Einstein & Spacetime!

In this lecture you will see what happens when you look at things moving very fast (at a significant fraction of the speed of light) and you will see that space and time can be radically changed in a very strong gravitational field. In fact, you will find that gravity is not really a force!

This revolutionary leap in our understanding of gravity and the nature of space and time was made by Albert Einstein. In the first two decades of the 20th century, Einstein laid out a new paradigm of gravity and motion in space and time.

Albert EinsteinAlbert Einstein

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Einstein and Relativity 3

Einstein (1879 – 1955) noticed that Newton’s laws of motion are only correct in the limit of low velocities, much less than the speed of light.

→ Theory of Special Relativity

Also, revised understanding of gravity

→ Theory of General Relativity

Einstein stated:the fundamental laws of physics do not depend on your location or motion

Two consequences of Special Relativity are a stationary observer will find (1) the length of a fast-moving object is less than if the object was at rest, and (2) the passage of time on the fast-moving object is slower than if the object was at rest.

However, an observer inside the fast-moving object sees everything inside as their normal length and time passes normally, but all of the lengths in the world outside are shrunk and the outside world's clocks are running slow.

3a

Time and space are relative to the motion of an observer and they are not independent of each other. Time and space are connected to make four-dimensional spacetime (three dimensions for space and one dimension for time).

This is not that strange---we often define distances by the time it takes light to travel between two points. For example, one light year is the distance light will travel in a year. To talk about an event, you will usually tell where (in space) and when (in time) it happened. The event happened in spacetime.

3b

Another consequence of Special Relativity is that nothing can travel faster than the speed of light. Any object with mass moving near the speed of light would experience an increase in its mass.

That mass would approach infinity as it reached light speed andwould, therefore, require an infinite amount of energy to accelerate it to light speed. The fastest possible speed any form of information or force (including gravity) can operate is at the speed of light.

The three strange effects of Special Relativity (shrinking lengths, slowing time, increasing mass) are only noticeable at speeds that are greater than about ten percent of the speed of light. Numerous experiments using very high-speed objects have shown that Special Relativity is correct.

3c

THE SPEED OF LIGHT = 1 Billion km/hr

3dThe speed of light is nature's speed limit!

And speeding tickets will be issued to violators!

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Time is not completely separate from and independent of space as you would ordinarily assume. In his Special Relativity theory, Einstein assumed that the fundamental laws of physics do not depend on your location or motion.

Two people, one in a stationary laboratory and another in a laboratory aboard a train or rocket moving in a straight line atuniform speed, should get the same results in any experiment they conduct. In fact, if the laboratory in the trainor rocket is soundproof and has no windows, there is no experiment a person could conduct that would show he/she is moving.

Special RelativitySpecial Relativity

Special Relativity Special Relativity (2)(2) 5

Einstein found that what you measure for length, time, and mass depends on your motion relative to a chosen frame of reference. Everything is in motion. As you sit in your seat, you are actually in motion around the center of the Earth because of the rapid rotation of the Earth on its axis. The Earth is in motion around the Sun, the Sun is in orbit around the center of our Galaxy, the Galaxy is moving toward a large group of galaxies, etc.

When you say something has a velocity, you are measuring its change of position relative to some reference point which may itself be in motion. All motion is relative to a chosen frame of reference. That is what the word “relativity” means in Einstein's Relativity theories.

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Special Relativity also predicts that matter can be converted into energy and energy in to matter.

By applying Newton's second law of motion to the energy of motion for something moving at high speed (its ``kinetic energy''), you will find that energy = mass × (speed of light)2. More concisely, this is Einstein's famous equation, E = mc2.

The energy ‘E’ of a motionless particle is not zero, rather it’s energy at rest is described by the equation above!

The ‘C’ is the speed of light and the ‘M0’ is the mass of the particle when at rest. This “simple” formula suggests that mass and energy are related and we will see later in the semester how nature can convert one into the other inside stars!

For example, suppose that we convert 1 kg of matter into energy, that would result in the release of an amount of energy equal to a 20-megaton nuclear bomb!

7Einstein deciding on which mathematical terms to use…

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Eureka!Eureka!E = MC2

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Einstein had put forward two postulates that are encapsulated inwhat is called The Special Theory of Relativity.

These two postulates essentially imply that whenever you move relative to someone else you enter into a distinct temporal universe. You will measure time differently from the other person and you will measure lengths differently from the other person.

Things, however, will conspire in such a way that everybody will measure exactly the same value for the speed of light. Things will also conspire to never allow causes to precede effects. Special relativity preserves the causality of all situations.

10Time Dilation

The different temporal universes that the theory of Special Relativity demands for moving observers will have times such that any moving observer's clock always ticks with dilated intervals. Hence a moving clock will always seem to run slow. Your clock, in your rest frame, will tick away, accumulating time much more rapidly than a clock in a fast moving spaceship passing by.

It is for this reason that people connect the time dilation effect with future time travel.

