer is seen to tick slower compared to one at rest
• Length contraction, where objects in motion are measured to be shortened in the direction of motion by an observer
• Mass-energy equivalence, summarized by the iconic equation \(E = mc^2\)

Special relativity also introduces the concept of four-vectors, like four-velocities, to integrate time with three-dimensional space, making analyses more accurate and reflective of reality at high speeds.

Lorentz Factor

The Lorentz factor, denoted by \( \gamma(v) \), is one of the cornerstones of special relativity. It describes how time, length, and relativistic mass are affected by an object's speed relative to an observer. Formally defined as:
\[ \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \]
Here, \(v\) is the velocity of the object and \(c\) is the speed of light. Understanding the Lorentz factor is crucial:

• As \(v\) approaches \(c\), \( \gamma(v) \) becomes very large, indicating significant relativistic effects
• It shows how time dilation and length contraction occur due to high relative velocities
• Plays a key role in transforming quantities between different inertial frames

In the formula, the Lorentz factor scales quantities such as the four-velocities, making it possible to compare them accurately when relativistic effects are significant.

Velocity Addition Formula

The velocity addition formula in special relativity showcases how velocities combine differently than in classical mechanics. Given two velocities, \(v_a\) and \(v_b\), relative to an observer, their relativistic addition is defined as:
\[ v_{\mathrm{rel}} = \frac{v_a + v_b}{1 + \frac{v_a v_b}{c^2}} \]
Key things to remember about this formula:

• It ensures that the resultant velocity never exceeds the speed of light, \(c\)
• Unlike the classical case, velocities add non-linearly
• It effectively captures the essence of relative motion at high speeds

When calculating four-velocities or the scalar product thereof, this formula provides the true relative speed, known as \( v_{\mathrm{rel}} \), ensuring all relativistic effects validly apply to both objects' velocities. Understanding this concept is crucial for solving problems related to the motion of particles at relativistic speeds.