The Age of Einstein by Frank W. K. Firk - HTML preview

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fundamental relation using purely theoretical arguments, long before experiments were carried

out to verify its universal validity. The heat that we receive from the Sun originates in the

conversion of its central, highly compressed mass into radiant energy. A stretched spring has

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more mass than an unstretched spring, and a charged car battery has more mass than an

uncharged battery! In both cases, the potential energy stored in the systems has an equivalent

mass. We do not experience these effects because the mass changes are immeasurably small,

due to the 1/c2 factor. However, in nuclear reactions that take place in nuclear reactors, or in

nuclear bombs, the mass (energy) differences are enormous, and certainly have observable

effects.

7. AN INTRODUCTION TO EINSTEINIAN GRAVITATION

7.1 The principle of equivalence

The term “mass” that appears in Newton’s equation for the gravitational force between

two interacting masses refers to

“gravitational mass”; Newton’s law should indicate this property of matter

F = GMGmG/r2, where MG and mG are the gravitational masses of the

G

interacting objects, separated by a distance r.

The term “mass” that appears in Newton’s equation of motion, F = ma, refers to

the “inertial mass”; Newton’s equation of motion should indicate this property of matter:

F = mIa, where mI is the inertial mass of the particle moving with an

acceleration a(r) in the gravitational field of the mass MG.

Newton showed by experiment that the inertial mass of an object is equal to its

gravitational mass, mI = mG to an accuracy of 1 part in 103. Recent experiments have shown

this equality to be true to an accuracy of 1 part in 1012. Newton therefore took the equations

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F = GMGmG/r2 = mIa

and used the condition mG = mI to obtain

a = GMG/r2.

Galileo had previously shown that objects made from different materials fall with the

same acceleration in the gravitational field at the surface of the Earth, a result that implies mG ∝

mI. This is the Newtonian Principle of Equivalence.

Einstein used this Principle as a basis for a new Theory of Gravitation. He extended the

axioms of Special Relativity, that apply to field-free frames, to frames of reference in “free

fall”. A freely falling frame must be in a state of unpowered motion in a uniform gravitational

field . The field region must be sufficiently small for there to be no measurable gradient in the

field throughout the region. The results of all experiments carried out in ideal freely falling

frames are therefore fully consistent with Special Relativity. All freely-falling observers

measure the speed of light to be c, its constant free-space value. It is not possible to carry out

experiments in ideal freely-falling frames that permit a distinction to be made between the

acceleration of local, freely-falling objects, and their motion in an equivalent external

gravitational field. As an immediate consequence of the extended Principle of Equivalence,

Einstein showed that a beam of light would be deflected from its straight path in a close

encounter with a sufficiently massive object. The observers would, themselves, be far

removed from the gravitational field of the massive object causing the deflection.

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Einstein’s original calculation of the deflection of light from a distant star, grazing the

Sun, as observed here on the Earth, included only those changes in time intervals that he had

predicted would occur in the near field of the Sun. His result turned out to be in error by

exactly a factor of two. He later obtained the “correct” value for the deflection by including in

the calculation the changes in spatial intervals caused by the gravitational field.

7.2 Rates of clocks in a gravitational field

Let a rocket be moving with constant acceleration a, in a frame of reference, F, far removed

from the Earth’s gravitational field, and let the rocket be instantaneously at rest in F at time t =

0. Suppose that two similar clocks, 1 and 2, are attached to the rocket with 1 at the rear end and

2 at the nose of the rocket. The clocks are separated by a distance . We can choose two light

sources, each with well-defined frequency, f , as suitable clocks. f is the frequency when the

0

0

rocket is at rest in an inertial frame in free space.

F (an inertial frame, no gravitational field)

y 2 constant acceleration, a, relative to F

Clocks at rest in rocket



1 Pulse of light emitted from 1 at t = 0

acceleration begins at t ≥ 0

x

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Let a pulse of light be emitted from the lower clock, 1, at time t = 0, when the rocket is

instantaneously at rest in F. This pulse reaches clock 2 after an interval of time t, (measured in

F) given by the standard equation for the distance traveled in time t:

ct = ( + (1/2)at2),

where (1/2)at2 is the extra distance that clock 2 moves in the interval t.

