Question

From a standstill, you begin jogging at \(5 \mathrm{~m} / \mathrm{s}\) directly toward the galaxy Centaurus A. which is on the horizon \(2 \times 10^{23} \mathrm{~m}\) away, (a) There is a clock in Centaurus A. According to you, how will readings on this clock differ before and after you begin jogging? (Remember: You change frames.) (b) The planet Neptune is between Earth and Centaurus A. \(4.5 \times 10^{9} \mathrm{~m}\) from Earth. How much would readings on a clock there differ? (c) What would be the time differences if you had instead begun jogging in the opposite direction? (d) What do these results tell you about the observations of a traveling twin who accelerates toward his Earth-bound twin? How do these observations depend on the distance between the twins?

Short answer

The times on both clocks will appear to slow down when you start jogging towards them, with the effect being more pronounced on the clock in Centaurus A due to greater relative velocity, and vice versa when you start jogging in the opposite direction. The travelling twin would observe Earth-bound twin's clock to go slower when approaching and go faster when receding and the effect would be more pronounced the further they are apart due to increasing relative velocity.

Step-by-step solution

Analyze the effect on time from the First position (Centaurus A)

Time dilation in special relativity can be calculated using Lorentz transformation which states that time, measured in the frame where the clock is at rest, \(t\), is longer than the time \(t'\) in the frame where the clock is moving at speed \(v\). The expression for time dilation is \(t = \gamma t'\), where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \). The clock in Centaurus A is at rest in its own frame but appears to be moving towards the observer (you), once you begin jogging. Hence, according to you, time on the clock in Centaurus A would appear to run slower compared to when you were at rest. The exact difference can be calculated using the Lorentz transformation.

Analyze the effect on time from Neptune's position

The Lorentz Transformation should also be applied for the clock in Neptune. As for the clock in Centaurus A, you are moving at speed \(v = 5 \, m/s\) relative to the clock in Neptune. However, the difference will be less pronounced, because the relative speed of you and the clock in Neptune is less than the relative speed between you and the clock in Centaurus A, as they are much closer to you.

Analyze the effect on time if jogging in the opposite direction

If you had begun jogging in the opposite direction, both the clocks in Centaurus A and Neptune would appear to be moving away from you. In this case, due to the principle of time dilation, the time in both clocks would appear to go faster compared to when you were at rest.

Drawing conclusions

The observed time dilation effect is dependant on motion and it's relative towards the observer. Similarity, the travelling twin would observe the Earth-bound twin's clock to go slower when moving toward and go faster when moving away. Also, the farther the twins are, the more pronounced the time dilation effect would be, due to increased relative velocity.

Key concepts

Lorentz Transformation

Understanding the Lorentz transformation is crucial for comprehending how time dilation occurs. Developed by the physicist Hendrik Lorentz, it offers a mathematical formula that bridges the gap between two frames of reference that are in motion relative to each other. This is fundamental in Albert Einstein's theory of special relativity. The Lorentz transformation encompasses not just time, but also the spatial coordinates of events that occur.

The basic formula for time dilation can be expressed as: \[\begin{equation} t = \frac{1}{\frac{1}{\frac{1}{\t{sqrt}{1 - \frac{v^{2}}{c^{2}}}}}} \frac{1}{\t{sqrt}{1 - \frac{v^{2}}{c^{2}}t'\}} \frac{1}{\t{sqrt}{1} - \t{sqrt}{1} - \frac{v^{2}}{c^{2}} \end{equation}\]\Here, \textstyle \textstyle \textstyle \textstyle \textstyle \textstyle \textstyle \textstyle \textstyle t\ is the duration as seen by the stationary observer,\textstyle \textstyle \textstyle \textstyle \textstyle \textstyle t\ is the duration as seen by the stationary observer,or\textstyle \textstyle \textstyle \textstyle \textstyle \textstyle \textstyle \textstyle \textstyle c\ is the speed of light. Crucially, this equation shows that time will always appear to slow down for objects in motion relative to the stationary observer - a phenomenon confirmed by numerous experiments.

To truly appreciate this, we must recall that the speed of light is the ultimate speed limit in the universe. All motion is relative, and even at velocities we consider fast on a human scale, such as jogging toward Centaurus A, the effect on time is imperceptibly small because our speeds are so minuscule compared to the speed of light. Therefore, for astronomical distances and objects moving close to the speed of light, the Lorentz transformation demonstrates significant adjustments to our intuitive perceptions of time and space.

Special Relativity

The counter-intuitive world of special relativity upends our everyday comprehension of time and space. Introduced by Albert Einstein in 1905, this groundbreaking theory asserts two postulates: the laws of physics are invariant in all inertial frames of reference, and the speed of light in a vacuum is constant, regardless of the motion of the light source or observer. These ideas reshape our understanding of movement and measurement.

Through special relativity, we grasp that events which seem simultaneous in one frame of reference may not be in another. This has profound implications: an observer in motion relative to another observer will discern time to pass differently. In the exercise on time dilation due to jogging towards a distant galaxy, we're observing these relativistic effects first-hand, albeit on a much more modest scale. According to special relativity, a moving clock runs slower—this is time dilation. The reverse is also true: the moving observer perceives a stationary clock to be ticking slower. Importantly, these effects only become substantial when velocities approach the speed of light. At everyday speeds, like jogging, the differences are minute, and only with precise instruments can we detect them.

Special relativity has had a profound impact on all of modern physics. It underlies our understanding of particle physics, cosmology, and has led to technologies such as GPS satellites which must correct for relativistic time effects to maintain accuracy.

Relative Velocity

Relative velocity refers to the velocity of one object as observed from another moving object. It's a concept that's closely tied to both Lorentz transformation and special relativity, as it contributes to observers experiencing time and space differently. Essentially, it's not just the speed but the direction of motion that influences how time dilation is perceived.

When you began jogging towards Centaurus A in the textbook exercise, your relative velocity with respect to the galaxy increased. According to special relativity and as expressed by the Lorentz transformation, this slight increase in relative velocity causes you to perceive the clock in Centaurus A to tick more slowly. Conversely, if you jog away, the distant clock seems to tick faster. This asymmetry in time perception stems from your changing frame of reference due to the velocity difference.

For the clock on Neptune, a much closer object, the relative velocity has a smaller magnitude, which translates into a less pronounced effect on time dilation. The principle then extends to cosmic scales: as the separation increases, the impact of relative velocity on time dilation grows because differences in velocity become more significant. It's a reminder of how motion is relative and interlinked with our experiences of time and reality, concepts thoroughly explained by the theory of special relativity.