equivalence principle, review
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Transcript equivalence principle, review
PH 301
Dr. Cecilia Vogel
Lecture 6
Review
Lorenz transformation
velocity transformation
Outline
relativistic momentum & energy
mass energy
Relativistic Momentum
p
mv
v
1
c
2
m (sometimes mo) is rest mass
measured when object is at rest
v is object’s velocity
p is the object’s momentum.
This quantity is conserved in all collisions,
reactions, etc
F=ma?
What is mass?
Mass tells how hard object is to accelerate.
F = ma is a classical equation
Generally F = dp/dt
is equivalent to F=ma,
if p=mv, with m constant, and a = dv/dt
Classically dp/dt=m dv/dt – requires the
same force, for the same rate of change of v
Not at high speed!
F ≠ ma
p
mv
v
1
c
2
mv
As speed changes, numerator and
denominator both change
dp/dt ≠ m dv/dt
As v gets close to c, object gets harder
and harder to accelerate
What is mass?
Rest mass (m) is mass measured
when at rest
property of the object
what you look up in text book
Relativistic mass (mrel) shows
how hard it is to accelerate the
object
increases with speed
p = mrelv
if use relativistic mass
mrel
m
v
1
c
2
Energy
Momentum increases with
speed, so does energy.
E = mrelc2
E
mc
2
v
1
c
When v= 0
Energy is not zero
rest energy = mc2
2
mc
2
Kinetic energy
is zero when v=0
K= mc2 - mc2
Momentum and Energy Units
Energy SI units:
J = kgm2/s2 = CV
Momentum SI units:
kgm/s
Mass SI unit
kg
Energy can also be given in eV
Momentum can also be given in eV/c
Mass can also be given in eV/c2
Momentum Change p
mv
v
1
To increase the speed of electron
c
from 0.1c to 0.2 c:
from 0.8c to 0.9 c:
MeV )(0.8c)
(0.511MeV c 2 )(0.1c)
(0.511
c2
p1
p3
2
1 0.1
1 0.82
p3 0.681 MeV/c
p1 0.05135MeV/c
p 4 1.055 MeV/c
p 2 0.1043MeV/c
p 0.374 MeV/c
p 0.053MeV/c
Same change in speed, but seven times as
much force needed.
2
Energy Change
E
mc 2
v
1
c
To increase the speed of electron
from 0.8c to 0.9 c:
from 0.1c to 0.2 c:
(0.511 MeV c2 )(c 2 )
E1
2
E3 0.8517 MeV
1 0.1
E1 0.5136 MeV
(K1 0.0026 MeV)
E 2 0.5215 MeV
E 4 1.1723MeV
E 0.32MeV
E 0.008MeV
Same change in speed, but forty times as
much energy needed!
2
Graphically
to ∞
It would require infinite force and
infinite energy to accelerate anything
with mass to the speed of light!
to ∞
Inertial Reference Frames
recall
special relativity (ch 2)
only true for
inertial ref frame is one in which
Newton’s Law of Inertia holds
not accel
no gravity (or “weak” like Earth’s)
To describe grav or accel
use General Relativity
Principle
Recall the first Postulates of Special Rel:
All laws of physics the same in all
inertial frames
const vel frame indistinguishable
from rest frame
The basic postulate of gen rel
The Principle of Equivalence
uniform gravity frame indistinguishable
from constant accel frame
equiv accel and g are equal in size, but
opposite dir
Equivalence
Artificial Gravity
rotating spaceship, with centrip accel = g
feels like home, earth’s grav
Virtual Reality
tilted chair has grav down and back
feels like grav down, and accel forward
Car accelerating forward
you pushed down and back in chair
cup falls down and back
fuzzy dice hang down and back
He balloon floats up and forward
all as if grav had a backward component
Effect on Light
Effect of grav on light identical to effect of
accel on light.
Light traveling perpendicular to grav field:
consider accel frame
you move a small dist rel to light at
first, then larger and larger distances
in your frame, light moves small
distance, then larger and larger
light follows curved path
see handout
Effect on Light
Effect of grav on light identical to effect of
accel on light.
Light traveling parallel to grav field:
consider accel frame
the frame is moving slowly when
light emitted
moving faster when received
as if sender and receiver are moving
relative to each other
Doppler shift
see handout
Gravity
Light’s path is bent by gravity,
light from a star behind the sun travels
a curved path, so appears at a different
place
gravitational lensing
Light retardation,
pulse sent to Venus
is slowed as it passes the Sun
all of time is slowed by grav not just the
freq of light
Black Hole
very large stars collapse
their gravity is stronger than the Pauli
repulsion of neutrons
the star continues to collapse
no known force can stop it
collapses to zero volume (?)
infinite density (?)
PAL – Classical and Relativistic
Momentum
In classical mechanics, momentum is
proportional to velocity, so if you double the
velocity, you double the momentum. Let’s see
how this goes at high speed:
1 a) By what factor does the momentum
change when an object’s speed is doubled
from 0.05 c to 0.1 c?
1 b) By what factor does the momentum
change when an object’s speed is doubled
from 0.3 c to 0.6 c?
1 c) The faster you go, the (worse or better)
classical physics fits reality.
PAL – Classical and Relativistic
Kinetic Energy
In classical mechanics, kinetic energy is
½mv2. Let’s see how this goes at high speed:
2 a) What is the percent error you get by using
the classical equation instead of the relativistic
equation for an object moving at 0.05 c?
2 b) What is the percent error you get by using
the classical equation instead of the relativistic
equation for an object moving at 0.6 c?
2 c) The faster you go, the (worse or better)
classical physics fits reality.
Spacebarn Paradox
The spaceship is 40 m long in its own
frame. The spacebarn is traveling at
about 0.866c relative to the ship, so it is
only 10.01 m long in the ship’s frame.
In the ship’s frame, the ship does NOT fit
in the spacebarn. In the ship’s frame,
will the ship get hit by the doors?