General Relativity for Grades 9 to 12

General Relativity for grades 9 to 12

Part 1: Masses curve space and space tells masses how to move. 1) Many students are holding some stretchy fabric so that the edges are level and the tension uniform. A dense mass is placed in the middle or a student underneath pulls down. a) Three different balls are released from similar positions. How is their motion similar to objects being dropped near the Earth? How is it different?

b) Get all three balls to orbit at the same time. How is their motion similar to the planets? How is it different?

c) If you were an amoeba on this surface, you would only be aware of two dimensions. You wouldn’t have a sense of up and down. You might explain the motion of the balls using ‘gravity’. How similar would the amoeba’s gravity be to ours?

d) This model seems to capture many aspects of how curved space can cause objects to orbit. What is its flaw?

2) Newton’s gravity can’t account for the way Mercury’s orbit precesses. Einstein’s GR can. a) Trace the cardboard orbit on a flat surface (Newton’s model). Trace the cardboard orbit on a balloon (Einstein’s model). Sketch what the orbits look like in each case.

b) There is a similar, but smaller, problem with orbits of Venus and the Earth. The problem was most noticeable for Mercury because Mercury is A) smaller B) denser C) hotter D) closer to the Sun 3) Alice steps off a ladder and falls to the ground where Bob is standing. A picture is taken at the start, halfway through the time and at the end. Space is up and time is across. Draw the diagram below left, on a whiteboard. Draw Alice and Bob`s positions at each of these times. Use a piece of tape to smoothly join Bob`s positions and another to join Alice`s. Which piece of tape is wrinkled and which way do the wrinkles point?

4) Repeat the exercise in the previous question on a beach ball. Sketch the result above. Which piece of tape is wrinkled and which way do the wrinkles point?

5) Suppose Alice stayed at the top of the ladder. How will her time compare to Bob`s? What does Einstein’s model say about time? Watch Alice and Bob: Can we travel through time? https://www.youtube.com/watch?v=HHYECUcfC1Y

6) What does GR have to do with the GPS? Watch GPS and Relativity https://www.youtube.com/watch?v=zQdIjwoi-u4 and/or GPS, relativity and nuclear detection https://www.youtube.com/watch?v=ky4RgRvVDoA Part 2: Masses curve spacetime and spacetime tells light how to move 7) Einstein predicted that light would be bent by curved space. Place a bowl upside down to represent the curved space. Take a piece of masking tape and press one third of its length on the table. Place the edge of the bowl in its path as shown below. Press the rest of the tape so it travels smoothly onto the bowl, along the bowl’s surface and off onto the table again. a)Draw the path of the light from the star, through the warped space to the Earth.

b) Where would people on Earth think the star was? Draw this on the diagram..

Earth star Sun

b) Which way will the light bend if the space is curved down, not up?

e)Where else will the star be seen?

f)To see this effect, you need to look at stars that appear to be near the Sun. That means that you need to look in the daytime. When could you see the stars?

8) Einstein predicted that the curved space could act like a lens. It would produce multiple images and partial or complete rings. The image shows a massive white galaxy. The arc around it is the warped image of a smaller galaxy that is much farther away. Suppose that the central galaxy was made entirely of invisible matter. What would be different in the image? What would be the same?

9) Suppose that you lived on the surface of a balloon and could only sense the two dimensions of the surface. What mathematical measurements could you make that would let you know that your 2-D space was curved into a third dimension? Part 3: High densities can collapse spacetime

10) In 1783 Rev. John Michell recognized that a large enough mass in a small enough space would result in a dark star – an object whose escape velocity was greater than the speed of light. What radius would our sun have if it turned into a black hole? (The mass of the sun is 2.0 x 1030 kg.)

