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Deriving the Gravitational Constant G

Today we will derive the gravitational constant G, also known as Newton's constant. Here are the variables we will be working with: ...

Monday, March 27, 2017

Using Quantum Physics to Solve Relativity Conundrums

Imagine you are driving down a lonesome highway. According to relativity theory, you could look at the situation in two ways: the highway is at rest and your car is moving; or, your car is at rest and the highway is moving, so are all the trees lining the highway, so are the buildings and the sky above--the whole earth is moving while you and your car sit still and watch it all go by.

But which way of looking at the situation is true? Could they both be equally valid? No. Here's why: the amount of energy needed to move the entire earth beneath your car far exceeds the amount of energy that is in your gas tank. Thus, conservation of energy is violated if we take seriously the notion that your car is at rest and it is the earth that is moving. Therefore, it is your car that is moving relative to the earth.

Now, consider Bob and Alice. Each are inside a separate spaceship. The two spaceships are at rest relatively speaking. Bob takes some measurements inside his spaceship; Alice does likewise. Both drink a potion that puts them to sleep. While they sleep, one of their spaceships is accelerated to speed v; the other remains at rest relative to the other. When Alice and Bob wake up, they must figure out which of their spaceships is going speed v. According to relativity theory, they won't be able to tell. If there is no friction and if there are no windows or portholes, both will perceive they are still at rest.

Both Alice and Bob experience the same laws of physics but ... not the same measurements. As I mentioned earlier, both took some measurements when both ships were at rest. One measurement was a momentum, given their respective masses. They each fired themselves out of a small cannon. They each measured the distance they flew and the time it took; i.e., the velocity. They calculated the momentum by multiplying mass and velocity. Using the same method, Alice now measures her velocity again. She notices no change. Bob, however, is in for a surprise. His velocity has slowed down. The momentum is conserved, so that means his mass has increased! Thus Alice and Bob deduce, pursuant to equation 1) below, that Alice's spaceship is at rest relative to Bob's. Bob's ship is the one moving at velocity v.

Suppose Alice and Bob can see inside each other's ship as Bob's ship passes by Alice's. According to relativity theory, Alice sees Bob's clock running slower and the length of Bob's ship has shrunk. Bob, however, sees Alice's clock running slower and it is her ship that has shrunk. Is Bob right? Or is Alice? Are they both right? Are they both wrong? Did one or both of their ships shrink? Is time really running slower for Bob and/or Alice?

Since his ship is moving faster than Alice's, Chances are it's Bob's ship that is experiencing time dilation and length contraction. But how can we be sure?

Let's assume Bob and Alice each have a box that tells time by firing two photons vertically, one up and one down. The photons have a velocity c and take time t to traverse the box. The vertical distance is ct:

When Bob's ship passes Alice's here's how they see each other's clock:

They each see the other's clock lagging behind their own. When Alice notices her photons have completed the distance (ct), she looks over at Bob's and notices his photons have not. (His have gone ct'.) Bob, of course, sees the opposite.

Alice and Bob rotate their clocks 90 degrees, so that the photons go horizontally.

Again, they see their clocks going a distance of ct. When they look at the other's clock, they see this:

The above diagram indicates that time appears distorted. Going from left to right, the distance is longer and time rate is faster, but going right to left, distance is shorter and time rate appears slower. Both Alice and Bob calculate the average time and distance. They both notice the photons in both clocks, on average, complete the distance simultaneously. Since Alice and Bob have concluded the other's clock is slower, there is only one way the photons could complete their respective distances simultaneously: the other's horizontal distance has shrunk to length ct'. (In the diagram below, each assumes the other is in motion while he or she is at rest.)

However, what Alice and Bob see and conclude in the above thought experiment depends on one's confirmation bias. Instead of seeing time slow down and length contract, Alice and Bob could see the other's clock (vertical space) grow taller and the other's time in sync with their own:

If the vertical space grows taller, that explains why the other's photons take longer to complete the distance. If the horizontal space does not contract for either ship, that explains why both sets of photons completed the distance simultaneously. Also the fact that both Alice and Bob see the other's ship having the relativistic effects suggests that what they see in the other's ship is some sort of optical illusion. It ain't really happening--or is it? To know the truth about relativity we must delve deeper. Quantum physics can assist us.

Let's address whether a spaceship in motion experiences horizontal contraction or vertical growth. We can think of the particles that make up a spaceship as waves. Waves have wavelength and amplitude. We can imagine each particle's wavelength contracting. This may cause the macroscopic ship length to contract. Or, we can imagine each wave's amplitude increasing--possibly causing the ship's height to grow. We begin our investigation by defining some variables:

Here's the math that can help us figure out what is going on:

From Hooke's law we derive equation 13) above. We mentioned earlier that Bob's mass increased. Equation 13) shows that an increase in mass is a function of increased amplitude and/or decreased wavelength. So Bob's spaceship may have really shrunk horizontally and/or grown vertically. By contrast, Alice measured no increase in her mass. We can assume when Bob sees her spaceship change shape, he's seeing an illusion, like a guy looking out a train window, watching the train station go by--or, maybe he sees no change in Alice's ship.

