Spacetime interval question involving a Pendulum in a Space Ship

  • #1
Chenkel
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Hello everyone, I was working on a thought experiment and came upon an inconsistent result that I thought maybe I could get some help with.

Firstly I am using natural units (c = unitless 1 i.e 1 light second per second) and I chose v = 3/5 to make gamma come out nicely (gamma equals 5/4 when v equals 3/5)

So the spaceship traveling at 3/5 light seconds per second has a pendulum at rest inside it with a period of 1 hertz (1 cycle per second) so I calculated the spacetime interval to be the squared distance between events minus the squared distance light travels between the events occurring, so that means the spacetime interval is equal to 0 - 1 = -1 light seconds.

In one second the spaceship travels 3/5 light seconds so relative to earth there is now a distance between the event of the pendulum coming back to its initial position of 3/5 light seconds.

So to discover the period of the pendulum relative to earth I solve the equation ##-1 = (\frac 3 5)^2 - (\Delta t)^2## So that means ##(\Delta t)^2 = \frac {9} {25} + \frac {25} {25} = \frac {34}{25}## so I have ##\Delta t = \sqrt {\frac {34} {25}}= 1.16##

So if the period of the pendulum in the spaceship is 1 second then the period of the pendulum that's on the spaceship relative to earth would be 1.16 seconds using my spacetime interval equations.

If I take the period of the pendulum on the spaceship (1 second) and multiply by gamma I should get the period of the pendulum on the spaceship relative to someone on earth which should be gamma, but gamma equals 1.25 using the time dilation formula, so these results seem inconsistent.

Any help would be appreciated, thanks in advance!
 
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  • #2
A pendulum won't work on a spaceship! It relies on gravity.
 
  • #3
In the Earth frame, ##\Delta x## is not ##v\times 1##, it's ##v\Delta t##.
 
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  • #4
PeroK said:
A pendulum won't work on a spaceship! It relies on gravity.
Can't a pendulum be made with springs?
 
  • #5
Chenkel said:
Can't a pendulum be made with springs?
A simple harmonic oscillator can be made with springs, but I don't think anyone would call it a "pendulum". "Pendulum" is a special case of "simple harmonic oscillator", not the other way around.
 
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  • #6
Chenkel said:
I was working on a thought experiment
Which looks just like all the other thought experiments you have already posted multiple threads on. Your basic issue in every single one has been the same: you still haven't grasped how to correctly transform between inertial frames. Your question here has nothing specifically to do with pendulums or springs or oscillators, and could be posed just as easily with light clocks--if you hadn't already posted a thread on light clocks.

Enough is enough. It is pointless to keep repeating the same explanations that you have already been given multilple times in multiple prior threads. If it didn't work with you then, it's not going to work with you now. You will have to somehow figure this out on your own. We have done all that we can to help you.

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  • #7
PeterDonis said:
A simple harmonic oscillator can be made with springs, but I don't think anyone would call it a "pendulum". "Pendulum" is a special case of "simple harmonic oscillator", not the other way around.
There are torsion pendulum clocks

https://en.m.wikipedia.org/wiki/Torsion_pendulum_clock
 
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  • #8
Dale said:
There are torsion pendulum clocks
Hm, yes, I wasn't thinking of a torsion spring, but it does qualify as a spring. :oops:

However, the basic point I made in post #6 remains the same.
 

FAQ: Spacetime interval question involving a Pendulum in a Space Ship

What is a spacetime interval?

A spacetime interval is a measure of the separation between two events in spacetime, combining both spatial distance and time difference. It is invariant under transformations and is given by the formula \( s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 \) in special relativity, where \( c \) is the speed of light.

How does the motion of a pendulum in a spaceship affect the spacetime interval?

The motion of a pendulum in a spaceship affects the spacetime interval by introducing time and spatial components that must be considered when calculating the interval between two events involving the pendulum. The pendulum's oscillation will contribute to the time elapsed (\( \Delta t \)) and the spatial displacement (\( \Delta x, \Delta y, \Delta z \)) of the events.

How do you calculate the spacetime interval for a pendulum in a spaceship?

To calculate the spacetime interval for a pendulum in a spaceship, you need to determine the time difference (\( \Delta t \)) between two events and the spatial distances (\( \Delta x, \Delta y, \Delta z \)) traveled by the pendulum between these events. Using the formula \( s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 \), you can compute the interval.

Does the velocity of the spaceship affect the spacetime interval of the pendulum's motion?

Yes, the velocity of the spaceship affects the spacetime interval of the pendulum's motion due to relativistic effects. If the spaceship is moving at a significant fraction of the speed of light, time dilation and length contraction must be considered, altering the perceived time intervals and spatial distances.

Can the spacetime interval be zero for a pendulum in a spaceship?

The spacetime interval can be zero if the separation between two events is such that the time component and spatial components balance out perfectly, resulting in a null interval. This typically occurs for events connected by light signals, but for a pendulum in a spaceship, achieving a zero interval would be highly specific and unlikely.

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