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Physics News
NASA's JUNO probe arrived in orbit around Jupiter this past weekend.
Juno captures the "Bow Shock", the shock-wave like part of
space where the stellar winds from the sun smack into Jupiter's
magnetic field.
Juno also captures the transition from being within the domain
of the sun's magnetic field and that of Jupiter.
What we've learned so far:
We reviewed Galilean Relativity and introduced the fact
that the predictions for the speed of light from Electromagnetism
do not depend on the relative velocities of observers.
While electromagnetism was seen as a great triumph, experiments
were beginning to find deviations from predictions and some physicists
such as Lorentz came up with mathematical patches to try to
"fix" classical Physics.
Maxwell's Equations predict light has a constant speed $c=3.0 \times 10^8 m/s$ no matter how fast
the observer's velocity is.
Einstein achieved a breakthrough by accepting what the mathematics and
physics was telling the world: that the constancy of the speed of
light, $c$, was so important to the universe that time and space would
be "warped".
What we've learned so far:
Let's say your birthday twin travels on an airplane, when they
land they are a tiny little bit younger than you because
moving clocks tick slower, i.e. their experience of time
proceeded slower than yours due to them traveling faster than you.
While your friend is in the air on the airplane, your
friend's measurement of the airplane
is not consistent with your measurement. You would find the airplane
to be just very slightly shorter than your friend measured it.
If your friend, on the plane in the air, saw that lightening had
struck the tip and the back of the plane simultaneously, you would
most definitely not see them strike simultaneously. In your spacetime
experience, they did not strike simultaneously, and you are
both right.
Principles of Relativity
The laws of physics are the same in all inertial frames
of reference (all observers not accelerating but may be
traveling at different velocities relative to each other.)
All such observers will measure the speed of light to be
$c$ no matter how fast they are going relative to another
observer.
Relativity of Length and Time:
Define the proper frame for an
object or event to be the frame in which
the object or event is at rest.
$$ L = \frac{L_p}{\gamma} $$
$$ \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} \geq 1 $$
Longest length is measured in the proper frame of the object
being measured.
$$ \Delta t = \gamma \Delta t_p $$
Shortest time interval is measured in proper frame
of event being timed.
e.g. muon decay, we detect more muons from space on the ground
than would be expected if time dilation did not exist. Because
they are going so fast they effectively get to live longer
in our spacetime experience.
Some consequential paradoxes
Lorentz transforms:
$$ x^{\prime} = \gamma \left( x - ut \right) $$
$$ y^{\prime} = y $$
$$ z^{\prime} = z $$
$$ t^{\prime} = \gamma \left(t-\frac{ux}{c^2}\right) $$
Tip: Use Galilean Transforms to orient yourself between
reference frames. For example, imagine a truck being the $S^{\prime}$
frame and yourself on the ground as the $S$ frame. You measure
a sign on the road to be five meters ahead of you. In the trucks
coordinate frame, the sign is $5 - ut$ behind it assuming it has
passed it already. If the truck is going $10 m/s$ for $2s$, then
the sign is $5 - 20 = -15 m$ or 15 meters behind it already.
We will try to consistently use $S$ as the frame at rest relative
to Earth and $S^{\prime} as the moving frame. However, in relativity,
it doesn't really matter as it is all relative.
Some thoughts about spacetime...
Relativity has us think about spacetime rather than space
and time, but in a way, we already do so...
Velocity Transforms:
Suppose a truck and observer on the ground start out at $t=0$ at $x=0$
and the truck travels $u=30m/s$ in the $+x$ direction.
The ground observer shoots a rocket in the $+x$ direction and
let's pretend it doesn't get pulled down by gravity. If the
ground observer measures the rocket to be going at $v_x = 100 m/s$
then how fast will the truck observe the rocket to be going?
$v^{\prime}_x =$ the velocity of the rocket as measured by the truck
$v_x =$ the velocity of the rocket as measured by ground observer
$u =$ the relative velocity of truck/ground frames (how fast the truck
is going relative to the ground observer or how fast the ground observer
is receding relative to the truck observer.)
Suppose an observer is moving towards, at speed $u$,
a wave source whose wave speed is $c$, then
classically the wavelength they would measure would be
$\lambda = (c - u)T_0$ where $T_0$ is the period of the wave.
The detected frequency is
$$ f = \frac{c}{\lambda^{\prime}} = \frac{c}{(c-u)T_0} $$
However, from relativity we know that
$$ T = \gamma T_0 $$
Some algebra work yields:
$$ f = \sqrt{ \frac{c + u }{c - u} }f_0 $$
where $f$ is the frequency observed by the traveler,
and $f_0$ is the frequency the source would measure itself
emitting.
The relativistic Doppler effect is used to measure of shifts in the frequency of light emitted by a moving astronomical object such as a galaxy. Light emitted by atoms and normally found in the extreme violet region of the spectrum is shifted toward the red end of the spectrum for atoms in other galaxies, indicating that these galaxies are receding from us. Edwin Hubble (1889–1953) found that most galaxies are moving away from us, indicating that the Universe is expanding. This surprised Einstein.
Big Bang:
Relativistic Momentum:
Relativity does not do away with Classical Physics, but rather,
enhances it to be consistent with Electrodynamics.
One of the key features of Classical Physics, something you should
always remember, is conservation of momentum and
conservation of energy.