If equation 13) is valid, we should be able to derive from it the Lorentz equation. Let's give it a go:

Equation 28) above confirms equation 13's validity. However, we still can't be sure what Alice sees when Bob's ship whizzes by. Bob's ship could have shorter length and/or taller height. Perhaps combining the Lorentz factor and quantum mechanic's wave function can help:

Take a close look at equation 37) above. Notice that the right side's numerator is cosine squared plus sine squared. This value always equals one. It is constant. This constant value represents the amplitude or height, so when velocity (v) increases, the amplitudes of all the particles in Bob's ship don't change. It's possible then that the height of Bob's ship remains constant. Only the denominator (cosine squared) can change. This represents the wavelength. So when Alice sees Bob's ship whiz by, she probably sees a shorter length and a constant height. Such a scenario is consistent with the following wavelength equation:

Equation 37b) makes no mention of wave amplitude--which implies the amplitude does not change when there's a change in momentum (p). Only the wavelength changes. Below we derive equation 41). If you check it against the Lorentz (equation 42) you will find it yields the same results--the wavelength shortens when velocity (v) increases.

The above math seems to have solved the riddle of what Alice really sees when Bob's ship flies by. However, the following math suggests that all the dimensions of Bob's ship shrink--not just the length. If we make the Lorentz equation a function of mass, the y and z dimensions shrink.

Could that be how it really works? Mass increases due to high velocity, and all three dimensions shrink? Not according to the definition of a wavelength. A wavelength is the distance along the direction of propagation ... If Bob's ship is moving along the x-axis, then we can imagine all the spaceship's particle-waves doing likewise. Thus their shortening wavelengths are along x--the ship's length. So the length of Bob's ship will likely be ct'. The height, depending on which clock is used, will be nct' or ct (nct'=ct; n is a coefficient). The width will also be some factor of ct'.

The diagrams below show the relationship between particle-waves and Bob's square clock. As velocity (v) increases, Bob's ship propagates further as expected, but the wavelengths of his ship's particles shorten, so his ship's length shortens. You can think of Bob's clock and ship as macroscopic particle waves propagating from left to right. The photon (see wavy diagonal line) is moving vertically, but its shorter wavelengths have no effect on the height, since it is the only particle moving vertically.

Equation 59) below expresses the Lorentz in terms of frequency, wavelength, mass, and time:

When energy is added, frequency increases from fo to f'. The wavelength shortens. The light-speed term (second term) stays constant, since wavelength and frequency inversely offset each other. The last term is the velocity v term. Here, mass increases, frequency increases to inversely offset the mass increase, so what's left to cause higher velocity? A shorter wavelength.

To further understand what's happening at the quantum level, we rely on a Hamiltonian (H) equation expressed in terms of mass, wavelength, amplitude, and time:

Equation 65) shows what happens to a typical particle-wave. Velocity is seen by Alice, not Bob. Bob can treat his spaceship's particles as though they are at rest. This is why the velocity term or kinetic energy term has time (t), Alice's time. Amplitude stays constant and, once again, velocity and kinetic energy are determined by wavelength. The last term is rest-mass energy. Here, when Bob's time (t') reduces, so must the wavelength (lambda): otherwise, light speed (c) would not remain constant.

After making the case above, can we conclude that Bob's spaceship actually contracts when whizzing by Alice? No. We have to take into account thermal expansion (see equations below). High velocity implies high kinetic energy; high kinetic energy implies high temperatures; high temperatures cause thermal expansion which could more than offset length contraction. At very high speeds Bob might notice he's sweating profusely. If his velocity is great enough, Alice might see a fireball going by. If she sees anything, she might presume she's at rest, since anyone moving at such high speeds might be dead.

Then again, thermal transfer to the cold surrounding space might provide Bob some relief. Equation 70) below gives us the rate of heat loss. We combine equation 70) with equations 68) and 69) to get equation 71) which tells us the thermal expansion of length L, taking into account the temperature decrease due to heat loss as well as any temperature gain due to high speeds.

Keeping relativity in mind, thermal expansion occurs when the molecules that make up Bob's spaceship move at higher velocities relative to each other. If the molecules only move in the direction Bob's ship is moving, their relative velocity to each other is zero--no thermal expansion occurs. If, on the other hand, they move in many directions, their relative velocity to each other is greater than zero--and thermal expansion could offset relativity's length contraction.

