Download Gravity and Relativity: The Interplay of Space, Time, and Matter and more Lab Reports Physics in PDF only on Docsity!
General relativity, I
Einsteinian special relativity has numerous profound consequences—including the
intermingling of space and time, the equivalence of mass and energy, and, combined with
quantum mechanics, the prediction of spin and antimatter. In special relativity, events occur in
the arena of space-time which may be coordinatized differently by different observers, but which
is otherwise immutable. Adding gravity to relativity provides an even more amazing result:
space-time becomes “organic,” taking its form from the matter and energy it contains. This is
Einstein’s general theory of relativity and it has the capacity to tell us about the past and future of
the universe. Embedded in the history of the cosmos are several chapters on the origins of matter.
Thus, relativity + gravity unites the structures of matter on the largest and smallest scales.
Newtonian gravitostatics
To begin the story of matter in the universe, we need to recall a few aspects of the
classical (static) theory of gravity due to Newton. Newton’s theory of gravity states that any two
point-like masses attract one another with a force that is proportional to the product of the masses
and inversely proportional to the square of their distance of separation. Thus, this force of
attraction can be expressed as
F
1 on 2
=! G
m
1
m
2
r
2
r. In this equation, r is the distance between the
masses, m 1
and m 2
, and
r is a unit vector pointing from m 1
to m 2
. The quantity G (the “universal
gravitational constant”) is the intrinsic strength of the gravitational interaction (i.e., independent
of which bodies are interacting). In conventional units G = 6.67x
Nm
2
/kg
2
—that is, a 1 kg
mass separated by 1 m from another 1 kg mass pulls on the second mass with an almost
nonexistent gravitational force of 6.67x
N. Gravity is a puny force. By way of calibration,
two protons repel each other with an electrical force k
E
q
2
/ r
2
, where k E
= 9x
9
Nm
2
/C
2
and q =
1.6x
C, and attract each other with a gravitational force Gm
2
/ r
2
, where m = 1.67x
kg.
The ratio of these forces does not depend on r since both forces! 1 / r
2
; the numerical value of
the ratio F grav
/ F
elec
is 8x
! Thus, compared with electrical interactions, gravity is
phenomenally weak. Wherever electrical interactions are essential—as in atomic, molecular, and
solid-state physics—gravity is always ignorable.
On the other hand, there are about as many electrons as protons in bulk matter and all of
the various electrical pulls and pushes from charges inside a bulk object on charges outside it tend
to cancel out. Such cancellation doesn’t occur for gravity, however. That’s because there is just
one kind of mass (apparently). So gravitational pulls outside a bulk body get increasingly
stronger the bigger the body is. Indeed, planets, stars, galaxies, and the whole universe are held
together by gravity; on such scales it is the electrical force that is insignificant.
The modern view of gravity (as well as all other forces) is that force arises from a field.
In the language of fields, the Newtonian force law is rewritten as
F
1 on 2
= m
2
g
1
, where
g
1
=! G
m
1
r
2
r. Here, the force is parsed into (a) m 1
making a field
g
1
everywhere around it
(though getting weaker the farther away from m 1
one goes) and (b) m 2
“coupling” to
g
1
. The
result is that m 2
feels the force
F
1 on 2
. In this view, m
2
doesn’t care gravitationally about m
1
; it
just cares about the field m 1
makes. There is a reciprocity arising from how the gravitational
force is defined: m
2
g
1
=! m
1
g
2
, that is, m
2
makes a field that m
1
couples to. Note that this
reciprocity is consistent with Newton’s Third Law:
F
1 on 2
F
2 on 1
Spherically symmetric distributions of mass create gravitational fields that, outside of
them, are identical to what a point particle of the same mass would produce if it were located at
the center of the distribution. Thus, the strength of Earth’s gravitational field is g ( r ) = G
M
E
r
2
where M E
is Earth’s mass and r > R
E
(Earth’s radius). At the surface, g ( R
E
) = g
E
is approximately
9.8 N/kg. Because R E
= 6.4x
3
km = 6.4x
6
m, we can “weigh” Earth: M E
= 6x
24
kg.
Example: Often, we are interested in “near-Earth” situations, like satellites in “low-Earth” orbit.
In these cases, distance is often reckoned as an altitude above Earth’s surface: r = R E
R
E
then g ( h ) = G
M
E
R
E
2
! GM
E
R
E
" 2
" 2 R
E
" 3
h
= g
E
( 1 " 2 h / R
E
). The altitude of the Hubble
Telescope, for example, is about 560 km so h / R E
= 560/6.4x
3
= 0.09, and Earth’s g at Hubble is
about 18% less than at the surface.
Though Earth’s gravity is approximately constant everywhere near the surface of Earth,
variations in other bodies’ gravity at Earth turn out to have important, noticeable effects. In
particular, the daily oceanic high and low tides are caused by the gravitational variations of our
moon and Sun.
Example: Our Sun has a mass = 2x
30
kg and a radius = 0.7x
6
km = 0.7x
9
m. Sun’s
gravitational field at it’s surface is therefore g S
= 272 m/s
2
(calculate it), about 28 times greater
than g E
. Earth is about 1.5x
11
m from Sun so g due to Sun at Earth is a factor of about
(0.7x
9
/1.5x
11
2
= 2x
less than at Sun’s surface, or about 6x
g
E
. Thus, if you want to
throw something that will escape Earth and get to Sun you have to throw it pretty hard.
Example: Earth’s moon has a mass = 7.4x
22
kg and is 3.85x
8
m away. So, g of Moon at
Earth is about 3.4x
g
E
(calculate it)—much less than g of Sun.
You’ve probably heard that the ocean tides on Earth are mostly caused by Moon. If
Moon’s pull at Earth is so much less than Sun’s how can that be? The answer has to do with how
rapidly g falls off with r. Imagine that the oceans form a uniform coating over Earth’s surface. If
you are an observer fixed to the center of Earth, you see both Sun and Moon pull the ocean
closest to them (i.e., the “front side”) more than they pull the center and less than they pull the
oceans on the “back side.” The difference between the front side and back side pulls is
g
0
r
2
r! R
E
2
! r
2
r + R
E
2
& 4 g
0
R
E
r , where r is the distance from the pulling body to
the center of Earth and g 0
is the gravitational field of the pulling body at Earth. If we plug in the
appropriate values, we find that this difference is more than twice as big for Moon as for Sun.
Any difference in gravitational fields strengths is called a “tidal field,” and Moon’s tidal field is
bigger than Sun’s.
Note that mass plays three roles in Newtonian gravity. Mass has an active gravitational
role: it makes field. It has a passive gravitational role: it couples to field. And it has the role of
inertia: force equals inertial mass times acceleration. It is not at all obvious that one kind of mass
is responsible for these three effects, but all attempts to experimentally distinguish between
g
inside
( r ) = GM ( r ) r
2
= GMr / R
3
= g
S
r R ,
where g S
is the gravitational field at the surface ( GM / R
2
). In other words, within the spherical
mass, g increases linearly from 0 at the center to a maximum, g S
, at the surface, then falls off as
1/ r
2
outside (i.e., g
outside
( r ) = GM r
2
= g
S
R
2
r
2
). The figure below depicts this behavior. The
difference between inside and outside will arise again when we consider Einstein’s theory of
gravity.
