Real Time Relativity. ANU PHYS1201 Lab

Real Time Relativity. ANU PHYS1201 lab. Updated 5 October 2008 1. References [1] realtimerelativity.org (RTR software available here) [2] http://www.anu.edu.au/Physics/Savage/TEE/site/tee/learn.html [3] http://www.phys.unsw.edu.au/einsteinlight/ [4] PHYS1201 Relativity notes. Section 10.

2. Apparatus Computer with programmable graphics card, Real Time Relativity software Internet access.

3. Aim The aim of this experiment is to develop an understanding of some of the concepts of Special Relativity and the effects of near light speed travel. This includes time dilation, length contraction, time delay, the Doppler effect, the relativity of simultaneity, the headlight effect and optical aberration.

4. Introduction Our perception of the world around us is limited to our everyday experience of space and time as observed moving at low relative velocities. However, Einstein showed that when one travels at a significant fraction of the speed of light relative to other objects, then the appearance of those objects can change dramatically. Indeed, the very concepts of space and time become relative. This is described in his famous 1905 paper about the theory of special relativity” “On the electrodynamics of moving bodies”, available on WebCT.

Since the publication of this paper, there have been many tests of the theory and it has been found to be completely accurate within the limits of experimental uncertainty. Relativity is not just an abstract theory of little importance to the real world. A good example of its application is in the measurement of absolute position using the Global Positioning System. A failure to account for relativistic effects (from both Special and General Relativity) would see the system become inaccurate at a rate of about 10 km/day.

Special relativity is based around two postulates: 1. The laws of physics are the same in all inertial reference frames

2. The speed of light (in a vacuum) is the same in all reference frames.

The first of these postulates indicates that there is no universal stationary reference frame and is a concept that has existed for a significant time. For example, if one is in an enclosed truck travelling at a constant velocity, there is no measurement that can be performed that can determine the speed of the truck (or indeed to determine that the truck is moving relative to the ground). With the discovery of electromagnetism, it was proposed that an aether existed which set the absolute reference frame in which light and other forms of electromagnetic waves propagated. However experimental measurements such as the famous Michelson-Morley experiment failed to detect the aether and the concept was eventually discarded.

1 Einstein’s insight was to realise that the speed of light is constant in every frame of reference. This is counterintuitive to our everyday perception of the world and leads to the many unusual effects that form part of special relativity. Our normal understanding of objects moving at relative speeds is probably best considered when travelling along a highway. A pedestrian observes two cars travelling past, one at 80 km/hr relative to the ground say, and a second, overtaking the first, at a speed of 120 km/hr relative to the ground. For the driver of the car travelling at 80 km/hr, the faster car moves past at a speed relative to their own speed of 40 km/hr. However, when dealing with light and relativistic velocities, the situation is different. An observer travelling at, for example, half the speed of light will still measure light to be travelling at c. As they accelerate to even higher velocities, they will still continue to measure the speed of light as c.

One implication is that when someone is moving relative to you their time is slower. This is called Time Dilation and is given by the formula: 1 Dt = g Dt0 where g = 2 1- v ( c2 ) 8 -1 v is the relative velocity. c is the speed of light in a vacuum – about 3x10 ms . t0 is the time between two events as measured by someone for whom they occur in the same place. t is the time between them for someone for whom they occur in different places – i.e. who is moving relative to the first person.

From this it follows that if something is moving relative to you its length contracts (in the direction of motion). This is called Length Contraction and is given by the formula: 1 Dx = g Dx0 where g = 2 1- v ( c2 ) v is the relative velocity. c is the speed of light in a vacuum. x0 is the length of the moving object. x is the length of the object when it’s stationary.

 Your notes provide a derivation and discussion of these formulas.

 The Through Einstein’s Eyes website (http://www.anu.edu.au/Physics/Savage/TEE) provides visualizations in video format.

 The Einstein Light webpage (http://www.phys.unsw.edu.au/einsteinlight/) has video introductions and derivations of these formulas.

You will find this lab easier and more enjoyable if you spend some time working through these references.

5. Pre-lab Problems 1. An object is 5 light-seconds away. How long does it take light from the object to reach you?

2. Three clocks are synchronised so that they read the correct time in their mutual rest frame, in which you are also at rest. One clock is right next to you, another is a light-second away, and the third a light-minute away. The clock next to you reads exactly midday. What times do you see on the faces of the other clocks when you look at them?

2 3. (a) Express a light-second per second in ms-1. (b) Express a light-second per second per second in ms-2, and in units of “gees”, the acceleration due to gravity, g = 9.8 ms-2. Use c = 3108 ms-1.

4. Does the Doppler effect increase or decrease the frequency of a light ray coming from: (a) in front, q < 90 degrees? (b) behind, q > 90 degrees?

