Problem 1 - College Physics

Question

A spaceship is traveling toward earth from the space colony on Asteroid 1040 \(\mathrm{A}\) . The ship is at the halfway point of the trip, passing Mars at a speed of 0.9\(c\) relative to Mars's frame of reference. At the same instant, a passenger on the spaceship receives a radio message from her boyfriend on 1040 \(\mathrm{A}\) and another from her hairdresser on earth. According to the passenger on the ship, were these messages sent simultane- ously or at different times. If at different times, which one was sent first? Explain your reasoning.

Step-by-Step Solution

Answer

The messages were not sent simultaneously. The message from Earth was sent after the message from 1040 A from the passenger's perspective.

Step 1: Understanding the Problem Context

The passenger receives two radio signals—one from 1040 A and another from Earth while passing Mars. The spaceship is traveling at 0.9c relative to Mars. We need to determine if these messages were sent simultaneously or at different times from the passenger's perspective.

Step 2: Applying the Concept of Simultaneity in Relativity

In special relativity, simultaneity is relative, meaning whether two events occur simultaneously can differ based on the observer's frame of reference. The passenger on the spaceship moving at a high velocity (0.9c) will experience time and simultaneity differently under these conditions.

Step 3: Using the Relativistic Doppler Effect

We can use the relativistic Doppler effect to see how the messages are perceived. Since the spaceship is moving toward Earth and away from 1040 A, the message from Earth would be blue-shifted (received earlier) while the message from 1040 A would be red-shifted (received later) from the passenger's point of view.

Step 4: Applying Lorentz Transformation

Using the Lorentz transformation, we determine the time at which each signal was sent. The transformation considers both time and distance, factoring in the speed of light and the relative motion of the observer on the spaceship. This allows us to calculate the time difference of the sending events from each point.

Step 5: Conclusion from the Calculations

According to Lorentz transformations, the message from Earth was sent later from the passenger's perspective because it is blue-shifted, implying closer synchronization with the passenger's current frame compared to the red-shifted message coming from the distant 1040 A.

Key Concepts

Simultaneity

In the realm of relativity, simultaneity is not as straightforward as it seems in everyday experiences. Normally, we think of two events occurring at the same time if we see or record them simultaneously.

However, Einstein's theory of relativity introduces us to the idea that simultaneity is not absolute. It can vary depending on the observer's state of motion. For someone moving at a significant fraction of the speed of light, like the passenger on the spaceship traveling at 0.9c in our exercise, the synchronization of events changes.

This means that what appears to be simultaneous events in one frame of reference (like Mars) might not be simultaneous in another (like the spaceship). This shift in simultaneity is due to the effects of relative motion on time perception, defined by the principles of special relativity.

Relativistic Doppler Effect

The relativistic Doppler effect is a phenomenon that modifies how we perceive the frequency of waves, such as light or radio signals, depending on the relative motion between the source and the observer.

When the observer is moving toward the source, the waves are compressed, leading to a "blue shift", making the signal appear to be of higher frequency. Conversely, if the observer is moving away from the source, the waves are stretched, causing a "red shift", giving the perception of a lower frequency.

In the context of the exercise, the spaceship heading towards Earth would experience a blue-shifted message from Earth, received earlier than another message from Asteroid 1040 A, which would be red-shifted because the ship is moving away from it. This helps explain the sequence of how the messages were received by the passenger on the spaceship.

Lorentz Transformation

Lorentz transformations are mathematical equations used to convert between two different frames of reference in relative motion within the context of the speed of light. These equations help in understanding how measurements of time and space differ for observers in different states of motion.

For instance, when the passenger on the spaceship receives two different radio signals, Lorentz transformations allow us to calculate not just when these signals were sent, but also the sequence of their sending from her moving frame. This is crucial because, at speeds close to the speed of light, time dilation and length contraction take effect, altering the perception of simultaneity and sequence of events.

In our problem, applying the Lorentz transformation helped affirm that the message from Earth was perceived as being sent later than from Asteroid 1040 A due to their relative motions and points of origin.

Speed of Light

The speed of light, denoted as \(c\), is central to the theory of relativity and several calculations involving time and space. In a vacuum, light travels at approximately 299,792,458 meters per second, a speed that remains constant regardless of the observer's motion.

This constancy of the speed of light is a cornerstone in Einstein's theory, implying that no matter how fast an observer is moving, light's speed remains unchanged. This characteristic has profound implications. For example, it ensures that observers, such as our passenger, perceive signals and events in a way that obeys relativistic physics, even at high velocities like 0.9c.

In our exercise, the speed of light acts as a limiting factor and a decisive element in the calculations involving simultaneity and sequence of events, reinforcing why observers on the spaceship and Earth might report different timings for sending and receiving messages.