Question

According to Bob on Earth. Planet Y (uninhabited) is 5 ly away. Anna is in a spaceship moving away from Earth a \(10.8 c\). She is bound for Planet \(Y\) to study its geology. Unfortunately, Planet Y explodes. According to Bob, this occurred 2 yr after Anna passed Earth. (Bob, of course. has to wait a while for the light from the explosion to arrive, but he reaches his conclusion by "working back ward.") Call the passing of Anna and Bob time zero for both. (a) According to Anna. how far away is Planet Y when it explodes? (b) At what time does it explode?

Short answer

The short answer would be that Anna would observe the explosion at the calculated time \(T\) and at a distance from Earth equal to \(10.8c \cdot T\). The exact values for these depend on the value calculated for \(T\) in Step 4.

Step-by-step solution

Problem Parameters

Establish that the velocity of Anna's spaceship relative to Bob is \(10.8 c\). The distance from Earth to Planet Y is 5 light-years (ly) as per Bob and the planet explodes 2 years after Anna passes Earth according to Bob.

Time of Explosion

Since the speed of light is '1 ly / yr', it would take 5 years for the light from planet Y's explosion to reach Bob on Earth. However, Bob says that the explosion occurred 2 years after Anna left Earth. Hence, the light took 3 more years to reach Bob after the explosion.

Lorentz Transformation

Now we switch from Bob's frame of reference to Anna's. We use the Lorentz transformation, which in this case simplifies to the time dilation formula. The time dilation formula is \(T = T_0 \cdot \gamma\), where \(\gamma = 1/\sqrt{1 - v^2/c^2}\) is the Lorentz factor, \(T_0\) is the stationary time (Bob's frame), and \(T\) is the time in motion (Anna's frame). Given that \(v = 10.8c\) (since Anna is moving), \(T_0\) is 2 years (when the explosion happened according to Bob).

Calculate Time

Plug \(v\) and \(T_0\) into the formula, and solve for \(T\) to find out when Anna would observe the explosion.

Calculate Distance

Since Anna is moving at a velocity of \(10.8c\) when she observes the explosion, she will have traveled for a distance of \(vt = 10.8c \cdot T\).

Key concepts

Lorentz Transformation

The Lorentz transformation forms the bedrock of Albert Einstein's theory of special relativity, which fundamentally altered our understanding of space and time. These mathematical equations describe how measurements of time, space, and even mass are related between two observers moving at a constant velocity relative to each other. The Lorentz transformation accounts for the fact that observers moving relative to one another will measure different distances, times, and even sequential ordering of events.

Understanding Lorentz transformation through an example

In our scenario with Bob and Anna, Bob measures the distance to Planet Y and the time of its explosion from his stationary reference frame on Earth. However, Anna is zooming through space at a significant fraction of the speed of light. To calculate when and how far away the explosion is according to Anna, we employ the Lorentz transformation. This allows us to convert Bob's 'stationary frame' observations into Anna's 'moving frame' observations accurately.

Time Dilation

The concept of time dilation is a direct consequence of the Lorentz transformation, which can leave students puzzled when they first encounter it. It states that time can run at different rates for observers who are in relative motion to each other. This isn't due to any mechanical fault in timepieces but is a feature of spacetime itself. According to special relativity, the faster you move through space, the slower you will move through time when compared to an observer at rest.

If Anna races away from Earth at incredible speeds, as she does in our textbook example, she would age more slowly from Bob's viewpoint. The time of events, such as Planet Y's explosion, will be stretched out (dilated) for her. So while Bob calculates the explosion to have occurred 2 years after Anna's departure, Anna, due to time dilation, would mark the explosion at a different moment on her spaceship's clock.

Light-Years

A light-year is a unit combining the concepts of both space and time, which makes it incredibly relevant in the field of astrophysics, and particularly in discussions involving relativistic physics. A light-year is the distance that light travels in one Earth year, about 9.46 trillion kilometers (or about 5.88 trillion miles).

Practical use of light-years in our exercise

In our exercise, the use of light-years provides a practical way to discuss the distance between Earth and Planet Y. Since we're told the planet is 5 light-years away, this translates to how long it would take for light, traveling at its constant speed in a vacuum, to cover the distance between these two points. This unit helps us conceptually bridge the gap between time and space, making it easier to apply the principles of relativistic physics, like the Lorentz transformation and time dilation, to real-world problems.

Reference Frames

Reference frames are perhaps one of the most crucial concepts in both classical and modern physics. They provide the context for observing and measuring phenomena. Imagining reference frames can be as simple as considering different viewpoints—a stationary person sees the world differently than someone on a moving train.

Reference frames in our exercise involve Bob, who is stationary on Earth, and Anna, who is moving. This distinction is essential because the laws of physics, though the same in form, can produce different measured outcomes based on the reference frame. For instance, the event of Planet Y's explosion has two different time measurements because Bob and Anna are in different reference frames. To reconcile these observations, we use the mathematical tools provided by Einstein's special relativity—showing us once again that the universe looks different depending on where you stand or how fast you're moving.