Einstein’s Gravity

Understanding space-time and the gravitational effects of mass

Albert Einstein (1879 - 1955)

One of the iconic figures of the 20th century, Einstein revolutionized our understanding of nearly all aspects of physics.

His impact stretched from the quantum revolution (the physics of the very small) to a new theory of gravity (and the physics of the entire cosmos).

In 1915, Einstein published his Theory of General Relativity in which he described a new way to think about gravity.

Relativity and Space-TimePart of Einstein’s genius was to consider all four dimensions as a whole: space-time.

His theory of Special Relativity (involving high speeds) and General Relativity (involving strong gravity) affect space and time together.

What do we mean by dimensions?

zero one two

three

The Dimension of TimeUnfortunately, no one can point in the direction of time, but Einstein’s Theory of Gravity treats time just like the three dimensions of space.

It is common to visualize space-time by removing one dimension and thinking about space as flat:

Newton vs. EinsteinAccording to Newton, an object travels straight because it experiences no force (such as gravity).

According to Einstein, an object travels straight because it is traveling in “flat” space-time.

Newton vs. EinsteinAccording to Newton, masses attract other masses through the mysterious force of gravity.

Newton’s theory would predict that gravity has no effect on light since light has no mass…

Einstein’s Theory of GravityIn Einstein’s theory, mass curves space-time.

Objects passing near a mass respond by traveling on curved paths through curved space-time.

According to Einstein, gravity is geometry!

Einstein’s Theory of GravityAccording to Einstein,

Mass tells space-time how to curve,

Space-time tells objects how to move.

Newton vs. EinsteinAccording to Einstein, gravity is not a force!

Einstein’s theory predicts that light should bend as it travels through curved space-time, too.

Einstein’s Theory of GravityUnfortunately, the details of Einstein’s theory are very mathematical and beyond the scope of this course (lots of tensor calculus concepts).

However, the results from Einstein’s theory can still be understood with everyday concepts.

For example, consider escape speed of an object:

Escape speed is the minimum upward speed needed for an object never to fall back down…

Escape Speed and CurvatureEscape speed is an indirect measure of curvature of space-time caused by a massive object.

To escape from the Sun’s surface, you would need a speed of more than 600 kilometers per second!

If the Sun shrank to Earth’s size (a “white dwarf”), you would need a speed of 6500 km/s (0.02c).

Escape Speed and Curvature

Why does the escape speed increase?

True, it’s the same amount of mass.

But now you are closer to every piece of mass, so the effect of the mass is magnified.

Now imagine you are 1 AU from the collapsing Sun. What happens to escape speed at this distance?

A. Escape speed increases greatly.

B. Escape speed decreases slightly.

C. Escape speed remains the same.

Escape Speed and CurvatureHas anything changed? Yes, the Sun’s size has.

But the curvature at 1 AU is still the same.

So the “strength of the Sun’s gravity” and the escape speed remains the same at 1 AU, too.

Imagine all the Sun’s mass suddenly collapsed to zero size. What would happen to Earth’s orbit?

A. Earth would fall directly into the Sun since the Sun would curve space-time much more now.

B. Earth would slowly spiral down into the Sun, eventually colliding with the collapsed Sun.

C. Earth would continue to orbit like normal.

What happens to the curvature at the surface of the completely-collapsed Sun?

At the surface, the curvature becomes extreme since a point on the surface is very close to all of the Sun’s mass!

When the escape speed exceeds the speed of light, no matter or light can escape from the object!

It has become a black hole.

Black Holes Don’t Suck!The conclusion: intense curvature (aka gravity) only exists very close to a black hole.

Space-time is still nearly flat very far away.

Because all of the black hole’s mass is so compact, other objects can get very close and experience very curved space-time. How close can you get?

That was one of the first things to be calculated using Einstein’s theory (Karl Schwarzschild, 1916):

1 solar mass equals 3 kilometers radius

Schwarzschild RadiusThis special distance provides a second definition of a black hole:

Any object entirely insideits own Schwarzschild radius is a black hole…

Sun: Schwarzschild radius of 3 kilometers

Earth: Schwarzschild radius of 9 millimeters

Moon: Schwarzschild radius of 0.1 millimeter

Squash the Moon down to the size of a sand grain, and it will become a black hole!

The Event HorizonThis special distance also defines the boundary between the black hole and the Universe outside.

This boundary is known as the event horizon.

You can’t see over the horizon on Earth and you cannot see inside the event horizon of a black hole. Its interior is truly hidden from the Universe!

In addition, anything passing inside the event horizon can never come back out:

A black hole is the ultimate trap…

Black Holes and SingularitiesThe curvature of space-time inside the black hole is so great that no object can resist moving inward toward the center, adding to the black hole’s mass.

Even the original object that collapsed to form the black hole will shrink to zero size: a singularity.

event horizon

Limits of KnowledgeThe trouble is, we cannot observe anything inside the black hole: no information can escape!

What really happens inside a black hole will remain a mystery until someone develops a theory of gravity that includes quantum mechanics.

?Huge mass, strong curvature

High density, small sizes

Falling InWhat are the effects of the intense curvature near a black hole?

Both space and time are affected by strong gravity:

• Gravitational redshift: Radial distances shorter in location of stronger curvature when compared tolocation of lower curvature.

• Gravitational time dilation: Time runs slower in location of stronger curvature when compared to location of lower curvature.

Falling In: Curved SpaceRadial distances near the black hole are shorter than they are farther from the black hole.

One meter here is longer than one meter here!

The true length of a meter can only be measured in flat space-time (no curvature, no mass nearby).

Falling In: Curved TimeTime suffers severe distortion close to the hole too.

While space is compressed close to the horizon, time is stretched.

Clocks run faster here than they do here.

Are Black Holes Real?

What do we know of real black holes?

In other words, how can you measure something you cannot see since it emits no light?

Black holes still have mass and they influence nearby objects with their gravity.

One of the best candidates of a black hole is called Cygnus X-1.

Cygnus X-1This star is the brightest x-ray source in the constellation Cygnus. At the position of the x-ray emission is a 9th magnitude blue star.

The spectrum of the star reveals a blue supergiant (B0I) in a 6 day orbit around an object that does not produce any detectable amount of light.

The supergiant star has a mass of 30-40 solar masses and the unseen companion must have a mass of 25-30 solar masses to explain the orbit.

This is well over the limit for neutron stars, so this unseen object is probably a black hole.

Cygnus X-1Here is an artist’s idea of what the system looks like. The x-ray emission is believed to come from a large, hot accretion disk orbiting the black hole.

Sagittarius A*At the very center of the Milky Way galaxy is a faint radio-emitting object.

Two groups, American and European, have been monitoring the stars near this spot using infrared telescopes (too much dust to see them optically).

They began in the 1990’s and still monitor the motions of these stars today.

The first complete analysis was published in 2008.

Sagittarius A*What both teams measured surprised everyone!

To explain the motions of the stars at the center of our galaxy, there must be a dark object:

• Mass of 4 million solar masses

• Size no bigger than 17 light-hours

There is no other physical explanation of the object than a black hole.

Not Just Black Holes!Any mass will curve space-time!

The strange predictions of Einstein’s theory of gravity have been confirmed in white dwarfs and neutron stars and right here on Earth using precision measurements.

You, in fact, test Einstein’s theory any time you use the global position system (GPS).

Without correcting for subtle effects on distances and times due to Earth’s mass, GPS would be no more accurate than a distance of 100’s meters.