Einstein’s Universe

Tuesday, February 5

Einstein – Newton smackdown!Einstein – Newton smackdown!

Two different ways of thinking about gravitygravity and spacespace.

The Way of Newton:The Way of Newton:

Space is staticstatic (not expanding or contracting) and flatflat.

(“FlatFlat” means that all Euclid’s laws of geometry hold true.)

“Objects have a natural tendency to move on straight straight

lineslines at constant speedconstant speed.”

However, we see planets moving on curved orbitscurved orbits

at varying speed.varying speed.

How do you explain thatthat, Mr. Newton?

“There is a force force acting on the planets – the force

called GRAVITYGRAVITY..”

The gravitational force depends on a property that we may call the

“gravitational massgravitational mass”, mg.

2

ggg r

mMGF

Fg = gravitational force Mg = mass of one object mg = mass of other object r = distance between centers of objects G = “universal constant of gravitation” (G = 6.7 × 10-11 Newton meter2 / kg2)

Newton gave another law that gives the acceleration in response to anyany force

(not just gravity)!

The acceleration depends on a property that we may call the

“inertial massinertial mass”, mi.

im / F a If a gravitational force is applied to an object with gravitational massgravitational mass mg and

inertial massinertial mass mi, its acceleration is

i

g2g

mm

rM

Ga

Objects falling side-by-side have the same acceleration (the same mg/mi).

Truly astonishing and fundamental fact of physics:

mg = mi

for every known object!

This equality is known as the “equivalence principleequivalence principle”.

The equivalence principle (which Newton leaves unexplained) led Einstein to devise his theory of

General RelativityGeneral Relativity.

Let’s do a “thought experiment”, of the kind

beloved by Einstein.

Two ways of thinking about a bear:

1) Bear has constant velocity, box is accelerated upward. 2) Box has constant velocity, bear is

accelerated downward by gravity.

Two ways of thinking about lightlight:

1) Light has constant velocity, box is accelerated upward. 2) Box has constant velocity, light is light is accelerated downward by gravityaccelerated downward by gravity.

Einstein’s insight:

Gravity affects the paths of photons, even though they have no mass!

Mass and energy are interchangeable: E = mcE = mc22

NewtonNewton EinsteinEinstein

Mass & energy are very different things.

Mass & energy are interchangeable:

E = mc2

Space & time are very different things.

Space & time are interchangeable:

part of 4-dimensional space-time.

Light takes the shortest distance between two points.

In flat space, the shortest distance between two points is a straight line.

In the presence of gravity, light follows a curved

line.In the presence of gravity,

space is not flat, but curvedcurved!

A third third way of thinking about a bear:

3)3) No forces are acting on the bear; it’s merely following the shortest distance

between two points in space-time.

The Way of Einstein:The Way of Einstein:Mass-energy tells space-time how to

curve; curved space-time tells mass-energy how to move.

The Way of Newton:The Way of Newton:Mass tells gravity how much force to exert; force tells mass how to move.

Objects with lots of mass (& energy) curve space

(& distort time) in their vicinity.

The Big Question:How is the universe curved

on large scales (bigger than stars, bigger than galaxies, bigger than

clusters of galaxies)?

That depends on how mass & energy are distributed on large scales.

The cosmological principle:The cosmological principle:On large scales (bigger than stars, galaxies, clusters of galaxies) the

universe is homogeneous and isotropic.

There are threethree ways in which space can have homogeneous,

isotropic curvature on large scales.

(Apologetic aside: describing the curvature of 3-dimensional space is difficult; I’ll show 2-d analogs.)

(1) This 2-d space is flatflat (or Euclidean):

(2) This 2-d space is positivelypositively curved:

(3) This 2-d space is negativelynegatively curved:

Measuring curvature is easy…in principle.

Flat: angles of triangle add to 180°

Positive: angles add to >180>180°°

Negative: angles add to <180<180°°

Curvature is hard to detect on scales smaller than the radius of curvature.

Flat = good approximation

Flat = bad approximation

Parallax (flat space)

February

August

p

p

d = Sun-star distance

da

a=Sun-Earth distance (1 A.U.)p = a/d

Parallax (positive curvature)

February

August

p

p

p < a/d

Parallax (negative curvature)

February

August

p

p

p > a/d

As d→infinity, p→a/R

Bright idea: The smallest parallax you measure

puts a lower limit on the radius of curvature of negatively curved space.

Hipparcos measured ‘p’ as small as 0.001 arcsec; radius of curvature is at leastat least 1000 parsecs.

We need BiggerBigger triangles to measure the curvature accurately!

L

d

=L/d (flat)

>L/d (positive)

<L/d (negative)

In a negatively curved universe,

The most distant objects we can see are the hot & cold spots on the

Cosmic Microwave Background.

Universe became transparent when it was 350,000 years old.

The largest hot & cold spots are about 700,000 light years across.

Hot & cold spots should be 1° across …if the universe is flat.

But we’ve measured the size of the hot & cold spots, and they are are 1° across!

Consistent with flatflat (Euclidean) space.

What Would Einstein Say?What Would Einstein Say?

Density > critical density: Positive curvature

Density < critical: Negative curvature

Since our universe is close to flat, the density must be close to the

critical density.

Thursday:

Midterm Exam

Bring your calculator and your favorite writing utensil.