Therefore,

t = (/c) + (a/2c)t2 ,

≈ (/c) if (at/2) << c

At time t, clock 2 moves with velocity equal to v = at ≈ a/c, in F.

An observer at the position of clock 2 will conclude that the pulse of light coming from clock 1

had been emitted by a source moving downward with velocity v. The light is therefore

“Doppler-shifted”, the frequency is given by the standard expression for the Doppler shift at

low speeds (v << c):

f′ ≈ f [1

0

− (v/c)]

= f [1

0

− (a/c2].

The frequency f′ is therefore less than the frequency f . The light from clock 1 (below) is “red-

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shifted”. Conversely, light from the upper clock traveling down to the lower clock is measured

to have a higher frequency than the local clock 1; it is “blue-shifted”.

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The principle of equivalence states that the above situation, in a closed system, cannot

be distinguished by physical measurements, from that in which the rocket is at rest in a uniform

gravitational field. The field must produce an acceleration of magnitude |a|, on all masses

placed in it.

y′ G is a non-accelerating frame with a uniform gravitational field present

G 2

Blue shift

Red shift Rocket at rest in G

1 Gravitational field

Massive body

x′

The light from the lower clock, reaching the upper clock will have a frequency lower than the

local clock, 2, by f g/c2, (replacing |a| by |g|), where g

0

≈ 10 m/s2, the acceleration due to gravity

near the Earth. The light sources are at rest in G, and no oscillations of the pulses of light are

lost during transmission; we therefore conclude that, in a uniform gravitational field, the actual

frequencies of the stationary clocks differ by f g/c2. Now, g is the difference in the

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“gravitational potential” between the two clocks. It is the convention to say that the upper

clock, 2, is at the higher potential in G. (Work must be done to lift the mass of clock 1 to clock

2 against the field).

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Consider the case in which a light source of frequency f (corresponding to clock 1) is

S

situated on the surface of a star, and consider a similar light source on the Earth with a

frequency f (corresponding to clock 2). Generalizing the above discussion to the case when

E

the two clocks are in varying gravitational fields, such that the difference in their potentials is

Δφ, we find

f = f (1 +

E

S

Δφ/c2)

(g = Δφ, is the difference in gravitational potential of the clocks in

a constant field, g, when separated by ).

For a star that is much more massive than the Earth, Δφ is positive, therefore, f > f , or in terms

E

S

of wavelengths, λ and , > . This means that the light coming from the distant star is

E

λS λS λE

red-shifted compared with the light from a similar light source, at rest on the surface of the

Earth.

As another example, radioactive atoms with a well-defined “half-life” should decay

faster near clock 2 ( the upper clock) than near clock 1. At the higher altitude (higher potential),

all physical processes go faster, and the frequency of light from above is higher than the

frequency of light from an identical clock below. Einstein’s prediction was verified in a series

of accurate experiments, carried out in the late 1950’s, using radioactive sources that were

placed at different heights near the surface of the Earth.

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7.3 Gravity and photons

Throughout the 19th-century, the study of optical phenomena, such as the diffraction of

light by an object, demonstrated conclusively that light (electromagnetic in origin) behaves as a

wave. In 1900, Max Planck, analyzed the results of experimental studies of the characteristic

spectrum of electromagnetic radiation emerging from a hole in a heated cavity (so-called

“black-body radiation”). He found that current theory, that involved continuous frequencies in

the spectrum, could not explain the results. He did find that the main features of all black-body

spectra could be explained by making the radical assumption that the radiation consists of

discrete pulses of energy E proportional to the frequency, f. By fitting the data, he determined

the constant of proportionality, now called Planck’s constant; it is always written h. The

present value is:

h = 6.626 × 10−34 Joule-second in MKS units.

