This tutorial is a visual study of the familiar Meterstick and the Hole 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 Meterstick and the Hole paradox may be defined as the following. Suppose that a meter stick is traveling at .866c in the x-direction. It would be 50% contracted so that the observer would say it is only half a meter long. And suppose a surface with a hole slightly larger than half a meter is traveling at .866c the y-direction. Suppose further that the meter stick and hole intersect in such a way that the meter stick passes through the hole without interference. If we switch reference frames to that of the meter stick, it will lose its length contraction and will now be a full meter long. How can it still pass through the half meter hole?
If you have not run this 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 “Launch” 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 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. Your list may vary. Scroll to and select Meterstick and the Hole.
In a few seconds, you will see two shapes appear in the scene. The orange shape performs the same function a as the meterstick. The green shape provides the hole, although in this case it is a notch. 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.
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 the Meterstick and the Hole example, click three times on Counterclockwise in the X-Navigation group. You will get a better view of the notch in the plate.
Click Reset Navigation to return to the original view.
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. 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 Meterstick and the Hole example, clicking Run will start the meterstick and plate moving on intersecting courses. The size and velocity of the two objects are such that, in relativistic physics, the meterstick will just pass thru the notch in the plate.
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. Click Stop and Reset now to make sure the simulation is ready for the next section.
Switching Reference Frames
One of the objectives of this simulation is to give you the opportunity to see how the position and shape of objects changes when observed from different reference frames. You can do this whenever the simulation is stopped and reset. Just below the Examples List is another selection box labeled Observer Reference Frame. The default reference frame is identified there. In that reference frame both the meterstick and the plate are moving. As required by the theory of relativity, the meterstick, moving in a direction parallel to its length is contracted along its length. The plate, moving in a direction perpendicular to its length has its thickness contracted. The objective of the example is to examine what will happen when the observer switches references frames to that of either the meterstick or the plate. In both cases the contractions currently being displayed will no longer apply. And yet the meterstick must still clear the notch or relativity will be inconsistent.
With the simulation Reset, click the down arrow of the reference frame selection box and notice that the meterstick and plate are listed there. Select the meterstick. (The number is a generic ID to help you keep track if more than one meterstick is in the scene.)
Notice that the meterstick is longer and the plate is now tilted. The meterstick is longer because it is now at rest with respect to you and it has lost its relativistic length contraction. The plate is tilted because it has acquired some x-directed velocity in addition to its original y-directed velocity. Run the simulation again and notice that the plate is now the only object moving and that it just clears the meterstick.
Click Stop and Reset. Then change reference frames to the plate. Note that the program picks the starting time of the simulation. Sometimes the objects are closer than you would expect as in this case. But the reference frame transformation is always true.
Now the plate has lost all its relativistic contractions. It is at rest with respect to you. Now it is the meterstick that is going to move diagonally toward the center of the plate. Compare the meterstick as it appears now with its shape when it was the observer reference frame. Notice that it is contracted along its length and thickness and its sides are tilted. So, when you run the simulation, the meterstick again just clears the plate.
No matter the reference frame of the observer, the meterstick and plate interaction is that they just clear each other.
Viewing and Changing Object Properties
When an object is inserted into the scene, it is provided with its own tab above the scene. The tabs allow you to view and change some of the object’s properties. When you are running a Newtonian simulation, clicking a properties tab will show you the Newtonian properties. When running a Relativistic simulation, clicking a properties tab will show you the Relativistic properties for the same object. If you have been following these instructions and have selected the Meterstick and the Hole example, there will be a properties tab for the meterstick and the plate. Click Reset and Reset Navigation. Switch to the default reference frame. Make sure that you are running a relativistic simulation by verifying that the checkbox labeled Relativistic at the bottom of the scene is checked.
Then click the properties tab for Meterstick-1. (The number suffix provides a unique ID in case you have more than one plate in the scene.) You will see several sets of information. At the top left are miscellaneous properties that may include fields for Id, name, description, rigidity, dimensions, color, etc. Notice that the meterstick’s x-size is .4. That’s 40,000,000m in this simulation. All of the objects in all the examples have very large sizes so as to exaggerate the time and contraction differences predicted by Special Relativity. To the right are sections for attached clocks and itineraries. The tutorial Pole in the Barn uses attached clocks. The Twins tutorial uses itineraries. Below are sections entitled Initial Conditions with respect to Observer and Current Relativistic Conditions with respect to Observer. Notice that the initial translation for the meterstick is x = -30,000,000m, y = 30,000,000m and z = 5,000,000m. Since the origin of the coordinate system is the center of the scene the scale of the scene is very large indeed! The meterstick’s initial velocity is Vx = 260,000,000m/s, Vy = 0m/s and Vz = 0m/s. Its rotation is zero for all axes, meaning it is inserted into the scene as originally drawn. Notice that the current relativistic conditions are identical to the initial. The simulation is not running so no time is passing and the meterstick has not yet moved.
Click the properties tab for the Notched Plate-2. Notice the initial translation and velocity are such that the centers of the meterstick and plate will intersect as you saw in the simulation. (The notched plate shows a rotation of 90 degrees about the z-axis. This is because the default shape has a vertical notch and this example requires a horizontal notch.)
Then click the properties tab for Meterstick-1. Change the x-dimension of the meterstick from .4 to .5. (Remember to Tab out of the table. If you don’t you should get a message reminding you to do so.) Then click the Simulation tab to return to the scene. Notice that the meterstick is now a little longer. Run the simulation and notice that the meterstick and plate collide. The simulation is equipped with a collision detector which may be programmed to pause the simulation and alert the user when a collision occurs. In the System Message Window, click OK. Then Continue.
The meterstick and plate will pass through each other. Click Stop and Reset and switch reference frames to that of the meterstick and run the simulation again.
Notice that the meterstick and plate again collide. But the two ends collide at different times with respect to this observer reference frame. This is another characteristic of Special Relativity. Two spacially separated events that occur simultaneously in one reference frame may not occur simultaneously in another.
Reset the simulation and switch reference frames to that of the plate. Then run the simulation again.
Notice that the meterstick and plate again collide. A truly precise analysis would indicate that the two ends collide at different times in this reference frame too. But the program indicates both ends collided simultaneously. This is a result of the level of precision at which the program is designed. Click the Properties tab. This tab contains several pieces of information about the way the simulation will run.
Of interest right now is the table of Collision Rules. The rule that is implemented says to alert the user if there is a collision between the meterstick and the plate. You may turn the rule on and off and run the simulation again. If there were more objects in the scene, you would be able to add more rules.
Try running the MeterStick and the Hole, Plus example or the Obstacle Course example. These examples extend the study of the basic MeterStick and the Hole with additional objects and geometries.