Question

A light beam moves at an angle \(\theta\) with the \(x\) -axis as seen from frame \(S\). Using the relativistic velocity transformation, find the components of its velocity when viewed from frame \(S\). From these. verify explicitly that its speed is \(c\).

Short answer

Applying relativistic velocity transformations, the components of the velocity of the light beam can be determined in a frame \(S'\). It can then be demonstrated that the resulting speed (magnitude of the velocity) in the \(S'\) frame is still \(c\), establishing the invariance of the speed of light across different inertial frames.

Step-by-step solution

Determine Velocity Components

First, determine the components of the velocity of the light beam in the frame \(S\) using the relative velocity transformation equations. If the velocity of light in frame \(S\) is \(c\), then the components are \(V_x = c \cos(\theta)\) and \(V_y = c \sin(\theta)\).

Use Relativistic Velocity Transformations

To find the components of the velocity when viewed from frame \(S'\), which is moving with a velocity \(v\) with respect to \(S\), we use the relativistic velocity transformations. If \(v\) is the velocity of frame \(S'\) relative to \(S\), then the transformed components of the velocity of light in frame \(S'\) are given by: \(V_x' = (V_x - v) / (1 - V_x v / c^2)\) and \(V_y' = V_y / \gamma(1 - V_x v / c^2)\), where \(\gamma = 1/ \sqrt{1 - v^2/c^2}\). Insert the values of \(V_x\) and \(V_y\) obtained in step 1 into these equations to obtain the components of the velocity of the light in frame \(S'\).

Calculate the Speed of Light

The magnitude of the velocity in the \(S'\) frame is given by \(V' = \sqrt{{V_x'}^2 + {V_y'}^2}\). Calculate this to verify that its speed remains \(c\) in all frames of reference.

Key concepts

Speed of Light

The speed of light in a vacuum is the ultimate speed limit according to the theory of relativity. Denoted as the symbol \(c\), it is approximately 299,792,458 meters per second (m/s). This speed is not just a limit on how fast light can travel, but it's the maximum speed at which all energy, matter, and information in the universe can travel.

It's important to note that no matter the motion of the source or the observer, the speed of light in a vacuum remains consistent. This was a fundamental observation by Einstein that led to the development of the special theory of relativity. In our example, when a beam of light moves at an angle, its velocity components in any inertial frame should always combine to equal the constant \(c\), irrespective of the relative motion between the source and the observer.

Lorentz Transformation

The Lorentz transformation equations are a set of equations that describe how, according to the theory of relativity, the space coordinates and time coordinates of two systems are related to each other. It becomes significant especially when the systems in question are moving at a high velocity relative to each other, close to the speed of light.

The Lorentz transformation takes into account the relative velocity between two frames of reference and includes a factor \(\textstyle \text{gamma (\( \textstyle\text{\textgamma} \))} \), which accounts for time dilation and length contraction as velocities approach \(c\). In our exercise example, these transformations are used to convert the velocity components of the light beam from one frame, \(S\), to another, \(S'\), which is moving at a velocity relative to the first.

Velocity Components

In physics, when we describe the motion of an object we often break down its velocity into components along the coordinate axes. For instance, if a vehicle is moving northeast, its velocity can be divided into a northward (y-axis) component and an eastward (x-axis) component.

Importance of Velocity Components in Relativity

In the context of relativity, breaking down velocities into components is crucial for applying the Lorentz transformation. For a light beam traveling at an angle \(\theta\) in frame \(S\), we can use trigonometry to find its components as \(V_x = c \cos(\theta)\) and \(V_y = c \sin(\theta)\).

By understanding each individual component, we can use Lorentz transformations to find how these components change when observed from a different reference frame that is moving at a relative velocity.

Theory of Relativity

The theory of relativity, developed by Albert Einstein, encompasses two theories: special relativity and general relativity. Special relativity focuses on the physics of moving bodies in the absence of gravitational forces and accounts for the constant speed of light for all observers. General relativity extends the theory to include gravity as a result of spacetime curvature around massive objects.

A key aspect of special relativity, relevant to our exercise, is the relativity of simultaneity, time dilation, and length contraction—all of which are encapsulated in the Lorentz transformation. These concepts underscore the idea that measurements of space and time are relative to the observer's frame of reference. This is why the speed of light remains constant regardless of how fast an observer is moving relative to the light source.