Observer S reports that an event occurred on the x axis of...

Observer $S$ reports that an event occurred on the $x$ axis of his reference frame at $x=3.00 \times 10^{8} \mathrm{~m}$ at time $t=1.50 \mathrm{~s}$. Observer $S^{\prime}$ and her frame are moving in the positive direction of the $x$ axis at a speed of $0.400 c$. Further, $x=x^{\prime}=0$ at $t=t^{\prime}=0$. What are the (a) spatial and (b) temporal coordinate of the event according to $S^{\prime} ?$ If $S^{\prime}$ were, instead, moving in the negative direction of the $x$ axis, what would be the (c) spatial and (d) temporal coordinate of the event according to $S^{\prime}$ ?

Key Concepts

Spacetime Coordinates

Spacetime coordinates combine spatial and temporal information to specify the location and timing of events in the universe. In the context of special relativity, events are described by coordinates in a four-dimensional spacetime continuum, and these coordinates change according to the observer’s frame of reference as determined by the Lorentz transformations. Recognizing this dual nature of events is fundamental to analyzing how different observers perceive the same phenomena.

Relative Velocity and Direction

Relative velocity, including its magnitude and direction, plays a crucial role in applying the Lorentz transformations. The sign of the velocity (positive or negative) determines the direction of the boost, which in turn affects the transformation of both spatial and temporal coordinates. Understanding the impact of relative motion, especially when switching the direction of travel, is key to correctly interpreting the results in different inertial frames.

Inertial Reference Frames

In special relativity, an inertial reference frame is one in which an observer is either at rest or moving at a constant velocity, meaning no acceleration is present. Observers in different inertial frames may measure different spatial and temporal coordinates for the same event, yet the underlying physical laws remain invariant. This concept is central to understanding how observers in relative motion perceive events differently.

Lorentz Transformations

The Lorentz transformations are the set of equations that relate the space and time coordinates of events as measured in two different inertial frames moving relative to one another at a constant velocity. They ensure that the speed of light is constant in all inertial frames and account for phenomena like time dilation and length contraction. This mathematical tool is essential for converting the coordinates of events between observers in relative motion.

Please give Ace some feedback