Download Time Dilation and Determining Eigen Values

PATRICK ABLES
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PART 1 TIME DILATION:
GPS, Relativity, and other
applications
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FACTORS CONTRIBUTING TO DILATION
OF SATELLITE TIME
• General relativity tells us gravitational fields
cause time dilation. Clocks on Earth tick
slower than clocks in satellites.
• Special relativity tells us rotating bodies cause
time dilation due to a difference in relative
velocities at different altitudes. Clocks in orbit
tick slower than clocks on Earth.
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GENERAL RELATIVITY
• Einstein’s field equations describe the
relationship between the presence of matter
and curvature of spacetime.
• In cases where the source of gravity has a
spherical distribution, there is a solution
called the Schwarzchild metric, which
determines the geometric properties of
spacetime outside the spherical mass.
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APPROXIMATING GRAVITATIONAL TIME DILATION FROM
GENERAL THEORY OF RELATIVITY, USING SPECIAL
THEORY OF RELATIVITY.
STEP 1: Let’s say, a hypothetical rocket travels from the surface of Earth, to a
height of 20,000km, at a constant g, equal to that of Earth. The gravity and relative
velocity is then calculated over 1 meter intervals following Newton’s Laws.
Newton’s Law of Universal Gravitation
Newton’s Second Law of Motion
Combine the two equations above and reduce
We use this equation as the acceleration of the
rocket, to simulate the gravitation force one would
experience at each interval.
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STEP 2: Recording the altitude and the velocity for each interval, we can
simulate a rocket moving at a constant acceleration, g, from the surface of Earth, r,
to an altitude, h = 1, 2, 3, …, 20000000 meters.
Where the v is calculated at each interval by,
STEP 3: Special Relativity defines time
dilation as:
is the time of the satellite.
is the time of the rocket at each interval, simulating that of Earth’s at a
given altitude, rest frame.
Subtracting the travel time of the rocket, we can
find the time dilation for each interval.
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SPECIAL THEORY OF RELATIVITY
TIME DILATION FROM ROTATION
v(r) is the difference in tangential velocities due to rotational motion.
r is the radius of Earth, and h is the height above its surface.
By special relativity,
the formula for time
dilation is:
(Where v is replaced by v(r))
Lorentz factor:
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TIME DILATION EFFECT ON EARTH
• To calculate the total time dilation, we must include the
dilation due to Earth’s gravity, and the dilation due to
rotational speed of the satellite.
• The nature of gravitational dilation is that time on the Earth is
slower than the time in the satellite.
• Inversely, for rotational motion, time in the satellite moves
slower because it moves very fast relative to Earth’s surface.
• For these reasons, to approximate the total time dilation on
Earth’s surface, we must subtract the dilation due to motion
from the dilation due to gravity.
• Relative to the satellite, the time dilation with respect to
height was graphed…
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The graph shows time dilation from the
satellite’s perspective, 20,000km above Earth
TIME DILATION FROM GRAVITY
TIME DILATION
TIME DILATION FROM ROTATION
TOTAL COMBINED DILATION
ALTITUDE ABOVE
EARTH’S SURFACE
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GPS SATELLITE SIGNALS
The time dilation from the satellite to the
surface was found to be:
(5.26 x 10^-10)sec – (8.97 x 10^-11)sec = (4.37 x 10^-10)sec
GRAVITATIONAL
DILATION
ROTATIONAL
DILATION
=
TOTAL
DILATION
While too small for slow moving observers to
notice, it’s a huge problem in calculations
where light is used to make measurements.
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• Since light travels at about 3.0 x 10^8 m/s,
compensation for the time dilation effect must
be made.
• For example, a signal is sent from a satellite to a
point on the surface of earth to determine a
person’s position.
