Question

Prove that if \(v\) and \(u^{\prime}\) are less than \(c\), it is impossible for a speed \(u\) greater than \(c\) to result from equation \((2-19 b)\). [Hint: The product (c \(\left.-u^{\prime}\right)(c-v)\) is positive.]

Short answer

Given that \(v\) and \(u'\) are less than \(c\), it is impossible for \(u\) resulting from the equation to be greater than \(c\) since it is concluded from the positive product of \( (c - v)(c - u')\), proving the statement.

Step-by-step solution

Begin the Proof

To start, you will need to accept the assumption that both \(v\) and \(u'\) are less than \(c\). This allows you to establish that \(c - v > 0\) and \(c - u' > 0\) as both \(v\) and \(u'\) are less than \(c\). Thus, the product of these two quantities, \( (c - v)(c - u')\), is positive.

Interpreting the Equation

Application of the given equation is important. According to the hint, the product of these quantities is equal to the product of the terms resulting from equation \((2-19 b)\). Since the product on the left is positive, the product of the terms resulting from the equation must also be positive.

Crystalize the Proof

Using the assumption that \(v\) and \(u'\) are less than \(c\), it can be concluded that \(u\), the speed resulting from the equation, is also less than \(c\). This is due to the fact that the product \((c - v)(c - u')\) is positive. As a result, \(u\) greater than \(c\) is not possible. This completes the proof.

Key concepts

Lorentz Transformation

Lorentz Transformation is a fundamental idea in the theory of Special Relativity, formulated by the scientist Hendrik Lorentz. It describes how measurements of space and time by two observers are related to each other in scenarios where the observers are moving at constant velocities relative to one another. Specifically, it's used to transform the coordinates of an event as seen in one inertial frame to the coordinates in another inertial frame.

A key aspect of Lorentz Transformation is that it accounts for the fact that the speed of light is the same in all inertial frames, irrespective of the motion of the light source or the observer. This is different from classical mechanics, where speeds can simply be added or subtracted depending on the direction of motion.

In mathematical form, the Lorentz transformations show how a set of coordinates \( (x, y, z, t) \) in one frame of reference can be transformed to another frame \( (x', y', z', t') \) using the equations:\

• \( x' = \gamma (x - vt) \)
• \( t' = \gamma (t - \frac{vx}{c^2}) \)

where \( v \) is the relative velocity between the frames and \( c \) is the speed of light in a vacuum, and \( \gamma \) is the Lorentz factor defined as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \).

The Lorentz Transformation preserves the spacetime interval \( s^2 = x^2 + y^2 + z^2 - c^2t^2 \), showing that this interval is invariant for all observers, underpinning the consistent value of the speed of light and the relativistic effects it entails.

Speed of Light

The speed of light, denoted by the symbol \( c \), is a universally constant value in the vacuum, approximately equal to \( 3.00 \times 10^8 \) meters per second. It plays a pivotal role in Einstein's theory of Special Relativity, serving as the ultimate speed limit for all energy, matter, and information in the universe.

Einstein's iconic equation \( E = mc^2 \) illustrates the relationship between mass \(m\) and energy \(E\), demonstrating that mass can be converted into energy, with the factor \( c^2 \) accounting for incredibly vast energies even from small amounts of mass. This relationship also indirectly hints at why massive objects cannot surpass the speed of light: as objects move faster, their relativistic mass effectively increases, requiring more and more energy to continue accelerating. Ultimately, infinite energy would be required to reach or exceed \( c \).

The constancy of the speed of light leads to phenomena such as time dilation and length contraction, where time appears to slow down and lengths to contract for objects moving at speeds close to \( c \). These concepts challenge our conventional understanding of time and space, establishing a foundation for modern physics that breaks with the classical mechanics view where time and space are absolute.

Relativistic Velocity Addition

When objects move at speeds close to the speed of light, the familiar rules of classical velocity addition no longer apply. Instead, we use the principle of Relativistic Velocity Addition to properly account for the relativistic effects on motion.

The Relativistic Velocity Addition formula is essential for ensuring that resultant velocities do not exceed the speed of light. This formula ensures that even if two objects are moving towards or away from each other at high velocities, their relative velocity doesn’t surpass \( c \). The formula is given by:\

• \( u = \frac{v + u'}{1 + \frac{vu'}{c^2}} \)

where \( v \) and \( u' \) are the velocities of two moving objects as observed from a stationary frame, and \( u \) is their relative velocity.

This equation inherently incorporates the effects of time dilation and length contraction, showcasing the interwoven nature of these effects within the fabric of spacetime. By ensuring that \( u \) remains less than \( c \), Special Relativity maintains consistency with the principle that nothing can travel faster than light. In practice, this makes calculations involving high-speed travel more complex, yet intimately related to the behavior of light and its unyielding speed limit.