Special Relativity Speedometer:

Transcription

1 Special Relativity Speedometer: Using Einstein s Theory of Special Relativity to Find the Velocity of a Rocket Colin Apke University of Colorado, Boulder

2 Purpose Demonstrate concepts of relativity Use those concepts to determine the speed of high moving objects. Rockets Satellites Interstellar Objects

3 The Speed of Light Light travels through space at a constant speed independent of the speed of the source or the observer. An observer is not able to discern from their surroundings that they are moving Terms to know: Frame of reference = System that someone inside a confined area observes C = speed of light constant

4 Concept of Simultaneity Though the speed of light does not change, other properties do.5c ~Time is different per frame of reference

5 Change in Time = Change in Speed Equation for time dilation: Time Dilation vs. Velocity t Earth = t v 1 Rocket 2 Rocket 2 c Velocity Time Dilation after 1 second Time Dilation after 1 seconds 1 km/hr 1 km/hr x 1-19 s x 1-17 s x 1-18 s x 1-16 s Speed you leave your driveway 1 km/min 1 km/min 1 km/s 1 km/s x 1-15 s x 1-13 s x 1-12 s x 1-1 s x 1-14 s x 1-12 s x 1-11 s x 1-9 s Speed of a 747 Airplane Speed of a GPS Satellite 1 km/s x 1-8 s x 1-7 s 1 km/s x 1-6 s x 1-5 s 1 4 km/s x 1-4 s x 1-3 s 1 5 km/s x 1-2 s s

6 The Clocks 1 Gigahertz = 1,,, times/second 1 Terahertz = 1,,,, times/second Dilation vs. Clock Speed Clock Speed 1 second 1 km/min 1 km/s 1 seconds 1 km/min 1 km/s 3 seconds 1 km/min 1 km/s 1 km/min 1 minute 1 km/s 5 minutes 1 km/min 1 km/s 1 minutes 1 km/min 1 km/s 1 GHz GHz GHz THz THz

7 Problems (Clocks) Errors in Oscillations: ~The faster the oscillator, the bigger the error ~Errors will start to mask the oscillation ~Could beat statistically Power: ~Fast Oscillators require large amounts of power and are complex ~Large batteries take up a lot of space ~Space on a rocket limits the maximum size Computers: ~Computer clock cycles are decreasing constantly as well as there size. ~Moore s Law: The number of transistors on a chip doubles every 2 years

8 Computers How to Make Computers Count: ~Computers follow instructions that take a certain amount of time ~An instruction loop: Loop is indefinite Size of the loop can very on the complexity of the program General Rule: no instruction should take longer then the dilation expected. ~Problems:.12 inches Distance light travels in a nanosecond: Distance light travels in a picosecond: 11.8 inches

9 Loops (Computers) Human Interaction vs. Computer Measurement Human Interaction: Loop could output a sine wave Dilation would cause interference Much better for slower speeds where the smallest difference will cause the most noticeable change Computer Measurement: Computer could transfer a jpg image Difference in pixels transferred = dilation More accurate using computer measurement

10 Results: Rocket To measure dilation, each computer must start and stop at the exact same time. Since time takes up a variable amount of time per reference frame, only a physical force should start and stop each clock First Possibility: Rocket For a rocket, dilation should not go in reverse. This means that the Earth clock should record more than the Rocket clock

11 Dilation (Results: Rocket) Apogee: Some dilation Velocity vs. Time Velocity vs. Time Launch: Maximum dilation Return: Maximum dilation Velocity Time Dilation vs. Time Dilation vs. Time Pre-launch: No dilation Landing: No dilation Dilation Earth-Clock: Rocket-Clock: Time

12 Better Solution (Results: Rocket) Instantaneous Velocity: Establish communication between the Rocket-clock and the Earth-clock Can create a dilation per unit interval More accurate description of rocket flight Limiting Factor: Size of the rocket Transmit vs. Receive

13 Results: Satellite Same basic concept for a satellite as was used for a rocket Satellite: dilation is easier to record since: Satellite is moving faster Satellite moving for a longer period of time Orbital Mechanics Velocity of a satellite is proportional to the altitude that it orbits at

14 Dilation (Results: Satellite) Circular Orbit: Constant velocity Variable Orbit: Variable velocity Velocity Dilation Dark Blue line: Velocity Pink Line: Dilation Case 1: Earth-Clock: Unit test period Satellite-Clock: Case 2: Satellite-Clock: Earth-Clock:

15 Even More Possibilities: Consider a celestial object: Different locations on the object spin at different velocities That difference can be measured by previous methods Day on a celestial object could be computed Radius could be computed Speed of planet at point 1 Speed of planet at point 2

16 Comparisons Relativity Speedometer *Complexity *Feasibility Strengths: *No direction required, just velocity *Good for high speeds or constant speeds *Applies everywhere Weaknesses: *Terahertz oscillator *Very precise measurements Other Methods: ~(GPS) ~(Accelerometers) Strengths: *GPS ~Widely used and available ~Much more applications *Accelerometers ~Cheap ~Small Weaknesses: *GPS ~May only work within a set limit ~Higher speeds may not work *Accelerometers ~Multi-dimensional analysis required

17 Goals Proof of Concept Possible for interstellar travel where very high speeds can be obtained Test to solve other problems that may not have been noticed in this write up

18 Questions?

19 Thank you for listening