Download Day 3 Sections S3.1-3 Spacetime, A New View of Gravity

Einstein’s Second Revolution
Movement on the surface of the Earth is
limited to two dimensions:
forward-backward and left-right
(or N-S and E-W)
However, that two-dimensional surface
is curved.
Satellites in space are NOT restricted to
two-dimensions
They can also go up and down
Then why does a satellite end up right
back where it started?
Gravity
It is mysterious:
How can the Earth “pull on” a satellite?
Could “curvature” have something to do with it?
Einstein’s Special Theory of Relativity made us realize that space and time are not separate
things.
Instead we talk about one thing – four-dimensional “spacetime”
And we refer to the “fabric of spacetime” because it can be “warped”.
spacetime
A good analogy is a rubber sheet that is warped by weights placed on it.
The heavier the weight, the more warping.
(This represents two dimensions: one of space and one of time)
YouTube video: Spacetime demonstration
Start at 0:35
End at 2:00
Thought experiment:
Albert says you’re accelerating away from him at 1g.
You say no, it’s him who’s accelerating away from you.
BUT – there is a test you can do to see who is really accelerating: release a ball
This leads to: The Equivalence Principle
The fundamental postulate of General Relativity
=
The effects of a gravitational field are the same as the effects of acceleration
Now compare these two situations:
2
1
Far from any planet or star
(no gravitational field)
In both, you are firing your engines and Al
is not.
In both, you feel weight and Al does not.
In the first you’re accelerating.
At the surface
of a planet
(strong gravity)
In the second, you’re hovering in a
gravitational field.
Let’s try to understand “spacetime” a little better
Space is 3-dimensional. Here’s how you demonstrate those three dimensions:
In that 3dimensional space,
objects can appear
different to
different observers:
But what about “spacetime”? It’s not just where something is located, but when:
A spacetime diagram
for the car
It shows the car’s
“worldline”
Spacetime diagrams show only 1 spatial dimension, but they also show the time dimension.
Spacetime diagrams are drawn from a particular perspective, or reference frame.
What is the shape of the worldline of an object at rest, when time is plotted on the vertical axis?
a) a horizontal line
b) a diagonal line going up to the right
c) a diagonal line going down to the right
d) a vertical line
e) a point
What does it mean for spacetime to be “warped” or “curved”?
What does it mean for spacetime to be “warped” or “curved”?
In grade school you learned the rules of “Euclidean” geometry
Consider what geometry would be like on a sphere
Those “rules” have now changed!
Real-life example: Planes fly the shortest route to save time.
From Chicago to Tokyo, the route should look like this, right?
Real-life example: Planes fly the shortest route to save time.
From Chicago to Tokyo, the route should look like this, right?
NO! It looks like this:
Because the surface of the Earth is a sphere, not a flat plane!
Consider what geometry would be like on a saddle
Those “rules” have changed again!
This is known as “hyperbolic” geometry
Straight worldliness represent objects
that are NOT accelerating
(they are free-floating
or free-falling)
This assumes no masses are present
Straight worldliness represent objects
that are NOT accelerating
(they are free-floating
or free-falling)
This assumes no masses are present
The presence of mass warps spacetime, so freefalling objects now follow curved paths
The geometry of spacetime is altered by mass
Straight worldliness represent objects
that are NOT accelerating
(they are free-floating
or free-falling)
This assumes no masses are present
The presence of mass warps spacetime, so freefalling objects now follow curved paths
The geometry of spacetime is altered by mass
Gravity is no longer a force that one object exerts on another.
We now say that mass warps spacetime, and spacetime dictates how objects move.
In general relativity, the orbital path of a planet
a)
is determined by the curved shape of spacetime that it is moving through.
b) never returns to the same point in four-dimensional spacetime.
c)
is the straightest possible path through four-dimensional spacetime.
d) all of the above
True statement: Light has no mass.
True or False: That means that light cannot be affected by gravity.
A) True
B) False
So far we’ve seen images of the warping of space (rubber sheet analogy).
What does it mean for time to be warped?
Use the Equivalence principle…
On an accelerating spaceship, the clocks at the front of the ship run faster than the clocks
at the rear of the ship
The clocks are synchronized when at rest, and designed to send flashes to each other every 0.01 s.
The observer at the front is accelerating away from the rear clock,
and so the flashes have further to travel.
The result is that the flashes from the rear arrive less frequently.
The observer at the rear is accelerating towards the front clock,
and so the flashes have a shorter distance to travel.
The result is that the flashes from the front arrive more
frequently.
According to the Equivalence Principle, the same thing must happen when at rest in a
gravitational field.
The clock that is “downhill” must run more slowly than the “uphill” clock.
Note that the observers agree on this result. They both say the uphill clock runs faster than the
downhill clock!
This has been observed in many forms, and agrees with the predictions of General Relativity!
Gravitational Redshift
Light from atoms near the surface of a large
mass (like the Sun) is at a longer wavelength
than light from atoms on Earth.
We’ve even checked it directly with atomic clocks separated by several floors:
the lower clock runs slower by an amount predicted by general relativity
If you want to age more slowly, would it be better to live on the top floor of a tall high-rise, or the
ground floor?
A) Top floor
B) Ground floor
Astro-Cash Cab!
Lucas
Aaron
Availya
Tyler
1) The general theory of relativity describes gravity as
a)
a distortion of the shape of four-dimensional spacetime.
b) a distortion of the shape of three-dimensional space.
c)
a force that acts through the action of photons at the speed of light.
d) a force that acts through the action of gravitons that travel instantaneously.
2) The equivalence principle states that
a)
all frames of reference are equivalent.
b) the speed of light will have the same value regardless of who measures it.
c)
the effects of gravity are equivalent to the effects of acceleration.
d) mass and energy are equivalent.
3) Astronaut Bill is hovering 15,000,000 miles above a neutron star while astronaut
Susan is down close the surface making measurements. When Bill compares his clock
to Susan’s, he says her clock is running slower than his.
What does Susan say when she compares her clock to Bill’s?
Bill’s clock is running faster than hers.
Bill’s clock is running slower than hers.
The two clocks are running at the same rate.
4) True / False
Planes always fly straight line routes to their destinations.