Solving the Spaceship Paradox: A New Explanation

In summary: C). I found##\Theta_{1,2}=\frac{d\beta_1}{dt}-\frac{d\beta_2}{dt}=\gamma^2 B\,\left( \frac{dB}{d\,t}\right)##.I suggest that there are three cases here, corresponding to ##\Theta_{1,2}##\Theta_{1,2}=0##, which is the case where there is no separation. This ties in with a number of other calculations.I rest my case.I don't think you rested your case. In fact I
  • #141
Please note that in the case at hand, the constant radar length measured by an inertial observer in the starting rest frame is not a rest length. The objects are accelerating in this frame, not at rest. The rest length of the object is length measured in its instantaneous rest frame (which is ambiguous here, because each end has a different family of instantaneous rest frames). However, whether you pick front or back instantaneous rest frames, you get increasing rest length for the congruence under consideration.

Thus, it does seem you have it backwards. Whether using radar distances or some other 'reasonable' definition of distance, rest length/distance is increasing. Distance (between ships) measured in the initial rest frame remains constant.
 
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  • #142
PAllen said:
...Two distant events are simultaneous for a give observer A making radar measurements if the proper time along A's world line of the midpoint of the signal round trips is the same.
That's clear enough.

The rest length of the object is length measured in its instantaneous rest frame (which is ambiguous here, because each end has a different family of instantaneous rest frames).
The ambiguity here is troubling. Maybe what I have been calling 'rest length' is more like a local ruler length. More to think about.

However, whether you pick front or back instantaneous rest frames, you get increasing rest length for the congruence under consideration.
I'll have to check that for radar measurements.

Good answers, and thanks for the reference. I have read that but not since I started on about radar in this thread.
 
<h2>1. What is the Spaceship Paradox?</h2><p>The Spaceship Paradox is a thought experiment that explores the concept of time dilation in the theory of relativity. It involves two identical spaceships traveling at different speeds and how time is experienced differently for each one.</p><h2>2. What is the traditional explanation for the Spaceship Paradox?</h2><p>The traditional explanation for the Spaceship Paradox is that time dilation occurs because of the difference in velocity between the two spaceships. The faster a spaceship travels, the slower time moves for that spaceship compared to a stationary observer.</p><h2>3. What is the new explanation for the Spaceship Paradox?</h2><p>The new explanation for the Spaceship Paradox proposes that time dilation is not solely based on velocity, but also on the acceleration of the spaceship. The acceleration causes a gravitational effect, which also affects time dilation.</p><h2>4. How does the new explanation solve the Spaceship Paradox?</h2><p>The new explanation takes into account the gravitational effects of acceleration, which were not considered in the traditional explanation. This additional factor helps to explain the observed time dilation in the thought experiment and provides a more comprehensive understanding of the phenomenon.</p><h2>5. What are the implications of the new explanation for the Spaceship Paradox?</h2><p>The new explanation challenges our current understanding of time dilation and the theory of relativity. It also has implications for space travel, as it suggests that the acceleration of a spaceship could have a significant impact on the passage of time for those on board.</p>

1. What is the Spaceship Paradox?

The Spaceship Paradox is a thought experiment that explores the concept of time dilation in the theory of relativity. It involves two identical spaceships traveling at different speeds and how time is experienced differently for each one.

2. What is the traditional explanation for the Spaceship Paradox?

The traditional explanation for the Spaceship Paradox is that time dilation occurs because of the difference in velocity between the two spaceships. The faster a spaceship travels, the slower time moves for that spaceship compared to a stationary observer.

3. What is the new explanation for the Spaceship Paradox?

The new explanation for the Spaceship Paradox proposes that time dilation is not solely based on velocity, but also on the acceleration of the spaceship. The acceleration causes a gravitational effect, which also affects time dilation.

4. How does the new explanation solve the Spaceship Paradox?

The new explanation takes into account the gravitational effects of acceleration, which were not considered in the traditional explanation. This additional factor helps to explain the observed time dilation in the thought experiment and provides a more comprehensive understanding of the phenomenon.

5. What are the implications of the new explanation for the Spaceship Paradox?

The new explanation challenges our current understanding of time dilation and the theory of relativity. It also has implications for space travel, as it suggests that the acceleration of a spaceship could have a significant impact on the passage of time for those on board.

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