5. Consider two incoming light rays with slightly different angles. Does the angular difference increase or decrease for rays coming from: (a) in front, q = 0 degrees. (b) behind, q = 180 degrees. See reference [4] for help.

6. Real Time Relativity Real Time Relativity (RTR) is a 3D virtual reality simulation that is able to show you what you would see if you were travelling at speeds near the speed of light. Because you are travelling at near light speeds in RTR the distances you cover are enormous. For example the Earth is 0.043 light seconds in diameter. Travelling at only 0.5c (0.5 x 3x108 ms-1 or about 1.5 x108 ms-1) you would travel 46.5 times this distance every second. In RTR you will see cube shaped clocks These clocks are 1 light second on a side – which is gigantic. Over 12,000 Earths could be stacked inside one of these clocks.

Comparative sizes of a RTR clock and the Earth

As you navigate a ship through the virtual reality, an orbiting camera provides lets you see where you are. You can turn off the Doppler Shift and Headlight effect as they can overwhelm the other effects (toggle these effects on and off with the control icons at top left of the screen).

To help read the time displayed on RTR’s clocks, even when they’re highly distorted, RTR adds labels with the observed time. These labels are not a consequence of relativity but an aid to help you read the time on the clock’s display.

RTR’s universe is a large, but finite, box. If you travel far enough in one direction you go out one side of the box and instantly reappear travelling in the same direction on the opposite side of the box. This is called a “toroidal universe”, and some people have suggested that our universe might be like this.

When you start RTR you see the Main Menu screen. The options are: “Scenarios”, “Controls”, and “Exit”. To return to this screen at any time, press the “Escape” key (ESC). Look at the Controls screen to see keyboard shortcuts for various functions. Select “Scenarios” to load a virtual world.

3 At the top left of the RTR screen are a number of clickable icons. The first takes you to the Main Menu screen. The adjacent icon toggles the visibility of the rest of the icons. These are toggles that control: the Doppler effect, the headlight effect, pausing, clock overlay, and clock zeroing. Pausing enables you to change your view direction, or inertial frame, while staying at the same event; the keyboard shortcut is “F2”.

To help you get your bearings and learn how to use RTR you can set the speed of light in RTR to be infinite. Toggle between an infinite and a normal speed of light by pressing the M key.

The mouse controls the rocket. Hold the left mouse button down while moving the mouse to steer the rocket.

The W and S keys control acceleration and deceleration. Holding the W key produces a constant proper acceleration of 0.2 light-seconds per second2, or 6.1×106 g. This is the acceleration you would feel if you were actually in the rocket (Would you survive?). However, your world frame acceleration decreases as you speed up. This fact prevents your speed exceeding that of light. When the acceleration is along the direction of motion, proper acceleration ap and world frame acceleration aw are related by

The mouse controls the orbiting camera when you hold down the right mouse button, so you can see around the rocket without having to change its direction. The F, B, L, R keys set your view direction to Front, Back, Left or Right relative the rocket’s velocity.

If you ever get lost you can reset to the starting configuration by using the O (Oh) key.

Exercise 1. Exploring RTR The aim of this exercise is for you to learn the interface, get to know your way around the world of RTR, and start observing some of the effects of moving near light speed.

Recall the instructions above for how you control your ship in RTR. Note: if you get lost or want to cut back on the weirdness – set the speed of light to be infinite by pressing the M key.

Load the “Racing course” scenario – press Escape to reach the menu screen. Try following the course by flying through all the hoops. Try it first with the speed of light being infinite (M key). Note that your rocket is still limited to speeds less than the speed of light. Next have a go as it really is with a finite speed of light (M key again). Don’t be too ambitious at first; get used to moving at half the speed of light, then accelerate gradually as you gain your relativistic “wings”. Which gives a faster trip – relativity on or off? Why?

Exercise 2. Looking at Reference Frames Consider a train moving through a station; from the inertial reference frame of the train, the station is moving, and from the inertial reference frame of the station, the train is moving. The relative speed of the train and platform are incredibly small in comparison to the speed of light

4 In RTR we can compare observations from reference frames moving at relative speeds close to the speed of light. Load the RTR scenario “Cube lattice”. You start facing a lattice of hollow cubes with respect to which you are at rest. Accelerate towards the cubes. When you change your speed you change your reference frame - and things can look very different from different reference frames.

Note three differences as observed from different reference frames. (You might want to turn on the Doppler effect by clicking the screen icon.) Describe what you observe and provide either a rough sketch or a screen printout illustrating each effect. Screen printouts should be secured into your lab- book.

Exercise 3. The effects of special relativity. The aim of this exercise is to identify most of the effects that can be seen in RTR, and understand them by connecting with the relevant theory.