What Alice sees when Bob's spaceship flies by can vary depending on the circumstances:

Sunday, March 12, 2017

The Essence of Time

What exactly is time? Is it just an abstract idea? Or does it exist independently of human imagination and perception? Atomic clocks reveal that the rate of time appears to run slower on the earth's surface than way out in space. Assuming time is a real entity, what is it made of? What is its essence? We start our investigation by defining some variables:

We know that nothing goes faster than light in a vacuum. If we add velocity (v) to velocity (c) we still get the speed of light (c).

Equation 1) above seems absurd. When we combine velocities, we should get a higher velocity than c ... unless ... the rate of time (t') shrinks. Equation 2) below works:

And from equation 2) we can derive the famous Lorentz equation:

When velocity (v) increases, time (t') shrinks, but something else happens that's also strange: mass (m') increases. We can verify this if we start with Einstein's energy equation below.

From equation 6) we can derive the relative mass equation:

Equation 13) confirms that when velocity (v) is increased, mass (m') increases. Now, let's take equation 10) and derive equation 14) below:

Equation 14) shows why increased velocity increases mass. When the velocity of a system increases, velocity (u) decreases. To conserve momentum, mass (m') must increase. We start with momentum (mc) and end up with (m'u). Of course m'u must always equal mc. But what exactly is this velocity u? I call it the velocity of time.

The rate of time is the relative speed (u) of a photon or (c^2-v^2)^.5. If a system is moving at velocity (v) and we assume that system is at rest, then the photons in that system may still appear to be moving at c, but relative to v they have slowed to velocity u. Since photons are bosons, their relative speed (how fast they carry force) will determine how fast or how slow the system evolves. If the system is your watch, your watch will noticeably slow down if it moves at a significant fraction of light speed. This suggests that time is real and not just a concept. After all, the original concept of time was that the rate of time is fixed.

Below is a Feynman diagram where velocity v is zero, so velocity u equals velocity c. At the beginning of time (t), two electrons colide. Next, a photon is emitted, then the electrons fly apart.

In the next diagram imagine that the entire diagram is moving through space at velocity v. If we assume the diagram is at rest, the emitted photon will be relatively slower and the measure of time will also be slower:

The electrons in the above diagram are moving faster, but photons can't increase their speed, so, relative to the electrons, the photon is slower. This seemingly slow moving photon is the time we measure or proportionate to the time we measure. Now, just for fun, what happens if the electrons go faster than light?

Time reverses! Your watch is now running backwards. In the diagram above, the particles fly apart, then comes the photon, and finally, the particles collide.

Before we conclude that time is (or is proportionate to) the speed of photons relative to the other particles in a system, let's look at how gravity impacts time, and take a closer look at light, i.e., electromagnetic waves. Once again we define some variables:

We include the variables for permittivity and permeability. Taken together, they determine the speed of an electromagnetic wave. A photon's relative speed (u) has its own corresponding permittivity and permeability. Free space is a vacuum. That is where light has a velocity of c. If the permittivity of free space, for example, had a lower value, light would go faster. Increasing velocity v causes permittivity and permeability to increase (so does additional mass/energy). As a result, EM waves (light) slow down or are relatively slower. The mathematical proof below provides further insight into the essence of time:

Equations 23) and 24) show that time is the reciprocal of frequency. Equations 24) and 26) define the rate of time (with variable t[sub o] set to 1) in one of two ways: The ratio of the relative speed (u) of a photon to light speed (c); or, the ratio of permittivities and permeabilities. Both ways are equivalent. To put it more simply, time (at the quantum level) is the measure of the rate bosons can carry force between particles. If that process is disrupted by high speeds or increased mass/energy, that process will slow down. Anything that is a function of that process will also slow down--including your watch.

Update: Quantum particle-waves are transverse waves: their oscillations are perpendicular to their propagation direction. The following is a mathematical proof that shows that the Lorentz equations above work for transverse waves.

Equation 37) is the formula used to calculate the velocity of a transverse wave. We were able to derive it from the Lorentz equation for relative mass. This shows that the two are intrinsically connected. Equation 39) predicts what we expect: increased mass (m') reduces the time rate (t').

Update: Here are a couple of equations that enable you to calculate time rate (t') when there is light moving between any two stars in a galaxy:

Diagram A above shows light moving between two points at a 90 degree angle to velocity v. The first term of equation 40) drops out and the equation reduces to the Pythagorean theorem used for the Lorentz transform. If the light and the galaxy are going in the same direction or opposite directions (zero degrees), then equation 40's second term drops out. The equation reduces to a linear relation between c and v which always equals c according to the math below:

Angles between zero and 90 degrees yield some combination of both equation-40) terms.

Diagram B above assumes the galaxy is rotating and moving to the right at varied angles to the direction of the light. We take the sum of all those angles to get the average angle. We calculate u and plug it into equation 41) to get time rate t'.