All you have to do is leave home in a very fast spaceship. Travel for a certain distance and then return to your home base. You will then be going into a future.

11In the picture below a rocket ship reference frame travels at 99% of the speed of light relative to some stay-at-home frame. The clock on the moving frame runs slower.

Less time is shown on the moving clock since its tick intervals have grown 'fatter' as it achieved its high speed.

12TWIN PARADOX

The twin paradox arises from the time dilation effect. Say there are twin brothers. One brother stays at home in a slow speed environment. The other brother goes away in an ultra-fast spaceship. The "slow" twin ages considerably (loses his hair). The younger twin returns still young (having all of his hair). This situation is called the twin paradox.

13In the example shown here, the fast twin leaves the Earth at 75% of the speed of light. He leaves as a baby. When he returns he has aged 37 years. The stay-at-home twin has aged 57 years.

Time dilation is a relative effect. Each person sees the other person as moving. Hence each person sees the other person's clock run slow. Each person is legitimately allowed to claim that the other person's clock is the slow clock.

The reason why the fast twin gets younger is that he does something that the stay-at-home doesn't do. To turn around, he has to slow down, turn, and then speed up again to get back to his home. It is this action, that the stay-at-home twin doesn't experience, that forces the time difference between the twins to be non-reciprocal.

Time dilation effect on GPS satellites 14

The time dilation effect is taken into account every day to help keep the atomic clocks on the 24 Global Positioning System (GPS) satellites that encircle the Earth in sync with Earth-based atomic clocks.

Since the satellites are moving relative to the atomic clocks that are on the ground they have dilated time intervals. The Global Positioning System requires that the satellite clocks be coordinated to a very high accuracy.

This is absolutely necessary to predict where the location is of people using the GPS receivers on the Earth's surface. Even though the special relativity effect is very very small, the atomic clocks are accurate enough to be affected by the time dilation effect.

General Relativity 15

A new description of gravity

Postulate:

Equivalence Principle:

“Observers can not distinguish locally

between inertial forces due to acceleration and

uniform gravitational forces due to the

presence of massive bodies.”

Part of Einstein's genius was his ability to look at ordinary things from a whole new perspective and logically follow through on the consequence of the insights he gained from his new perspective.

Equivalence (1)(1)

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He proposed an experiment involving two elevators: one at rest on the ground on the Earth and another, far out in space away from any planet, moon, or star, accelerating upward with an acceleration equal to that of one Earth gravity (9.8 meters/second2). (Modern readers can substitute ``rocket ship'' for Einstein's elevator.)

17If a ball is dropped in the elevator at rest on the Earth, it will accelerate toward the floor with an acceleration of 9.8 meters/second2. A ball released in the upward accelerating elevator far out in space will also accelerate toward the floor at 9.8 meters/second2. The two elevator experiments get the same result!

Oh no!I’m in

trouble!

18Equivalence Equivalence (2)(2)

Einstein used this to formulate the equivalence principle that would be the foundation of General Relativity.

It states that ``there is no experiment a person could conduct in a small volume of space that would distinguish between a gravitational field and an equivalent uniform acceleration''.

A consequence of this is that if an elevator is falling freely toward the ground because of gravity, an occupant inside will feel weightless just as if the elevator was far away from any planet, moon, or star. No experiment would help you distinguish between being weightless far out in space and being in free-fall in a gravitational field.

Gravity and Acceleration

According to Einstein, gravity is either an acceleration or a distortion in space and time.

When accelerating, you feel an inertial force.

When on the surface of a planet, you feel gravitational forces.

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20Curvature in Spacetime

Gravity according to general relativity:

“Mass tells spacetime how to curve, and the curvature of spacetime (gravity) tells mass how to accelerate”.

(or more simply put…A gravitational field is a curvature of spacetime caused by the presence of a mass).

20aCurvature in Spacetime

Matter distorts spacetime :: 21

• Matter distorts spacetime like weights on a taut rubber sheet.• The greater the mass, the greater the distortion of spacetime.

Mass and Spacetime 22

• According to Newton, all bodies with mass exert a gravitational force on each other.• even Newton had problems accepting this concept of “action at a distance”

• General relativity removes this concept.• mass causes spacetime to curve• the greater the mass, the greater the distortion of spacetime• curvature of spacetime determines the paths of freely moving objects

• Orbits can now be explained in a new way.• an object will travel on as

straight a path as possible through spacetime

The Strength of Gravity 23

• The more that spacetime curves, the stronger gravity becomes.• Two basic ways to increase gravity/curvature of spacetime:

• increased mass results in greater curvature at distances away from it• curvature is greater near the object’s surface for denser objects

• for objects of a given mass, this implies smaller objects

• All three objects impose the same curvature at a distance.

• White dwarf imposes steeper curvature at Sun’s former position.

• Black Hole punches a hole in the fabric of spacetime.

• Nothing can escape from within the ‘event horizon’ of a Black Hole.

