Question

Derive the following expressions for the components of acceleration of an object, \(a_{x}^{\prime}\) and \(a_{y}^{\prime}\), in frame \(S^{\prime}\) in terms of its components of acceleration and velocity in frame S. $$ \begin{array}{c} a_{x}^{\prime}=\frac{a_{x}}{\gamma_{v}^{3}\left(1-\frac{u_{x} v}{c^{2}}\right)^{3}} \\ a_{y}^{\prime}=\frac{a_{y}}{\gamma_{v}^{2}\left(1-\frac{u_{x} v}{c^{2}}\right)^{2}}+\frac{a_{x} \frac{u_{y} v}{c^{2}}}{\gamma_{v}^{2}\left(1-\frac{u_{x} v}{c^{2}}\right)^{3}} \end{array} $$

Short answer

The components of acceleration in the frame S' are given by the Lorentz transformation equations as: \[a_{x}^{\prime}=\frac{a_{x}}{\gamma_{v}^{3}\left(1-\frac{u_{x}v}{c^{2}}\right)^{3}}\] and \[a_{y}^{\prime}=\frac{a_{y}}{\gamma_{v}^{2}\left(1-\frac{u_{x}v}{c^{2}}\right)^{2}}+\frac{a_{x} \frac{u_{y}v}{c^{2}}}{\gamma_{v}^{2}\left(1-\frac{u_{x} v}{c^{2}}\right)^{3}}\]. The expansions offer an understanding of how motion in one frame can be translated into motion in another frame using the concepts of special relativity.

Step-by-step solution

Understanding the Given

An object is observed in one frame S and moves with a velocity \(u\) and acceleration \(a\). The same object is observed in another frame S' which is moving with a velocity \(v\) relative to S. The task is to express the components of acceleration, \(a_{x}^{\prime}\) and \(a_{y}^{\prime}\), as seen from the frame S' in terms of the components of the velocity and acceleration as seen from frame S.

Lorentz Factor

In special relativity, the Lorentz factor or Lorentz term \(\gamma_{v}\) is the factor by which time, length, and relativistic mass change for an object moving with relativistic velocities. Note that \(\gamma_{v}\) is always greater than or equal to 1.

Acceleration Transformation

In this context, the object's acceleration \(a\) is calculated relative to the frame S, and the frame S' is moving with velocity \(v\) in the x-axis direction. The transformation equations for acceleration in different frames S and S' are given above. These can be derived using the Lorentz transformation equations for velocity and acceleration, taking into consideration that the velocity of the frame S' relative to S is \(v\) in the x-axis direction.

Key concepts

Lorentz Transformation

When discussing the fundamental aspects of special relativity, Lorentz transformations play an integral role. They describe how measurements of space and time by two observers moving relative to each other are related. These transformations demonstrate that observers in different inertial frames may measure different distances, times, and even different sequences of events.

Imagine you're on a train moving at high speed and you measure the time it takes for light to bounce between two mirrors. For an observer standing on the platform as you whiz past, the light not only travels up and down between the mirrors but also horizontally due to the train's motion. The Lorentz transformation mathematically relates your measurements on the train to those of the observer on the station platform. It depends on the relative velocity between the two frames of reference and the speed of light. These equations are the backbone of understanding how time dilation and length contraction occur at relativistic speeds.

Special Relativity

Special relativity is a groundbreaking theory of physics formulated by Albert Einstein. It reshaped our understanding of space, time, and energy at high speeds—approaching the speed of light. The theory is based on two postulates: the laws of physics are the same in all inertial frames, and the speed of light in a vacuum is the same for all observers, regardless of their relative motion.

This theory leads to several counterintuitive outcomes, such as time dilation, where time can pass at different rates for different observers based on their relative velocities; and length contraction, where the length of objects decreases when they move at high speeds relative to an observer. Also, mass and energy were found to be equivalent, leading to the famous equation E=mc^2.

Relativistic Velocities

Under the umbrella of special relativity, velocities don't simply add together. Imagine two spaceships moving towards each other at significant fractions of light speed. If they were cars on a road, you would add their speeds to get the relative velocity. However, because they're moving so fast, their relative speed is calculated using the relativistic velocity addition formula. This prevents any object from exceeding the speed of light, upholding the constancy of light's speed as dictated by special relativity.

Relativistic velocity transformation equations, derived from the Lorentz transformations, enable us to compute how one observer would perceive the speed and direction of an object moving relative to another observer. These equations reflect the concept that space and time are not separate but part of a four-dimensional space-time continuum, linked by the speed of light.

Lorentz Factor

The Lorentz factor, often symbolized as \( \gamma \), quantifies the amount of time dilation, length contraction, and mass increase experienced by an object moving at a significant fraction of the speed of light. It is a function of velocity given by \( \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \), where \( v \) is the relative velocity of an inertial frame of reference and \( c \) is the speed of light.

The Lorentz factor approaches infinity as the velocity \( v \) approaches the speed of light \( c \) which explains why objects with mass cannot reach or exceed the speed of light: it would require an infinite amount of energy. In the context of the exercise, \( \gamma_v \) is critical for transforming the acceleration from one frame of reference to another because it captures the profound influence of relativity on moving objects.