This tutorial is a visual study of the familiar Twins Paradox.†
Please Note:† Relativity is built on and modifies Newtonian Physics.† These tutorials do not attempt to teach the user Newtonian Physics.† They assume the user already knows Newtonian Physics
The Twins Paradox may be defined like this.† Given a pair of twins, one is a stay-at-home kind of person and the other is an astronaut.† The astronaut takes a space trip.† The trip is modeled as
∑ A quick change to an inertial reference frame moving at relativistic speed away from the earth,
∑ A long period coasting in the reference frame,
∑ A quick change to an inertial reference frame moving at relativistic speed toward the earth,
∑ A long period coasting in the reference frame,
∑ A quick change to the reference frame of the earth and reunion of the twins.
Since Special Relativity says that time for a body moving at relativistic speed with respect to an observer runs slower, the stay-at-home twin expects the astronaut to return younger.† However, Special Relativity also says that no inertial reference frame is special so the astronaut considers the stay-at-home twin to be the one with the relative velocity.† So wouldnít the astronaut expect the stay-at-home twin to be younger?
If you have not run the simulation application before, please read the System Requirements document at http://relativitysimulation.com/Documents/SystemRequirements.html.† If you are comfortable that your system satisfies the requirements, or just want to try it and see what happens, go to http://relativitysimulation.com and click the ďLaunchĒ button. †The first time you run the application, it may take a minute or two to load.† When it is successfully loaded, you will see the blank simulation scene below.†
The above picture was taken on a computer running Internet Explorer 8 on Windows Vista.† What you see on your computer may vary.† Notice the two checkboxes at the bottom center of the simulation window.† Simulations may be run using either Newtonian or Relativistic physics.† The default is relativistic and that is what this tutorial is for.††
Selecting a Predefined Example
At the bottom right is a selection box labeled Examples List.† Clicking the selection box will display a list of examples.† The easiest way to use the application is to select a predefined example from this Examples List.† The list may vary.† Scroll to and select Celebrated Twins.†
In a few seconds, you will see three rows of clocks (plus one) inserted into the scene.† After the scene is populated with an example, new tabs will appear above the scene.† The tabs will be explained in the section on Viewing and Changing Object Properties. †This scene should be familiar to those who have seen diagrams in textbooks illustrating relative time.† Notice that the clocks have no numbers on them.† They are intended to show only relative time.† The readings for the middle set of clocks are all the same.† The red clock hand points straight up for all of them.† The middle row of clocks is intended to show the time at equally spaced distances in the observerís reference frame.† For this example that would be the Earth reference frame.† The readings for the top and bottom rows of clocks are not the same.† The clocks themselves are contracted in the x-direction.† The top row shows the time at equally spaced distances for a reference frame moving at relativistic speed to the right.† The bottom row is for a reference frame moving to the left.† Recall that the Twins Paradox concerns twins, one of whom takes a round trip space flight while the other stays on Earth.† The blue clock in the middle of the observerís row belongs to the earthbound twin.† The lone blue clock just above it is the traveling clock, belonging to the astronaut twin.†
Navigating through an Example
At the bottom left of the scene are navigation buttons.† These buttons allow you to look around the scene.† The buttons are in three groups with a reset button above.† The Reset Navigation button will cancel all navigation commands and present the scene to you as first inserted.† Buttons in the X-Navigation group affect your view by changing your orientation with respect to the x-axis of the scene.† Similarly, there are buttons for the y-axis and z-axis.† Clicking the Left button, for instance, will move the objects in the scene a bit to the left.† If your browser is not showing you as much of the scene as you would like, clicking the Out button will zoom you out a bit and show you more.† To view the scene from a different angle, try clicking a Clockwise or Counterclockwise button.† If you have selected Celebrated Twins, the example is loaded with all the clocks in a single plane.† To verify this, in the Y- Navigation group, click the Clockwise button four times.†
Then, in the Z- Navigation group, click the Clockwise button three times.†
Click Reset Navigation to return to the scene as inserted and then in the Z- Navigation group, click In or Out until your computer screen displays about as many of the clocks as the following picture.
Running an Example
To run an example, at the bottom of the scene, click the Run button.† When running, the objects in the scene will move according to the velocities and rules specified for them in their respective properties tabs.† Note that if you have inserted objects into the scene yourself instead of selecting an example, the objects are initially inserted with no velocity and no rules.† So clicking the run button will not do anything.† If you have selected the Celebrated Twins example, clicking Run will start the top row of clocks moving to the right and the bottom row moving to the left.† The middle row will remain at rest with respect to you and the astronautís clock will make a round trip, first joining the row of clocks moving to the right, then joining the row moving to the left and finally returning to rest above the Earth clock.
Stopping an Example
If a scene is running, you will notice that the Run button has changed its name to Stop.† Click it to stop the simulation.