Conservation of momentum and relativity lead to the finding that:
$$ \vec{p} = \frac{ m \vec{v}}{\sqrt{ 1- \frac{v^2}{c^2}}} $$
When $v$ is much smaller than $c$, $p \approx mv$.
Relativistic Force:
Because Newton's Second Law says that $F = dp/dt$,
$$ F = \frac{ma}{\left( 1 - \frac{v^2}{c^2}\right)^{\frac{3}{2}}} $$
$$ a = \frac{F}{m} \left( 1 - \frac{v^2}{c^2}\right)^{\frac{3}{2}} $$
As $v \rightarrow c$, $a \rightarrow 0$.
The faster you travel, the less effective your applied force is.
It is impossible
for anything to exceed the speed of light.
Sorry science fiction fans, we will never explore the stars
in person$^{*}$.
$^*$ Actually, there is still a little bit of hope with General
Relativity.
Relativistic Kinetic Energy:
A further implication of the refinement of the definition
of momentum is that energy is also redefined.
Work-Energy theorem says
$$K = W = \int^{x_2}_{x_1} F dx = \left(\gamma - 1 \right) mc^2$$
Using the binomial theorem, $(1+x)^n = 1 + nx + n(n-1)x^2/ + \cdots$,
$$ K = \frac{1}{2} mv^2 + \frac{3}{8} \frac{mv^4}{c^2} + \cdots$$
So when $v$ is much smaller than $c$, $K = \frac{1}{2} mv^2$ works fine.
Rest Energy:
The form for relativistic kinetic energy has a component that
is not dependent on speed:
$$ K = \left(\gamma - 1 \right) mc^2 = \gamma mc^2 - mc^2 $$
We call this the "Rest Energy". It is an energy inherent in
all objects that have mass. $E_0 = mc^2$.
The implication is now that Energy and Matter separately aren't
conserved, but that together they are; and they can be converted
back and forth.
Total Energy:
The total energy of a massive particle that isn't in a potential
field, i.e. only has kinetic energy and the inherent rest
energy is:
$$ E_{TOT} = K + E_0 = \gamma mc^2 $$
Combining this with the momentum equation $p$ and with some algebra,
$$ E^2 = \left( mc^2\right)^2 + \left( pc\right)^2 $$
For particles without mass such as light, $E = pc$, and for particles
with mass that are at rest, $E = mc^2$.
Converting matter to energy
General Relativity:
$E = mc^2$ implies that there is something special about mass.
Mercury wasn't behaving as expected:
In our Solar System, the largest perihelion shift is 560 arcsec/century for
Mercury. Most of the shift can be attributed to perturbations from other planets, but
after all the known planets are taken into account a shift of 43 arcsec/century remains
unexplained. Historically, some people speculated that there might be another planet
closer to the Sun than Mercury that caused the additional perihelion shift. The
hypothetical planet was called Vulcan.
Einstein extended special relativity to general relativity, which
finds that gravity can be better seen as a warping of space and time.
Postulates of General Relativity
All the laws of physics hold true for any frame of reference, inertial or not.
Principal of equivalence: gravitational mass is equivalent to inertial mass.
Stated another way, acceleration due to non-gravitational force in
a gravity free part of space
and acceleration due to gravitation are effectively equivalent.
General Relativity
Equivalence principal:
Consequences General Relativity
Time and space are warped by massive bodies (gravity):
Consequences General Relativity
Time and space are warped by massive bodies (gravity):
Gravity "Lenses" light from stars and other galaxies
These arcs are actually the lensed light of galaxies that lie
behind the cluster of normal looking galaxies in the pictures.
The huge gravity well of this galaxy cluster "lenses" light around
it, providing us views of dozens of galaxies (albeit a warped view)
that we otherwise wouldn't be able to see.
General Relativity predicts additional Doppler-like affect
This has consequences for GPS satellites...we must apply the above
adjustment as well as time dilation adjustment from special relativity
or else your GPS will send you into the ocean.
The gravitational redshift: electromagnetic radiation originating from a
source that is in a gravitational field is reduced in frequency (redshifted)
when observed in a region at a higher gravitational potential (further
away from the gravitational source[s]).
The gravitational blueshift: electromagnetic radiation originating from a
source that is far away gravitational field is increased in frequency (blueshifted)
when observed in a region with higher gravitational field. e.g. the frequency
of a GPS satellite.
General Relativity predicts additional Doppler-like affect
General Relativity also implies gravity slows clocks
October 1971: Joseph Hafele and Richard Keating fly atomic clocks on
airplanes around the Earth. Results in $ns$:
General Relativity predicts blackholes
If gravity warps spacetime and alters the trajectory of
light, then a dense enough gravitational source
should be able to "trap" light. GR predicts such "black holes".
General Relativity also implies gravity slows clocks
7 minutes $\rightarrow$ 10 minutes in (feel free to watch entire):
A problem we will return to: GR predicts a singularity; quantum mechanics
does not. Both theories perform spectacularly well in their own domains. Where
they intersect, in a black hole, we only have?
In our Solar System, the largest perihelion shift is 560 arcsec/century for
Mercury. Most of the shift can be attributed to perturbations from other planets, but
after all the known planets are taken into account a shift of 43 arcsec/century remains
unexplained. Historically, some people speculated that there might be another planet
closer to the Sun than Mercury that caused the additional perihelion shift. The
hypothetical planet was called Vulcan.