• (3.0 x 10^8 m/s) x (4.37 x 10^-10) x 24 x 3600s
• Over the course of ONE day, the position as
calculated by the satellite, would be off by:
11,318m after a time dilation of 37.7 microseconds
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Time Dilation in a Blackhole
TIME DILATION
Time dilation for a gravitational body, equal to the mass of Earth. If the
sphere were to be compressed, e.g. the surface moves closer to the center
of mass, a blackhole is formed, with infinite time dilation at its center.
EARTH’S
SURFACE
TIME DILATION FROM GRAVITY
TIME DILATION FROM ROTATION
TOTAL COMBINED DILATION
ALTITUDE Hirophysics.com
ABOVE SURFACE
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SCHRODINGER EQUATION
The time independent, one dimensional
Schrodinger equation is given as:
The constants can be unit;
This is known as Hartree atomic units
The equation can now be simplified as:
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Shooting Method
• The basic idea behind the shooting method is to convert a
boundary value problem (BVP) into an initial value problem
(IVP). One parameter is introduced for each missing initial
condition.
• The boundary conditions that are not initial conditions serve
as the constraints to determine appropriate values for the
parameters. That is, given an initial guess for the parameters,
an iterative solver is used to find values of the parameters
that produce (approximate) solutions that satisfy the
boundary conditions.
• This idea is applied to Schrodinger’s equation, using Hartree
atomic units.
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HAMILTONIAN
Where the first term is the kinetic energy and the
second term is the potential energy.
The Hamiltonian used by
shooting method is given as:
Note: When n = 2, the Hamiltonian will be in harmonic oscillation
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Ground State Energy
E vs N (Ground-state)
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N
Wave Functions
Where n is a positive integer, each Hamiltonian
is inserted into the Schrodinger Equation to find
the corresponding wave functions, by solving for
PSI. Each respective wave function then can be
graphed, with respect to X, to provide a visual
for the behavior of the wave function of each
Hamiltonian.
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PSI vs X (Ground-state)
N=2
N = 10
PSI
N = 40
X
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PT SYMMETRIC HAMILTONIAN
One would not expect a theory defined by a nonHermitian Hamiltonian to be physical, because the
energy levels are not likely to be real, and the time
evolution is not likely to be probability-conserving
(unitary).
However, theories defined by PT –symmetric, nonHermitian Hamiltonians can have positive real energy
levels and can exhibit unitary time evolution. Such
theories are acceptable quantum theories.
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PARITY & TIME TRANSFORMATIONS
• The idea of parity transformation comes from
the concept of space inversion.
• Under the conditions of parity transformation,
X becomes –X, and P becomes –P.
• Since space and time is considered to be the
same, we also have time inversion to consider.
t becomes –t, and i becomes –i.
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PT SYMMETRIC HAMILTONIAN
EQUATION
The general form of the PT symmetric
hamiltonian is:
Where n is greater than or equal to 0.
The Hamiltonians used for the purpose of
presenting are:
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• Some PT symmetric equations yield non-real
eigenvalues; however, for even n values, we
can mathematically get real, physical energy
values.
• Making the potential energy negative, and
using the same PSI equations for non-PT
eigenvalues, we investigated what the wave
function might look like in terms of the
eigenvalues found…
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ENERGY
E vs N (Ground-State)
N
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Wave Functions
PSI
N=8
N = 12
N = 14
X
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Conclusion
• We have established a prototype time dilation
model for a blackhole using special relativity
• We found that the eigenvalues of higher order
PT symmetric Hamiltonians are the same,
where N=12, and for each successive N.
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FUTURE RESEARCH
• Since the eigenvalues for the PT symmetric Hamiltonians at
N=12, and each successive N, are the same, it seems we have
reached a limit that the magnitude can attain.
• Future research is needed to confirm the validity of these
results, before we can assume the values to be real values.
• The PT symmetric wave functions do appear interesting. The
graphs for N=12 and N=14, while having the same eigenvalue,
don’t have the same wave function.
• Future research is needed to investigate the nature of the
oscillation in the wave functions.
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