Continuing an Example
When stopped, the Stop button will change its name to Continue.† Click it to continue the simulation from where it stopped.
Resetting an Example
Clicking the Reset button will reset the objects in the scene to their initial positions ready to run again.
Try running, stopping and resetting the simulation several times and note the following.† When the simulation is running, the top and bottom clocks are running at half the speed of the middle, the observer string.† The astronaut clock runs at half speed also during its trip but switches to full speed when it comes to rest with respect to the observer string.† If you stop the simulation as soon as the astronaut gets back to earth, the astronautís clock will read half the elapsed time as the earth clock.†
Switching Reference Frames
One of the objectives of this simulation is to give you the opportunity to observe the movement of objects from different reference frames.† You can do this whenever the simulation is stopped and reset.† If you have selected the Celebrated Twins example, for instance, it is initially inserted into the scene with you, the observer, in a default reference frame.† This default also happens to be the reference frame of the Earth and its string of clocks.† That is, the Earth clock and its string are at rest with respect to you and the other two strings are moving.† Just below the Examples List is another selection box labeled Observer Reference Frame.† The default is identified there.† Click Reset.† (If Reset is not active, try clicking Stop first.)† Then Click the down arrow of the reference frame selection box and select Eastward Moving Frame-2.† (The numbers after the object names are generic IDs useful if you have more than one of the same objects in the scene.)†
The top row of clocks is now at rest with respect to you.† The readings are all the same.† The clocks and the spacing between them have lost their contraction.† The Earth row is now moving to the left.† Their readings are different.† They and the spacing between them have gained contraction (50%).† Since the astronaut starts on earth, the astronautís clock is contracted too.† The Westward Moving Frame clocks are still moving to the left but at a much higher velocity.† Their contraction is greater too (about 86%).†† Now, when you run the simulation you will be observing the astronautís trip from the point of view of the Eastward Moving Frame.† Try it.† The first thing that will happen is that the astronaut will jump to the observerís reference frame and the astronautís clock will lose its contraction.† The string of clocks in the Earth reference frame will be running at half the rate of the astronautís clock.† This may seem contradictory, but be patient.† The space trip is far from over.
When the astronaut makes the turnaround by jumping to the Westward Moving Frame, the astronautís clock will acquire the contraction and clock rate of the clocks in that reference frame.† At that point in the trip, from the point of view of the Eastward Moving Frame, the astronautís clock is now ahead of the Earth clock.† The astronaut is older.† But the clock rate for the reference frame in which the astronaut is now traveling has a much slower clock rate.† It is hard to see, but the earth clock reading will catch up and pass the astronautís clock during this phase of the space trip.
The astronautís clock, the Earth string and the Westward Moving Frame string will all disappear to the left.† You will have to click the Right button in the Navigation group many times to see the final stage where the astronaut returns to earth.†
When the astronaut finally rejoins the earth reference frame, click Stop and note that the elapsed times for the earth and the astronaut are the same for an observer in this reference frame as for an observer in the Earth reference frame.†
If you like you can switch to the Westward Moving Frame and observe the same result.† But the more interesting simulation is from the point of view of the astronaut.† Click Reset, Reset Navigation and switch to the Astronaut reference frame.
Now the observer will follow the astronaut.† Click Run and note that the Eastward Moving Frame immediately looses it contraction, all its clocks now agree and they are running at the same rate as the astronautís.† The Earth reference frame, on the other hand gains contraction, its clocks no longer agree and they are running slower than the astronautís clock.†
Next, Eastward Moving Frame will regain contraction and the Westward Moving Frame will lose it.† The Earth reference frame will continue to be contracted and its clocks will continue to run slow.†
But when the astronaut returns to Earth, the astronautís clock is behind the Earth clock by the same amount that you saw when running the simulation from other reference frames.†
What happened?† Rerun the simulation and keep an eye on the Earth clock.† The Earth clock is behind the Astronaut clock for the portion of the trip when the astronaut is in the Eastward Moving Frame.† However when the astronaut switches to the Westward Moving Frame, the Earth clock jumps ahead.† It jumps so far ahead that the entire return trip, where it is running slower, is not enough to allow the astronaut clock to catch up.† That behavior is not realistic.† Time is a continuum.† It doesnít jump for anybody.† But the problem itself is specified unrealistically.† It is given that the astronaut switches from relativistic speed in one direction to relativistic speed in the opposite direction instantaneously.† And that canít happen either.† A realistic statement of the Twins example would be to specify some magnitude of acceleration which creates the turn around.† And it is during that acceleration that the astronaut would observe the Earth clock to run very fast creating the situation you see in the final stages of the trip.
To get a better understanding of what happens during the astronautís turnaround, run the example Twins, no instant velocity change (the tutorial has the same name).† There are also two examples and tutorials where the astronaut and earth exchange light signals.† Exchanging light signals is a way for the twins to prove who is aging faster without having the astronaut land back on earth and compare clocks.