11) In 1919 Karl Schwartzschild used Einstein’s equations of general relativity to find the radius at which a star would become a black hole. His answer was exactly the same as the calculations made 136 years earlier! This radius is called the Schwarzschild radius and not the Michell radius because Michell got the right answer for the wrong reason. Which is the wrong reason? He assumed that light particles

A) will be slowed down by gravity. B) have mass C) have energy given by E = ½ mv2 D) all of the above

12) Spacetime around a black hole is strongly curved. The Schwarzschild radius defines a region known as the event horizon. What would the sky look like near the event horizon? Watch Alice and Bob: Can we travel through time? http://www.youtube.com/watch?v=HHYECUcfC1Y

13) What is the biggest difference between a black hole and a dark star? The black hole A) won’t let light escape B) is bigger C) mostly empty D) sucks in nearby matter

14) How is a black hole different from a hole in a box? The black hole A) won’t let light escape B) is bigger C) is darker D) looks like 3-D hole

15) The strongest candidate for a black hole is the supermassive black hole Sagitarius A* at the centre of our galaxy. The closest measured approach is 1.8 x 1013 m and the star is moving at 4.9 x 106 m/s. Go to the Dark Heart of the Milky Way http://www.einstein- online.info/spotlights/milkyway_bh

a) How much mass is needed to cause the star to orbit like that? How many suns is this?

b) How big is the Schwarzschild radius for this mass? Which planetary orbit is it similar to? A) Mercury (~1010 m) B) Mars (~1011 m) C) Uranus (~1012 m) D) Eris (~1013 m) c) Is this definitely a black hole? What will the Event Horizon telescope show? Part 4: Disturbing spacetime produces gravitational waves 16) Masses alter the shape of spacetime and this means that accelerating masses should produce waves that transfer energy away. Therefore orbiting objects should slow down and spiral into the centre. The 1993 Physics Nobel Prize went to Hulse and Taylor for showing this with a pair of neutron stars. How well is this modelled by balls orbiting on the stretchy fabric?

17) Direct evidence for gravitational waves was first found in 2015. Watch Gravitational Waves Detected http://www.nytimes.com/2016/02/12/science/ligo-gravitational-waves-black-holes-einstein.html?_r=3

a) How are the gravitational waves different from sound waves?

b) Why are the waves so hard to detect?

c) How were they detected?

d) What is the next step?

18) Go to http://www.blackholehunter.org/ and try to hear the characteristic ‘chirp’ among the noise. a) How does the signal change in amplitude and frequency??

b) What can the signal tell us about the two orbiting objects? 19) Use the templates C1 to C6 to find the best match to signal 2, a ‘chirp’ signal. For each template, discuss how well the signal matched it in terms of frequency, amplitude and phase. Part 5: Spacetime stretches 20) The spectral lines of the galaxies tell us what elements the stars contain and how they are moving relative to us. Use the red shift of the lines to determine the velocity of each galaxy. Ca-K Ca-H reference lines Distance Velocity Galaxy (x1020 km) (km/s) 3925 3950 3975 4000 4025 4050 Å NGC 7.7 2100 1357 NGC 13.6 3147 NGC 3.4 3368 NGC 22.1 5548 NGC 10.0 6764 NGC 19.7 6745 Velocity Scale (km/s) 0 2000 4000 6000 a) Plot the distance to the galaxy on the y-axis and the speed on the x-axis. ) m k

0 2 0 1 x (

c 30 n a t s i d 25

1 2 3 4 5 6 7 Speed (x 106 m/s) b) Calculate the slope in seconds and then in years. c) What does the slope mean?

21) Why are all of the galaxies moving away from us? Do we have galactic bad breath? a) Overlap transparency and paper version of dots so that one dot overlaps perfectly. What do the other dots look like?

b) Inflate a balloon but do not tie it. Draw some random dots on it and draw a wave with a constant wave length. Allow the balloon to deflate and inflate. What happens to the dots? What happens to the light wave?

c) A long series of elastic bands have been linked together. Paper clips have been added to represent galaxies. Each group will choose a galaxy and measure the distance to the Milky Way. Then the universe will be stretched and the measurements will be repeated. What will you see if you plot the original distance (vertical) against the change in distance?

d) Astrophysicists say that the galaxies are not moving away from us. How do the three models suggest a different explanation the red shift of the galaxies?

e) The theory that explains the red-shift of the galaxies is called the Big Bang. How is the Big Bang different from an explosion?

Part 6: How does this material fit the curriculum? Which parts are a good match for grade 9 (Astronomy), grade 11 (Kinematics, Forces, Waves), grade 12U (Fields, Modern Physics) and grade 12 SES (Astronomy).