1. Light and Gravity

3. Three Simple Questions
How do small coasting objects move under the influence of a massive body (in particular, as they move away from it)?
What happens if we change the size of the massive body?
How does light itself move near a massive body?

4. Start with Moving Objects: Newtonian Dynamics
Note the important proviso: we launch the rocket with a sudden acceleration (using an engine) but it quickly shuts off, after which the rocket is coasting, affected only by gravity (“ballistic” motion).
Can a rocket ‘escape’ from the Earth’s surface? If so, how?

5. Consider Ballistic Motions: [that is, where objects are affected only by gravity]
Here, a thrown baseball (say)

6. In General. the Path Depends on the Initial Speed…

7. ...and Direction

8. Escape
Velocity
To escape entirely, and never fall back, the projectile has to leave the surface with a velocity greater than vesc – the escape velocity.

9. Exactly What Velocity is Needed?
Here’s the equation: vesc2 = 2 G M/R
(M = mass of planet; R = radius of planet)
This makes sense!
bigger M = more atoms in total, stronger pull of gravity, so you have to be moving faster as you leave the surface if you hope to escape
smaller R = more densely packed. Standing on the surface, you are closer to the center of the planet and feel stronger gravity, so you have to be moving faster to escape from there

10. Example: Escaping the Earth’s Surface
M=6x1024 kg R = 6400 km
Escape velocity = 11 km/sec (~40,000 km /hr)
[A bullet from a 44 Magnum travels at about 0.5 km/sec,
so it will inevitably fall back.]

11. Suppose the Earth Was Bigger [say, 50000 km in radius]
You are now much farther away from many of the Earth’s atoms (so you would weigh less!) and vesc = 4 km/sec (Escape is much easier)

12. Equivalently, Hovering at Rest 50,000 km Above the Present Earth
As before, vesc = 4 km/sec
All that matters is how far you are from the centre of the distribution of atoms. You don’t have to be touching the surface (and rockets don’t have to ‘push against the Earth’ to do their thing!)

13. But What If…
We shrink the Earth to the size of an olive (say, R ~ 1 cm) in a huge hydraulic press, but stay where we are, hovering in space

14. It Doesn’t Matter
Just as before, vesc = 4 km/sec Again, all that matters is how far you are from the centre of the distribution of atoms.

15. But If We Descend to the Surface…
Standing on its surface, you are now very close to all of its atoms, and feel an enormous cumulative gravitational force We discover that vesc = c (the speed of light) And for even denser objects (squash the Earth a bit more!) vesc > c

16. For the Super-Dense Earth:
Any body we launch upwards will have less than escape velocity. Presumably it will rise up, come to a halt, and fall back.

17. That’s Indeed How Rockets Would Behave
After the engines shut off, the rocket continues its upward motion, but loses speed as it goes.
Since it started off with less than the escape velocity, eventually it comes to a halt and falls back to Earth.

18. The Key Question
If the Earth were very dense, would light act in the same way? Can light ever come to a halt?

19. Laplace Thought So! He believed that if you could
hover near such a super-
dense object, you might see
light rising, coming to a halt,
and falling back…

20. But It’s Not That Simple!
The behaviour of light is more subtle than that!
To understand that, we need to consider some of the manifestations of Einstein’s general theory of relativity – and what gravity really is.

21. Let’s Think More About Light
Does it always travel in “straight lines”? We tend to think so, and indeed we tend to rely on this ‘fact’ in any number of circumstances.

22. Playing Pool? Select a Straight Cue by Rolling it...

23. Alternatively, Look Along It
This approach assumes that light travels in straight lines.

24. Does Light ‘Curve’?
When we look along the cue, we are assuming that it does not. In day to day life, we take the motion of light to define ‘straight lines.’

25. Another Application: Bullets Change Path; Light Does Not The eyepiece looks ‘straight at’ the target; the rifle is aimed a little high (depending on the range) to compensate for gravity.

26. In Reality, However, The Trajectory of Light is Also Affected by Gravity!
Because of its enormous speed, however, the ‘fall’ of light as it moves near the Earth is very tiny, so we can usually think of light as moving ‘in straight lines.’ But this will not be the case when the gravity is locally very strong!

27. Critical Point No.2:
Unlike a thrown ball, light never slows down or comes to a halt, no matter your frame of reference
For all observers, in any location or any state of motion, light appears to pass by at the speed of light.
In other words, if you are ‘hovering’ just outside a black hole, you will not see light rise up, come to a momentary halt, and fall back in the way that a spent rocket would.

28. A Fresh Look
Together, these facts require a completely fresh way of thinking about light, space, and time itself! – Einstein’s General Relativity

29. Einstein’s Deep Insight
Einstein did not say “…because of gravity, light follows a curved path (like a thrown ball) as it moves through flat [undistorted] space.” Instead, he said that space itself is distorted by the presence of matter, and light follows the ‘simplest path’ it can (called a ‘geodesic’) through these dips and wiggles.

30. This is a New Way of Thinking About Gravity
Gravity should not be thought of as a force, but rather as a distortion of space itself (Moreover, as we will see, this also affects the rate at which time itself passes!)

31. Comparing the Two

32. Newton’s Laws Are Usually “Good Enough”
Newton’s “law of gravity” is a perfectly useful and effective way of calculating forces, motions, and so on -- unless you are in a region where space is strongly distorted
For instance
relatively close to a very large mass,
or when matter is locally quite densely compressed: neutron stars, black holes.

33. In Principle, Though: the Effect is Universal!
Your very presence in the room causes a [tiny] distortion in the fabric of space and time. So too does every atom, every mote of dust, and so on… not just big things like stars! Newton’s laws are merely good approximations!

34. So: Gravity as Geometry - an analogy

35. Golf Balls on a Green Note how the trajectory curves.
Note how the trajectory curves.
What would it look like from directly overhead? The ball changes direction quite dramatically.

37. Ballistic Motions Through Space

38. Analogous Motion (But Beware of Friction!)

39. The Sun in Various Stages

40. Why Should Gravity Affect Light?
Matter distorts space, and light moves through space along the distorted contours.
Because light is moving so fast, it undergoes only tiny changes in direction in everyday circumstances. We don’t notice this!
(Compare lobbing a baseball to home plate and throwing a fastball – or imagine putting a golf ball really fast!)

41. Einstein’s Formulation
The Theory of General Relativity (1915)
“Matter tells space how to curve; the curvature of space tells matter [and light] how to move.”