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Over the past two weeks,

we’ve seen how Special

Relativity introduces some

very interesting twists on

our conventional view of

time – from the relative

nature of simultaneity to the

distorted passage of time on

rapidly moving spacecraft.

But we’ve noted already in this class that space and time are

deeply intertwined with each other by our motions through the

universe – motions that are largely driven by gravitational forces.

In 1915, Einstein revolutionized our view of Newtonian gravity

and space with his theory of General Relativity, just as Special

Relativity had changed our view of Newtonian ‘universal’ time.

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A key concept from the

theory of General Relativity is

the “Equivalence Principle”,

which can be envisioned in

the following way: imagine

you are in a rocket ship freely

floating in deep space, far

away from the gravitational

influences of other objects.

Under these conditions, you

would feel ‘weightless’.

However, if you accelerate

your ship by firing its rockets,

the back of the ship will come

up to meet your feet and press

against them, and you will

feel that force. Further, if you

accelerate at precisely

9.8m/s2, the force of the ship

on you will be exactly the

same as the force of Earth’s

surface gravity, and it will

feel exactly as though you

were standing on the Earth.

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According to General Relativity, these two situations – falling

under the influence of gravity, and accelerating due to other forces

are completely equivalent for the observers inside these spaceships.

Earth

Now let’s see this in action – suppose your rocket ship has a laser

pointer on one side, and you fire it horizontally across the room

while the rocket is accelerating. Because it takes some time for

light to cross the distance, during which the ship is moving, the

beam strikes the wall on the other side at a lower point.

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Earth

The Equivalence Principle says that the curved path of the photons in

the accelerating rockets must also be seen if the rocket is just sitting

in a gravitational field on Earth – so light itself is affected by gravity!

…heavy…

But light has no mass, so how

could gravity affect it?

Einstein’s General Relativity

suggests that gravity is really

a distortion of spacetime,

caused by objects with mass.

Because light – like

everything else in our

universe – moves through

spacetime, this means that

light is affected by gravity –

even though photons

themselves have no mass.

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According to relativity, gravity warps spacetime in a manner analogous to way a person who stands on a trampoline distorts the surface of the trampoline. The person makes a “pit” on the rubber sheet that affects the motion of other objects on the trampoline.

Near a massive object, even the path of light – forced to move on

the surface of the spacetime “trampoline” – will be curved, and

can be made quite distorted if the curvature is sufficiently steep.

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More massive objects make deeper, more severe distortions in

spacetime – and we often refer to these distortions as

‘gravitational wells’. The Earth’s gravitational well is much

deeper than the Moon’s, but much less deep than that of the Sun.

Large masses like the Sun can bend light powerfully and affect

what you see like a lens does – except this is a gravitational “lens”.

In fact, these sorts of shifts in the positions of background stars,

seen first in the solar eclipse of 1919, were among the earliest

bits of observational evidence supporting Einstein’s theories.

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Now, given that General

Relativity predicts warping of

spacetime, it should be no

surprise that it predicts some

strange things about time –

so-called Gravitational Time

Dilation. To picture this

simply, imagine you are

falling in your spaceship

under the influence of Earth’s

gravity and you pass two

clocks that are stationary with

respect to the Earth.

As you pass the first clock,

it will seem to be moving

with respect to you, and as

you know from our work

with Special Relativity,

clock 1 will appear to be

moving slow compared to

your clock on the ship –

and of course, an observer

‘stationed’ on the Earth

clock would say it was you

moving, and your clocks

that were running behind.

Clock 1

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However, because of the

acceleration due to gravity,

as you pass clock 2 you

will be moving even faster,

and so clock 2 will appear

to move more quickly past

you and its clocks will

seem to be running even

slower still! And of course

an observer at that clock

would say the same about

your clocks!

Clock 2

This agreement between observers is key, and makes it

clear that gravitational fields affect time as well as space in

a self-consistent way – the deeper you go into a

‘gravitational well’, the slower time passes for you.

Clocks run

slower

here…

…and

faster here.

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This effect is very real and

practically important –

because of their orbits far

above the Earth’s surface,

for example, the clocks of

GPS satellites lose almost

38 microseconds a day

compared to Earth clocks.