Planck’s great discovery was the beginning of Quantum Physics.

In 1905, Einstein was the first to apply Planck’s new idea to another branch of Physics,

namely, the Photoelectric Effect. Again, current theories could not explain the results. Einstein

argued that discrete pulses of electromagnetic energy behave like localized particles, carrying

energy E = hf and momentum p = E/c. These particles interact with tiny electrons in the

surface of metals, and eject electrons in a Newtonian-like way. He wrote

E = hf and E = p c

PH

PH

PH

PH

The rest mass of the photon is zero. (Its energy is all kinetic).

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If, under certain circumstances, photons behave like particles, we are led to ask: are

photons affected by gravity? We have

E = m I c2 = hf ,

PH

PH

PH

or

m I = E /c2 = hf /c2.

PH

PH

PH

By the Principle of Equivalence, inertial mass is equivalent to gravitational mass, therefore

Einstein proposed that a beam of light (photons) should be deflected in a gravitational field, just

as if it were a beam of particles. (It is worth noting that Newton considered light to consist of

particles; he did not discuss the properties of his particles. In the early 1800’s, Soldner actually

calculated the deflection of a beam of “light-particles” in the presence of a massive object!

Einstein was not aware of this earlier work).

Let us consider a photon of initial frequency f , emitted by a massive star of mass M ,

S

S

and radius R. The gravitational potential energy, V, of a mass m at the surface of the star, is

given by a standard result of Newton’s Theory of Gravitation; it is

V(surface) = − GM m/R.

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It is inversely proportional to the radius of the star. The negative sign signifies that the

gravitational interaction between M and m is always attractive.

S

Following Einstein, we can write the potential energy of the photon of “mass” hf /c2 at

PH

the surface as

V (surface) = − (GM /R)(hf /c2).

S

PH

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The total energy of the photon, E

is the sum of its kinetic and potential energy:

TOTAL

E

= hf STAR + (

hf STAR /Rc2 ,

TOTAL

PH

−) GMS PH

= hf STAR (1

/Rc2).

PH

− GMS

At very large distances from the star, at the Earth, for example, the photon is essentially beyond

the gravitational pull of the star. Its total energy remains unchanged (conservation of energy).

At the surface of the Earth the photon has an energy that is entirely electromagnetic (since its

potential energy in the “weak” field of the Earth is negligible compared with that in the

gravitational field of the star), therefore

hf EARTH = hf STAR (1

/Rc2)

PH

PH

− GMS

so that

f EARTH/f STAR = 1

/Rc2,

PH

PH

− GMS

and

Δf/f ≡ (f STAR

EARTH) /f STAR = GM /Rc2 .

PH

− fPH

PH

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We see that the photon on reaching the Earth has less total energy than it had on leaving

the star. It therefore has a lower frequency at the Earth. If the photon is in the optical region, it

is shifted towards the red-end of the spectrum. This is the gravitational red-shift. (It is quite

different from the red-shift associated with Special Relativity)

Schematically, we have:

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f STAR To Earth f EARTH

PH

PH

Radius R

Mass M Blue light emitted Light red-shifted

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Massive Star far from field of star

7.4 Black holes

In 1784, a remarkable paper was published in the Philosophical Transactions of the

Royal Society of London, written by the Rev. J. Michell. It contained the following discussion:

To escape to an infinite distance from the surface of a star of mass M and radius R, an

object of mass m must have an initial velocity v given by the energy condition:

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initial kinetic energy of mass ≥ potential energy at surface of star,

or

(1/2)mv 2

0 ≥ GMm/R (A Newtonian expression).

This means that

v 0≥ √(2GM/R).

Escape is possible only when the initial velocity is greater than (2GM/R)1/2 .

On the Earth, v 0≥ 25,000 miles/hour.

