PHYS 182 Lecture Notes - Lecture 11: Bohr Model, Spacetime, Neutron Star
PHYS182: Our Evolving Universe
2017-10-17 LEC 11
GW170817 (gravitational wave + date)
• Collision of 2 binary black holes: Merged rapidly, produced a short signal of large amplitude
o When black holes come together, they are black and you can’t see anything
• Neutron stars: stars that have exhausted their fuel, final stage of a number of stars
• Whenever there are things orbiting each other in a binary system, it gives off gravitational waves
as the system loses energy
• But if you observe neutron stars colliding, what might you see?
• Aug. 17: an event was seen by the gravitational wave detector that looked like merging neutron
stars
• Telescopes saw light of different electromagnetic frequencies coming toward us
• This is the Birth of Multi-Messenger Astronomy
o 40Mpc distance, in galaxy NGC4993
o 2 neutron stars: m1 = 1.6 solar masses, m2 = 1.2 solar masses
o Merging of neutron stars was observed in X-ray, gamma ray, UV, optical, infrared
o The spectrum was taken, spectral lines were observed that correspond to gold, titanium
• What did we learn from this?
o Neutron star coalescence occurs & gives rise to a kilonova
o Kilonova: a locus of production of heavy elements (everything heavier than Fe)
o Gamma ray bursts are due to coalescence of neutron stars
o We see light travelling at the same rate as gravitational waves à confirms Einstein’s
general relativity
• The light appeared for different lengths for the different wavelengths
o The gravitational peak and peak of electromagnetic radiation from the event are different
§ When the radii collide, this is the maximum peak of gravitational waves, but only
the point where the photons start to be emitted
Finishing up Special Relativity
• We have discussed time dilation (at
!
v, time slows down)
• Length contraction (see notes)
• Mass increase: an object at !v has an increased mass
o Energy normally associated with mass is velocity dependent
o F = ma
§ Imagine you start at rest, and you slowly accelerate due to a constant force
§ The velocity will increase, but it can never be higher than the speed of light
§ The acceleration has to decrease otherwise the velocity will be higher than c
§ This means that the mass has to be increasing
o Something that has mass can never travel at the speed of light
o Anything that travels at the speed of light has no mass
• Since mmoving represents energy, mrest also represents energy
o E = mc2 represents rest energy
• Light as energy, and as a consequence you can get mass conversion
o To any particle, there is an anti-particle that has all of the same properties with opposite
charge
o E.g. for an electron, this equivalent is a positron
§ If you collide these, matter collides with antimatter to form energy (in the form
of light)
SPACE AND TIME
the orbiting stars in binary systems has also been measured,
with the same result: The speed of light is always the same.
Experimental Tests of Special Relativity Although
we cannot yet travel at speeds at which the effects of relativity
should be obvious, tiny subatomic particles can reach such
speeds, thereby allowing us to test the precise predictions of
the formulas of special relativity.
In machines called particle accelerators, physicists accel-
erate subatomic particles to speeds near the speed of light
and study what happens when the particles collide. The
MATHEMATICAL INSIGHT 2
Formulas of Special Relativity
We found a formula for time dilation in Mathematical Insight 1.
Although we will not go through the derivations, it is possible to
find similar formulas for length contraction and mass increase. The
three formulas are
time
(moving frame) =time
(rest frame) *
B
1-av
cb2
length
(moving frame) =(rest length) *
B
1-av
cb2
moving mass =(rest mass)
B
1-av
cb2
There’s also a simple formula for velocity addition. Suppose you
see Al moving at speed v1 and Al sees a second object moving rela-
tive to him at speed v2. By our old common sense, you would see
the second object moving at speed v1 + v2. However, the speed you
actually see is
speed of second object =v1+v2
1+av1
c
*v2
cb
EXAMPLE 1: Al is moving by you at 0.99c in a spaceship that is
100 meters long at rest. How long is it as it moves by you?
SOLUTION:
Step 1 Understand: This is a length contraction problem. Because
you see Al moving, his spaceship’s length should be shorter than its
rest length of 100 meters.
Step 2 Solve: We use the length contraction formula with Al’s speed
of 0.99c, or v/c=0.99, and rest length = 100 m:
length
(moving frame) =(rest length) *
B
1-av
cb2
=(100
m) *21-(0.99)2=14
m
Step 3 Explain: Al’s spaceship is 100 meters long at rest, but only
14 meters long when he is moving by you at 99% of the speed of light.
EXAMPLE 2: A “super fly” has a rest mass of 1 gram but is capable of
flying at 0.9999c. What is its mass at that speed?
SOLUTION:
Step 1 Understand: This time we are asked about mass, so we need
the mass increase formula and expect the moving mass to be greater
than the rest mass.
Step 2 Solve: We us e t he m ass i ncre ase for mula w ith t he f ly’s re st
mass = 1 g and v/c=0.9999:
moving mass =(rest mass)
B
1-av
cb2
=1
g
21-(0.9999)2
=70.7
g
Step 3 Explain: At a speed of 0.9999c, or 99.99% of the speed of
light, the mass of a fly that is 1 gram at rest becomes 70.7 grams, or
more than 70 times its rest mass.
EXAMPLE 3: Al is moving toward you at 0.9c. Your friend Jackie
jumps into her spaceship and, from your point of view, goes in
Al’s direction at 0.8c (see Figure 15). How fast will Al see Jackie
approaching?
SOLUTION:
Step 1 Understand: According to Al, your speed is v1=0.9c.
Jackie’s speed relative to you is v2=0.8c in the same direction. We
therefore need the velocity addition formula to find Jackie’s speed
relative to Al.
Step 2 Solve: We substitute the given values v1=0.9c and v2=0.8c
into the velocity addition formula:
Jackie>s speed
(relative to Al)
=v1+v2
1+av1
c
*v2
cb
=0.9c+0.8c
1+(0.9 *0.8)
=1.7c
1.72 =0.988c
Step 3 Explain: Al sees Jackie moving toward him at 0.988c, or
almost 99% of the speed of light. Notice that, as we expect, Jackie’s
speed relative to Al is faster than yours but slower than the speed of
light.
colliding particles have a great deal of kinetic energy, and
the collisions convert some of this kinetic energy into mass-
energy that emerges as a shower of newly produced parti-
cles. Many of these particles have very short lifetimes, at the
end of which they decay (change) into other particles. For
example, a particle called the p+ (“pi plus”) meson has a life-
time of about 18 nanoseconds (billionths of a second) when
produced at rest. But p+ mesons produced at speeds close to
the speed of light in particle accelerators last much longer than
18 nanoseconds—and the amount longer is always precisely
the amount predicted by the time dilation formula.
Document Summary
When the radii collide, this is the maximum peak of gravitational waves, but only the point where the photons start to be emitted. = (rest length) * b1 - a v cb 2 moving mass = (rest mass) Imagine you start at rest, and you slowly accelerate due to a constant force. The velocity will increase, but it can never be higher than the speed of light. The acceleration has to decrease otherwise the velocity will be higher than c. If you collide these, matter collides with antimatter to form energy (in the form of light) Summary: length & time are observer dependent, mass is equivalent to energy. Chapter 5: quantum revolution & nature of matter. There are discrete solutions; i. e. only for certain energies does a solution exist. Time independent: stabilization of matter, the electron is replaced by an electron wave function.