Please match each numbered description below with an observation in RTR. You can use different scenarios, although all effects may be found in the “Relativity of simultaneity” scenario. Describe what you observe and provide a rough sketch or a screen printout illustrating each effect. Screen printouts should be secured into your lab-book.

Some of these effects are subtle and may be difficult to observe; try looking at objects from different locations and at different speeds (including zero). When you are observing clocks it can be helpful to observe multiple clocks. Remember “F2” is pause; it may be useful if you find things flying by too fast.

The following two effects may be toggled on and off using the screen icons as they can overwhelm everything else you see.

1. Headlight effect: Light is concentrated in the direction of motion, which seems brighter, while everything around appears dimmer.

2. Doppler Effect: an effect of motion on the relative frequency of waves. It is observed in the changing pitch of sound waves from a passing ambulance.

5 With light a change in frequency is observed as a change in colour. Relativity also gives the transverse Doppler effect, due to time dilation, where frequency is changed perpendicular to the direction of motion as well as in the direction of motion.

3. Light delay: light has a finite speed – this means it takes light time to move from an object to an observer. When a star 8 light-years away is observed the light received left the star 8 years before it is observed, thus showing its appearance from 8 years earlier. Remember the size of RTR world when looking for light delay.

4. Aberration: Aberration is the difference in the angle of incoming light rays measured by relatively moving observers, this is analogous to the aberration of rain you see when driving.

Rain falling at 60 km/h Rain falling at 60 km/h Train moving at 0 km/h Train moving at 30 km/h

Rain falling at 60 km/h Rain falling at 60 km/h Train moving at 60 km/h Train moving at 250 km/h

6 www.fourmilab.ch/cship/aberration.html

Einstein derived the aberration, or angle-change, formula for incoming light rays. If the angle of incidence in the world frame with respect to a particular direction is , then the angle of incidence for an observer moving along this direction at a velocity v is given by

1 2 q ў жc - v ц q tan = з ч tan 2 иc + v ш 2

5. Terrell Rotation: objects moving appear rotated; this is a combined effect of aberration and length contraction.

A tram at rest. Images from movies in A tram in the same location but travelling at reference [2]. high speed to the left appears to be Terrell rotated, or equivalently, shortened and sheared.

7 6. Distortion of objects:

The finite speed of light leads to some interesting distortions in the appearance of objects when moving at high speeds. One can see the sides of an object moving past at relativistic speeds, even when one is directly in front of the object. This occurs because the time the light takes to reach the observer is dependent on the distance from the observer. The figure shows the case of a rectangular object moving past at a relativistic speed. Light emitted from points A and B on the object reach the observer at the same time, as they are the same distance from the observer. However, light from the side AD can also be seen, albeit from an earlier time. Light from point D’ (point D at a earlier time) can reach the observer as the object “gets out of the way” of the beam of light. Hence at the instant that the observer sees light from points A and B, the observer also sees light from point D’ and thus views the side of the rectangle as well as the front.

This also distorts objects in directions perpendicular to the relative motion. When you are moving horizontally directly towards an object oriented vertically, such as a pole, its top is further away from you than its middle (at the same height as you). Hence, at any instant, light from further up the pole must be emitted at an earlier time, when the pole was further away, and the pole appears thinner. There are similar curvature effects for objects to the side of the direction of motion.

7. Length Contraction: moving objects shorten in the direction of motion. Using angular size as a measure of length can do this. The basic requirement is that the positions of the ends of the object be observed at the same time. This means they must be the same distance from you, so that the light delays are the same. How can this be achieved? Remember that length contraction only occurs in the direction parallel to the relative motion.

2.6x108ms-1 or 0.85c

6ms-1 or 7.2x10-5c

Zebra running; notice the faster Zebra on the right has length but not height has been contracted.

8. Time Dilation: time is slower for moving objects, so clocks that are moving run slower.

9. The relativity of Simultaneity: events that are simultaneous in one reference frame are not necessarily simultaneous in another reference frame.

8 Exercise 4. Verifying Time Dilation. (Extra credit) In this exercise you are required to verify that time for moving clocks slows down according to the time dilation equation:

1 Dt = g Dt0 where g = 2 1- v ( c2 )

You may only use the real world time (i.e the time on your watch) and observations of the times displayed on clocks, as if you were travelling in space with a clock and observing external clocks.

You need to ensure that you think about the relativistic effects and the effect of your viewpoint on this task.

Plan what data you need, how you should collect it, how you should interpret it and use it.

Experiment. You may need to try a few different approaches to find one that is effective.

Describe your experiment, noting the specifics of what you do. Provide tables of your data and the analysis required to verify the Time Dilation formula.

The Racing Course revisited

Can you fly the race course faster now that you understand more about special relativity?