Problem 61 - University Physics with Modern Physics

Question

Frame \(S^{\prime}\) has an \(x\) -component of velocity \(u\) relative to frame S, and at \(t=t^{\prime}=0\) the two frames coincide (see Fig. 37.3\()\) . A light pulse with a spherical wave front is emitted at the origin of \(S^{\prime}\) at time \(t^{\prime}=0 .\) Its distance \(x^{\prime}\) from the origin after a time \(t^{\prime}\) is given by \(x^{\prime 2}=c^{2} t^{\prime 2} .\) Use the Lorentz coordinate ransformation to S, and at \(t=t^{\prime}=0\) the two frames coincide (see Fig. 37.3\()\) . A light pulse with a spherical wave front is emitted at the origin of \(S^{\prime}\) at time \(t^{\prime}=0 .\) Its distance \(x^{\prime}\) from the origin after a time \(t^{\prime}\) is given by \(x^{\prime 2}=c^{2} t^{\prime 2} .\) Use the Lorentz coordinate ransformation to be spherical in both frames.

Step-by-Step Solution

Answer

The light pulse has a spherical wavefront in both frames S and S' due to Lorentz transformations.

Step 1: Understand the Problem

The problem involves two frames of reference, S and S', where frame S' is moving with velocity \( u \) with respect to frame S. A light pulse is emitted from the origin of frame S' with the wavefront being spherical.

Step 2: Frame of Reference

The key to solving this problem is to understand that the light pulse maintains its spherical nature (a constant speed of light) in both frames. This means we have to show that if \( x'^2 = c^2 t'^2 \) in frame S', it should also take a spherical form in frame S.

Step 3: Apply the Lorentz Transformation

The coordinates \((x, t)\) in frame S relate to coordinates \((x', t')\) in frame S' through the Lorentz transformation:\[ x = \gamma (x' + ut') \quad \text{and} \quad t = \gamma (t' + \frac{ux'}{c^2})\]where \( \gamma = \frac{1}{\sqrt{1-u^2/c^2}} \) is the Lorentz factor.

Step 4: Substitute and Simplify

Substitute \( x' = ct' \) in the transformation equations. Simplifying those:\( x = \gamma (ct' + ut') = \gamma t'(c + u) \) and\( t = \gamma (t' + \frac{uct'}{c^2}) = \gamma t'(1 + \frac{u}{c}) \).Notice that after simplification for spherical coordinates, it holds \( x^2 - c^2 t^2 = 0 \), representing a sphere of radius \( ct \) in frame S.

Step 5: Conclusion

The Lorentz transformation confirms the consistency of the speed of light: the light pulse wavefront is spherical in both the frames S and S' since they both satisfy the equation \( x^2 = c^2 t^2 \), maintaining the spherical nature.

Key Concepts

Spherical Wavefront

A spherical wavefront describes the shape of the wave of light as it expands from a single point in space. Imagine dropping a stone in a calm pond and watching the ripples expand outward in perfect circles. Similarly, when a light pulse is emitted from a point source, like the origin of a frame of reference, it spreads out equally in all directions. This creates a spherical shape around the initial point of emission.
In the problem, you have two frames of reference: S and S'. A light pulse emitted from the origin in frame S' results in a spherical wavefront. This means that at any time \(t'\), the distance \(x'\) from the center, or origin, to any point on the wavefront is given by the relationship \(x'^2 = c^2 t'^2\).
This consistent representation emphasizes the uniform nature of light's propagation in any frame, adhering to the same speed and shape regardless of motion.

Frames of Reference

Frames of reference are essentially different perspectives from which an observer can measure the position and time of an event. In physics, understanding how different observers perceive physical events is critical. In our problem, we are discussing two frames: S and S'.
Frame S is the stationary reference point, while frame S' moves at a constant velocity \(u\) with respect to frame S. Both frames are initially aligned, meaning they coincide at \(t = t' = 0\).
As the light pulse is emitted from frame S', observers in both frames should see the event as preserving the laws of physics. This is achieved today through Lorentz transformations, which relate the coordinates of events in one frame to another, ensuring that physical phenomena, like the speed of light, are consistent.

Speed of Light Consistency

The concept of constant speed of light is central to Albert Einstein's theory of relativity. Regardless of the observer's frame of reference, the speed of light remains \(c\), approximately \(299,792,458\) meters per second.
This problem uses Lorentz transformations to verify that even when frames are moving relative to each other, the speed of light does not change. This is evident when exploring the equations given. For both frames, S and S', the light pulse's distance relationship \(x^2 = c^2t^2\) remains unchanged. This confirms that the light pulse emits a spherical wavefront in both frames without deviation in speed.
This concept emphasizes the universality of physical laws, showing that the speed of light is an unchanging truth across different frames of reference, no matter the relative motion. It underscores the principle that the essence of observable events is consistent across the universe.