That may not sound like

much – but if unaccounted

for, the location services

on your phone would be

off by more than 6 miles!

But as we’ve mentioned, the gravitational well of the Earth is

really quite shallow compared to stars – to see truly

impressive spacetime distortions, we need to turn to the

ultimate gravitational pits in nature, Black Holes.

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These objects are believed to form when very high-mass stars run

out of materials to fuse in their core, and can no longer support

themselves against gravity. Their mass is compressed to an

infinitely small point, with an infinitely deep gravitational well.

We call such an object a black hole, because its gravitational

strength is enough to prevent even light from escaping.

Animation of the formation

of a black hole during a

“High-Mass Star Supernova”.

Animation of an alternate formation model, in which the supernova doesn’t

“blow away” most of the star’s mass – the black hole forms inside the star,

which quickly collapses into it. These “collapsars” may be responsible for

Gamma-Ray bursts, the brightest explosive events we currently know of!

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The “event horizon” of a

black hole is the location

where the escape velocity

from the black hole is

equal to the speed of light

– so no ‘events’ from

inside that boundary can

be seen by outside

observers. All of the mass,

however, is located at the

singularity at the black

hole’s center – there is no

mass or other material at

the event horizon itself.

© 2004 Pearson

Education Inc.,

publishing as Addison-

Wesley

The geometry is a bit

more complicated for

the interior of a

charged or rotating

black hole – including

multiple ‘event

horizons’ and a ring-

shaped singularity.

Baffling and difficult

to observationally

constrain? You bet!

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A ‘relatively’ realistic model of an approach to and through a

charged black hole’s interior, courtesy A. Hamilton (UC Boulder)

Near a black hole, time is greatly affected by the warping of

spacetime. If we launched a probe into a black hole, as it

approached the event horizon time would appear to slow for

the probe, as seen from an outside observer – e.g., it might take

50 min of time on the mother ship for 15 min to elapse on a

probe closer to the black hole.

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But note what this means – if the probe then returns to the

mother ship, the mother ship will have experienced more

time passing than the probe, which is another way of

saying that the probe traveled into the future of the mother

ship! Time travel using black holes is totally doable, you

just need to get really close to the event horizon!

Unfortunately that’s more than

a bit tricky – near the event

horizon, the difference between

the gravitational force felt by

an astronaut’s feet becomes

significantly stronger than that

felt by her head! Radial

differences in the gravitational

field are even worse, and

squeeze the astronaut (or

probe) from the sides, so that

any physical object would be

“spaghettified”! Ouch!

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But that’s not necessarily

the end of the story – it

turns out to be the case

that the more massive a

black hole is, the less its

tidal forces at its event

horizon. For the most

massive of black holes, the

so-called supermassive

black holes that live in the

hearts of galaxies, these

forces are completely

manageable!

So could such massive black holes be used to travel through

time and space, in the manner suggested by the recent

Interstellar movie? That is very, very unclear – naturally

occurring black holes are almost certainly one-way streets to a

painful death as you encounter the singularity at their centers.

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However, General Relativity also predicts the possibility of

‘wormholes’ – phenomena similar to black holes in some ways,

but which specifically connect two different regions of spacetime.

However, the “natural” wormholes predicted by physics – if they

exist in nature at all – are extremely small and short-lived, and

therefore not so effective at moving spaceships around!

It may be possible to create larger wormholes, but the ends of

such space-time portals would necessarily be physically close

to each other when they’re formed. You would then have to

physically drag one of the two endpoints to where you wanted

the ‘exit’ to be – meaning you have to fix the long-distance

travel part before you use the wormhole! No shortcuts here!

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(Unless, of course, you find one already laid here by

some other aliens… Let me know if you find one!)

Such wormholes also suggest at least the possibility of

traveling backwards in time – which is something that

suggests so many bizarre possibilities that most

physicists believe it to be fundamentally impossible,

even if nothing at this point clearly rules it out!

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Because they link the very

large with the very small,

resolving these questions

about the nature of black

holes, wormholes, and

spacetime travel using them

requires us to understand

how gravity operates on the

very smallest of scales – and

that’s where will turn our

attention next time as we

start to explore time on the

quantum scale!