# Day 14

Mechanics

Nov 14, 2013 (4 years and 8 months ago)

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Announcements

Exam 3 is Thursday April 11

Tentatively will cover Chapters 7, 8 & 9

Sample questions have been posted

The main equations of special
relativity

Time Dilation:

Length Contraction:

2
2
1
c
v
L
L
p

2
2
1
rest
t
t
v
c

Proper Time: time measured stationary relative to the “event”

Your watch always tells proper time for you

Proper length: length measured stationary relative to the “event”

A meter stick you carry always measures the proper

length of objects that are stationary with respect to you

Rest mass: mass measured stationary with respect to the object

A scale you carry always measures the rest mass of

objects that are stationary with respect to you

Boost factor (
G
) from the Lorentz transformation.

2
2
1
1
c
v

G
Check out links on Special Relativity at
http://www.apsu.edu/astronomy/cosmology
-
-
and
-
applets

Other Equations of Relativity

Relativistic Momentum

Relativistic Mass

Relativistic Energy

2
mc
E

0
0
2
2
1
m v
p mv m v
v
c
   G

0
0
2
2
1
m
m m
v
c
  G

2
2
0
0
2
2
1
m c
E m c
v
c
  G

2
2
1
1
v
c
G 

The Classic Doppler Effect

The classical Doppler Effect is only radial: only motion away
or towards the observer affect the observed frequency

The Relativistic Doppler Effect

1
1
1
1
1
2
2

z
c
u
c
u
c
u
c
u
obs

The relativistic Doppler Effect includes the stretching
or contracting of waves due to relative motion
and

time dilation due to the relative motion.

Note that the relativistic Doppler Effect occurs even
perpendicular to the direction of motion

Space
-
time
Diagrams

Anything inside
could influence
you (below you)
or you could
influence it
(above you).
Anything outside
cannot affect you

The Space
-
time Interval is
invariant. i.e.: it is the same for all
inertial reference frames

2
2
2
x
t
c
s

Note the minus sign
difference from the
Pythagorean
theorem.
This means

s
2

can be
negative (space
-
like)

Play with Applets on
Special Relativity

The Twin
away if you
consider that the
traveler is in two
different inertial
reference frames

Terence stays at home while Stella
makes a trip to Alpha
Centauri.

Website

The “Special” of Special
Relativity was constant velocity

How do we deal with situations
where there isn’t constant
surface of the Earth where there
general case?

It took Einstein 10 years to figure
out General Relativity

The problem with Newton’s gravity:
the anomalous precession of the
perihelion of Mercury

The observed precession is
5600 arcseconds per
century. 5025 arcseconds is
due to the Precession of the
Equinoxes of Earth, 532 is
due to the effects of Venus,
Earth, Jupiter and all the
other planets on
Mercury.
That leaves 43 arcseconds
per century unaccounted for
by Newtonian
gravitation
and mechanics
.

Newton’s Universal Gravity is a force
law with no propagation speed

r
r
m
m
G
F
ˆ
2
2
1

There is nothing in the force
equation that says how fast it
is transmitted. In fact, it
assumes an instantaneous
transmission of the force.

An instantaneous transmission is in direct violation of
Special Relativity

Special Relativity was for “inertial”
accelerating reference frames?

The Weak Equivalence Principle

Any acceleration is
indistinguishable
from gravity

Is “freefall” an inertial reference frame?

On Earth’s surface we are
not

in
an inertial reference frame

The two “fictitious” forces we feel
are the Coriolis force and the
centrifugal force

The fictitious forces are invoked to
explain the observed accelerations
when we are in a non
-
inertial
reference frame

The Strong Equivalence Principle

All inertial and freely falling
reference frames are equivalent

Recall from Special Relativity: All laws of physics
are the same in all inertial reference frames

A consequence of
the equivalence
principle is the
bending of light by
massive objects

Time is changed by gravity

Gravity slows time. The stronger the gravity, the slower time
flows. Unlike Special Relativity, you can tell whose clock is slow.
The one that is deeper in the gravity well is the slower one.

Gravitational
Redshift is another
consequence of
equivalence

Light loses energy as it “climbs” out of a gravity well. This
results in the light being redshifted by gravity.

General Relativity deals with
geometry

Euclidian Geometry: Flat

Spherical Geometry

Hyperbolic Geometry

The Metric is the formula for the
distance between two points

2
2
2
2
y
h
y
x
gf
x
f
r

2
2
2
2
2
2
)
(cos

R
R
r
Euclidian geometry

Spherical geometry

Hyperbolic geometry

2
2
2
2
1
2
1
2
22
2
2
2
2
1
2
1
2
12
2
2
2
2
1
2
2
2
11
1
1
1
1
1
x
x
x
a
g
x
x
x
x
a
g
x
x
x
a
g

So what kind of geometry do we
use in General Relativity?

Riemannian geometries are locally flat. On a small
enough scale they are Euclidian

The Earth is
a good
example of a
Riemannian
Geometry

The mathematics of
Riemannian geometry