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PATRICK ABLES

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PART 1 TIME DILATION:GPS, Relativity, and other

applications

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• 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.

FACTORS CONTRIBUTING TO DILATION OF SATELLITE TIME

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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.

Newton’s Law of Universal Gravitation

Newton’s Second Law of Motion

Combine the two equations above and reduce

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.

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 RELATIVITYTIME DILATION FROM ROTATION

(Where v is replaced by v(r))

r is the radius of Earth, and h is the height above its surface.v(r) is the difference in tangential velocities due to rotational motion.

Lorentz factor:

By special relativity, the formula for time dilation is:

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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

ALTITUDE ABOVE EARTH’S SURFACE

TIME DILATION FROM GRAVITY

TIME DILATION FROM ROTATION

TOTAL COMBINED DILATION

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GPS SATELLITE SIGNALSThe 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

GRAVITATIONALDILATION

ROTATIONALDILATION

TOTALDILATION- =

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 BlackholeTime 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.

TIME DILATION FROM GRAVITY

TIME DILATION FROM ROTATION

TOTAL COMBINED DILATION

EARTH’SSURFACE

ALTITUDE ABOVE SURFACE

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SCHRODINGER EQUATIONThe 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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E vs N (Ground-state)

N

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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)

X

N = 2

N = 10

N = 40

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PT SYMMETRIC HAMILTONIAN

One would not expect a theory defined by a non-Hermitian 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, non-Hermitian 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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E vs N (Ground-State)

N

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Wave Functions

X

N = 8N = 12N = 14

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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.