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• Perihelion advance(in particular, of Mercury’s orbit)

Other Effects of General Relativity 25

26Precession of Mercury’s Orbit

• Newton’s law predicted that the orbit of Mercury should change.• due to gravitational influence of the planets• this change was measured in the 1800s• but Newton’s law could not account for the

exact change which was observed• the discrepancy between observation and

theoretical prediction was real

• Einstein knew of this discrepancy and used general relativity toexplain it.• Newton’s law assumed that time was absolute & space was flat• but when Mercury is closest to the Sun, time runs more slowly & space is

more curved• Predictions of general relativity matched the observations exactly!

Solar Eclipse Experiment 27

The bending of light by the gravitation of massive bodies has indeed been observed:

During total solar eclipses:

The positions of stars apparently close to the sun are shifted away from the position of the sun.

→ New description of gravity as curvature of spacetime!

28Bending of Light under Gravity

• Light will always travel at a constant velocity.• therefore, it will follow the straightest possible path through spacetime• if spacetime is curved near a massive object, so will the trajectory of light

29diagram:

Gravitational lensing:30

Lensing Basics 31

A Normal Lens

Bending of light

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Gravitational lensing:

Gravitational Lensing 33

Double image

Einstein Ring

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Lensing of a Castle

The Smithsonian on the Mall

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Lensing of a Castle

And we place a Black Hole with the mass ofSaturn in the middle of the Mall

Gravitational Lensing 36

• Today, more examples of gravitational lensing have been seen.

An Einstein Ringgalaxy directly behind a galaxy

An Einstein Cross

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The various types of gravitational lenses usually involve light paths from quasars & galaxies being bent by intervening galaxies & clusters.

38Science Fact or Fiction?• Do the theories of relativity prohibit interstellar travel?

• we can not travel faster than the speed of light (c)!• but what if we made the distance to our destination shorter?

• We might tunnel through hyperspace in a wormhole.

• A wormhole connects two distant points in the Universe.

• Or perhaps we could warp spacetime so that two locations of our choosing could touch momentarily.

• None of these ideas is prohibited by our current understanding of physics.• Most scientists are pessimistic about the possibilities.

• wormholes would also make time travel possible, with its severe paradoxes• For the moment, the Universe is safe for science fiction writers!

39Wormholes and Time Machines

• Einstein discovered that GTR predicts the possibility that black holes could connect our universe to another “parallel” universe via an Einstein-Rosen bridge.

• Such a bridge is called a wormhole.

• A wormhole also could be connected to another part of our own universe.

• However, theories suggest that these wormholes collapse as soon as they are formed.

Wormholes and Time Machines 40

• Also, since traversing a wormhole means that you are emerging at a different spacetime domain than the one you started with, you could start at present time and emerge at a time in the past (or the future) - time travel!

What have we learned? 41

• What is the primary topic of the general theory of relativity?• The general theory of relativity is primarily a theory of

gravity, stating that the force of gravity arises from distortions of spacetime.

• What is spacetime?• Spacetime is the four-dimensional combination of space and

time that forms the “fabric” of our universe.

42What have we learned?

• What is the equivalence principle?• The effects of gravity are exactly equivalent to the effects of

acceleration. • What do we mean by dimensions?

• Each dimension represents an independent direction of possible motion. In three-dimensional space, the three dimensions of length, width, height are perpendicular to one another. A four-dimensional space has a fourth dimension perpendicular to all three of the others. We cannot visualize this, but it can still exist.

• Does spacetime differ for different observers?• No. Both time and space differ for observers in different

reference frames, but the combination of spacetime is the same for everyone.

What have we learned? 43

• How does mass affect spacetime?• Mass causes spacetime to curve, and the curvature of spacetime

determines the paths of freely moving masses.• How would an ordinary star, a white dwarf, and a black

hole of the same mass differ in spacetime?• Because all three objects have the same mass, far from their surfaces

they would affect spacetime in precisely the same way. Up close, however, the white dwarf would distort spacetime much more than the ordinary star, and the black hole would distort spacetime somuch that it essentially would form a bottomless pit—a true hole in the universe.

• According to general relativity, how does gravity affect time?• Time runs slower in places where gravity is stronger.

44What have we learned?

• How have experiments and observations verified the predictions of the general theory of relativity?• Observations of the change of Mercury’s orbit match that

predicted by Einstein’s theory. Observations of stars during eclipses and photos of gravitational lensing provide spectacularconfirmation of the idea that light can travel curved paths through space. Gravitational redshifts observed in the light of objectswith strong gravity confirm the slowing of time predicted by general relativity.

New Termsnatural motionviolent motionacceleration of gravitymomentummassaccelerationvelocityinverse square lawfieldcircular velocitygeosynchronous satellitecenter of massclosed orbitescape velocityopen orbitangular momentumenergyjoule (J)

special relativitygeneral theory of relativity

End of lecture unit!