PHYS 202

Lecture 20Professor Stephen Thornton

April 11, 2006

Reading Quiz

Black holes

1) are imaginary objects, because they cannot be detected.

2) are objects that absorb all light and then emit light over all wavelengths.

3) can be inferred by the effects they exert on neighboring bodies.

4) are made by the “black death” plague ofthe middle ages.

Answer: 3Black holes absorb all light that reaches them. They do not emit light. They can only be indirectly observed by effects they cause on neighboring bodies – for example, by a neutron star rotating around a black hole.

Last time – started relativityUp to 1895Special relativityTime dilationTwin agingLength contractionRelativistic addition of velocities

Today

Relativistic momentum, mass, energy.Einstein mass-energy relation.AntimatterPETGeneral relativity – bending of lightSpacetime warp, black holes

Conceptual Quiz:Will the stick simultaneously fit into the manhole? Which possibility is correct?

1) Possibility 12) Possibility 23) The stick will not fit into

the hole.

According to Postulate 1 all laws of physics are the same in all inertial frames of reference.

The conservations of linear momentum and energy are still valid.

Momentum will appear different.

2 2***

1 /mvp mvv c

γ= =−

2 2

11 /v c

γ =−

Relativistic Momentum

Remember 1%-5%

There is some controversy as to whether we can write p = mv and consider a relativistic mass m = γ m0. Most physicists, and I am one of them, do not subscribe to the concept that mass increases with velocity.

Mass is an invariant quantity and has the same mass m = m0 irrespective of speed.

We will simply have different equations for momentum and energy. p mvγ=

Energy

0

20

2rest 0 0

2 2 2total rest 0 0 0

If we use calculus, we find

( )

We still use the work-energy theorem to relatekinetic energy and work.

( 1)

Rest energy (or simply )

( 1)

dp dF m vdt dt

K m c

E m c E

E K E m c m c m c

γ

γ

γ γ

= =

= −

≡

= + = − + =

Famous Einstein relation2

rest 0

2total 0

E m c

E m cγ

=

=

Sun

What happens when we squeeze the spring?

The best explanation is to say that the energy increases. E = m0c2 because K = 0. In this case we would say the rest mass m0increases. It does so because of energy, not because it is moving.

We also have a relation between energy and momentum. A careful comparison shows

2 2 2 2 40

0

***Note that for light, 0,and .

E p c m cm

E pc

= +

=

=

Work Problems 29-3629-4829-59

Conceptual QuizThe mass of a cup of steaming Starbucks coffee is

1) larger2) smaller3) the same4) no way to know

than it is when cold.

Answer: 1

Because of the famous Einstein energy relation, E = mc2, the mass of the hot cup of coffee is greater.

Conceptual Quiz:Consider an electron moving at a speed of v = 105 m/s. The rest mass of an electron is m0. Which of the following answers most correctly describes the kinetic energy of the electron to within 1% accuracy?

1) m0v2/22) m0c2(γ - 1) 3) Both equations work. 4) Neither equation works.5) Not enough information is given.

Answer: 3

An electron with speed v = 105 m/s is so much smaller than c that we can use either equation because γ = 1.000006.

Generally v/c ≈ 0.05 before we need to use relativistic results.

Electron–Positron Annihilation

What is antimatter?

Positron emission tomographyA radioactive chemical containing a positron emitter (e+) like 15O, 11C, 13N, 18F is injected into the body to study a particular object.

PET scans

Proton-antiproton interaction

The most efficient form of energy conversion known is for antimatter.

Dan Brown’s Angels and Demons discusses antimatter bomb that will destroy the Vatican. Watch out for the Illuminati!

Top: gravityBottom: acceleration

Person in enclosure cannot tell whether gravity or acceleration is acting on them and the ball.

Principle of Equivalence:All physical experiments conducted in a uniform gravitational field and in an accelerated frame of reference give identical results.

Light pulse in elevator.

As elevator moves up, light pulse goes down.

A Light Experiment in Two Different Frames of Reference

The Principle of Equivalence

Einstein proved that light must be bent in a gravitational field.

General RelativityAlbert Einstein published his special theory of relativity in 1905 while a patent clerk in Bern, Switzerland.

He published his general theory of relativity in 1916. How could it be proven?

Gravitational Bending of LightNormally we cannot see positions of stars near the sun to see if they are apparently moved, but during an eclipse by the moon, we can. Fortunately, there was an eclipse in 1919 in the southern hemisphere.

Bending of Light Near the Sun

Eddington mounted expeditions to Africa and South America in 1919 to attempt to prove whether Einstein was correct. A few months later, Einstein was the world’s best known scientist. It is said that only a handful of physicists actually understand the general theory of relativity.

Einstein cross

Schwarzchild radiusE

E

2

2 escape speed from Earth

Body of mass and radius :

2

2

No light inside this radius can leave the body

e

e

GMvR

M R

GMv cR

GMRc

=

= =

=

BLACK HOLE!!

Black hole evidence

Prediction

Actual Hubble image of M87 galaxy

Warped Space and Black Holes

Light rays are bent near gravity.

Warped space!

Black holes can do this.

A neutron star orbiting a black hole can emit gravity waves.

Laser Interferometer Gravitational Wave ObservatoryHow LIGO Detects a Gravity Wave

LISA – ESA and NASA

Planned for launch in 2011. Three satellites will be in space to detect gravity waves.

Gravity Probe B – launched April 2004Gravity Probe B is a relativity gyroscope experiment to test two extraordinary, unverified predictions of Albert Einstein's general theory of relativity.

1) The curved spacetime, or geodetic, effect- Einstein's theory predicted that the presence of a mass in space, such as the Earth, would warp local spacetime, creating a dip or curve in spacetime.

2) The frame-dragging effect- A few years after Einstein published his theory, physicists Lense and Thirring predicted that the rotation of a mass in space would twist or drag the local spacetime frame around it.

To observe and measure these effects, Gravity Probe B launched four sophisticated gyroscopes into low-Earth polar orbit(400 miles) in April 2004. After calibration, it took data for almost a year until June 2005. No results have yet been reported. Whilein orbit, each gyroscope's spin axis was monitored as it traveled through local spacetime.

Work Problem 29-73

Conceptual Quiz:The theory of relativity limits the maximum speed that an object can have. Is there a similar restriction on the maximum energy or momentum?

1) no for both2) yes for energy, no for momentum3) no for energy, yes for momentum4) yes for both

Answer: 1

There is no restriction on either.

Conceptual Quiz:We observe a clock in a moving rocket runing slow. What happens to the clock if the rocket reverses direction and moves with the same speed.

1) clock still runs slow.2) clock runs fast.3) clock is normal.

Answer: 1Clock still runs slow.