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For a star of given mass M, the escape velocity increases as its radius decreases. Michell

considered the case in which the escape velocity v reaches a value c, the speed of light. In this

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limit, the radius becomes

R

= 2GM/c2

LIMIT

He argued that light would not be able to escape from a compact star of mass M with a radius

less than R

; the star would become invisible. In modern terminology, it is a black hole.

LIMIT

Using the language of Einstein, we would say that the curvature of space-time in the

immediate vicinity of the compact star is so severe that the time taken for light to emerge from

the star becomes infinite. The radius 2GM/c2 is known as the Schwarzschild radius; he was

the first to obtain a particular solution of the Einstein equations of General Relativity. The

analysis given by Michell, centuries ago, was necessarily limited by the theoretical knowledge

of his day. For example, his use of a non-relativistic expression for the kinetic energy (mv2/2)

is now known to require modification when dealing with objects that move at speeds close to

c. Nonetheless, he obtained an answer that turned out to be essentially correct. His use of a

theoretical argument based on the conservation of energy was not a standard procedure in

Physics until much later.

A star that is 1.4 times more massive than our Sun, has a Schwarzschild radius of only

2km and a density of 1020 kg/m3. This is far greater than the density of an atomic nucleus. For

more compact stars (R

< 1.4 M ), the gravitational self-attraction leads inevitably to its

LIMIT

SUN

collapse to a “point”.

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Studies of the X-ray source Cygnus X-1 indicate that it is a member of a binary system,

the other member being a massive “blue supergiant”. There is evidence for the flow of matter

from the massive optical star to the X-ray source, with an accretion disc around the center of

the X-ray source. The X-rays could not be coming from the blue supergiant because it is too

cold. Models of this system, coupled with on-going observations, are consistent with the

conjecture that a black hole is at the center of Cygnus X-1. Several other good candidates for

black holes have been observed in recent studies of binary systems. The detection of X-rays

from distant objects has become possible only with the advent of satellite-borne equipment.

I have discussed some of the great contributions made by Einstein to our understanding

of the fundamental processes that govern the workings of our world, and the universe, beyond.

He was a true genius, he was a visionary, and he was a man of peace.

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Appendix

The following material presents the main ideas of Einstein’s Special Relativity in a mathematical

form. It is written for those with a flair for Mathematics.

A1. Some useful mathematics: transformations and matrices

Let a point P[x, y] in a Cartesian frame be rotated about the origin through an angle of

90°; let the new position be labeled P′[x′, y′]

+y

P′[x′, y′]

P[x, y]

-x +x

-y

We see that the new coordinates are related to the old coordinates as follows:

x′ (new) = −y (old)

and

y′ (new) = +x (old)

where we have written the x’s and y’s in different columns for reasons that will become clear,

later.

Consider a stretching of the material of the plane such that all x-values are doubled and all y-

values are tripled:

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3y P′[x′, y′] = P′[2x, 3y]

y P[x, y]

x 2x

The old coordinates are related to the new coordinates by the equations

y′ = 3y

and

x′ = 2x

Consider a more complicated transformation in which the new values are combinations of the

old values, for example, let

x′ = 1x + 3y

and

y′ = 3x + 1y

We can see what this transformation does by putting in a few definite values for the

coordinates:

[0, 0] → [0, 0]

[1, 0] → [1.1 + 3.0, 3.1 + 1.0] = [1, 3]

[2, 0] → [1.2 + 3.0, 3.2 + 1.0] = [2, 6]

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[0, 1] → [1.0 + 3.1, 3.0 + 1.1] = [3, 1]

[0, 2] → [1.0 + 3.2, 3.0 + 1.2] = [6, 2]

[1, 1] → [1.1 + 3.1, 3.1 + 1.1] = [4, 4}

[1, 2] → [1.1 + 3.2, 3.1 + 1.2] = [7, 5]

[2, 2] → [1.2 + 3.2, 3.2 + 1.2] = [8, 8]

[2, 1] → [1.2 + 3.1, 3.2 + 1.1] = [5, 7]

and so on.

Some of these changes are shown below

y

y′

New axes and grid-lines are oblique

x′