Perhaps you are familiar with some of the more exotic and far-out theories that attempt to explain why gravity is so weak compared to, say, electromagnetism. One of my favorites involves gravitons hiding in extra dimensions. The logic is as follows: if gravitons weren't hiding in extra dimensions, gravity would be much stronger. It's analogous to someone screaming at the top of his lungs. The intensity of the sound is very strong until you hide that person away in a broom closet and lock the door.
As much as I like this theory, it has a problem that can't be hidden in a closet or extra dimension: there is no empirical evidence of extra dimensions. That begs the question: is it possible to explain gravity's weakness without using extra dimensions? Yes! When doing physics, I prefer taking an Occam's razor approach and using concepts and ideas that are testable and/or have been tested and confirmed. The end result is not as exciting as extra dimensions--but more likely to be true.
Let's begin with Einstein's field equations:
To simplify matters we equate the left side of equation 1 to K:
In equation 3 above, we have energy density (T44) multiplied by a constant that consists of G and c. Why do we need such a constant? Why can't we get by using equations 4 and 5 below?
The constant in question is a very small number and substantially reduces the value of K, the spacetime curvature. Perhaps equations 6 through 8 can shed some light on the subject:
Imagine energy (E) interacting with a cubic meter of spacetime. Equation 6 shows that spacetime energy density (Ts) is a tiny number. Since curved spacetime causes gravity, and since spactime's energy is so weak, is it any wonder that gravity is weak? Let's see if the math agrees. Equations 7 and 8 are the average radius of a nucleus and the nucleus volume respectively. Why are these numbers important? Imagine energy (E) added to a cubic meter of spacetime. To see how that energy interacts with spacetime, it's easier to take all spacetime's energy and reduce it to a particle. We do the same with the added energy. So we have a particle interacting with another particle within a cubic meter. Spacetime's total energy is equivalent to a proton's. We can imagine it having an approximate radius of e-15 meters and a volume of e-45 cubic meters. Our energy particle will have the same dimensions.
To provide a simple visual demonstration of the interaction between matter and spacetime, let's pretend that the volume of our spacetime proton is .25 cubic meters and that the energy particle we add has a .25 probability of interacting with it.
Because the probability of interaction is .25, the average energy ends up being .25 the total energy. To get the proper value for K, we would need to multiply E/m^3 by a constant of .25. So K is weaker than E/m^3. To illustrate the point further, let's distribute the energy evenly throughout the cubic meter. This time it will be a field of isotropic energy interacting with our spacetime proton:
At any given time, only 25% of the energy field interacts with the spacetime proton. The remaining 75% contributes nothing to the value of K. To drive this point completely home, let's cut up the spacetime proton and evenly distribute it within the cubic meter:
As you can see, it matters not how we distribute the matter energy and spacetime energy. In the case above, only 25% of the total matter energy interacts with the spacetime energy. Our constant of .25 remains constant. Now, let's stop pretending and let's replace the .25 constant with the volume of a nucleus, which is approximately e-45 m^3.
If e-45 is the right constant, then that means only e-45 of the energy contained in T44 contributes to K. Let's check it:
It looks as though it's in the ballpark of the actual constant used in Einstein's field equations. So it's highly plausible that gravity is weak due to energy interacting with very weak spacetime energy.
Step one: Beginning with Einstein's field equations, derive the Scharzschild radius (equation 13 below):
Next, we call on Pythagoras and a right triangle to derive a basic metric equation (equation 15 below):
Using the same right triangle we derive the Lorentz factor (equation 19 below):
Now check out equation 20:
Because of equation 20, we can make a substitution and derive equations 22 and 23:
Equations 22 and 23 allow us to make more substitutions. The result is something that resembles the Schwarzschild metric (equation 24):
Here's the actual Scharzschild metric:
We can replace equation 24's cdt' with dr (differential radius) to get the following:
It would be great if the middle term (vdt)^2 had a plus sign instead of a minus sign in front of it. With some trigonometric slight of hand we change the minus sign to a plus. The result is equation 29:
So far we've used a triangle with only two space dimensions. We are one dimension short, but we can fix that:
Each dimension in space is a hypotenuse of a right triangle with two other dimensions which can replace the hypotenuse. We make a final substitution and we get the Schwarzschild metric (equation 32). Schwarzschild used spherical coordinates. For clarity and to help you visualize this type of coordinate system, I provide the diagrams below. The first two diagrams show the front and side view of a sphere of spacetime with a mass in the center. The position in spacetime is given by the radius (r), the first angle (top diagram), and a shorter radius (rsin[first angle]) and the second angle (bottom diagram).
The variables used in the Schwarzschild metric, however, are differential--a tiny piece of the radius and each angle. The value of each space variable is indicated in red below:
If we take the limit of these variables we get a point in spacetime indicated by the red dot in the diagram below:
The objective is to figure out the spacetime curvature in that tiny (red) region of space. To solve the field equations, we need to know the metric tensor components; i.e., the g's.
We can find the value of each g component within the Scharzschild metric:
Thus the metric tensor is as follows:
With the information we have, we can derive equation 42 below.
We can solve for R44--the spacetime curvature--by plugging in the mass (m) of a star, planet or black hole; the volume (V) of the space, mass and energy within an imaginary sphere with radius (r); and radius (r).
According to the textbooks, the principle of equivalence asserts that gravity and inertia are one and the same--not similar, proportionate or related to each other, but the same. According to the principle, you could kidnap some scientists by smothering each of them with an ether rag, place them in a rocket ship with an acceleration of 9.8 meters per second per second. When they regain consciousness, they won't be able to tell if the gravity they feel is from the earth or the rocket ship--there are no windows for them to look outside.
As the video above says, they can drop a pen and it will fall to the floor like a pen on earth. They are free to do whatever experiment they wish--and they won't be able to distinguish earth's gravity from the rocket's acceleration? Hmmm ... let's take a closer look at how a rocket propels itself:
According to equation 1) above, rocket fuel and oxygen enter the nozzle at a certain mass-flow rate and velocity, then exit the nozzle at a higher velocity and final mass-flow rate. Add to that the pressure difference. The total is the thrust force. Recalling Newton's third law: this is the action. The equal and opposite reaction is the rocket accelerating.
Equation 2) shows that the perceived gravity inside the rocket is acceleration (g). Since action and reaction are equal and opposite, we can set the thrust force equal to the total mass of the rocket (passengers, supplies, fuel included) times the acceleration g:
Doing a bit of algebra gives us equation 4)--the function g. Let's compare equation 4) to equation 5) below:
Equation 5), of course, is Newton's law of gravitation. Let's assume the scientists aboard the rocket are aware of equations 4) and 5). They're also aware of equation 1) that shows fuel mass flowing out the rocket's nozzle. How hard would it be for them to figure out that they are in a rocket ship and not in a sealed box on earth?
Not hard at all! In fact, all they have to do is weigh themselves repeatedly over time. The rocket's total mass (m) is decreasing thanks to the burning fuel exploding out the rocket's back end. This would cause the rocket's acceleration (g) to increase. So when the scientists weigh themselves they notice over time they are gaining weight (scientists' mass * increased g). Either their Weight Watchers diet is failing or they are in an accelerating rocket.
Unlike the rocket, earth's mass is virtually constant. If the scientists weighed themselves there, they would notice little or no change (assuming they didn't quit their diet).
Now, take another look at equations 4) and 5). Notice how mass (m) is in equation 4's denominator, but equation 5) has a mass variable in its numerator. Any change in mass would yield opposite results when comparing the two equations. If mass increases, the acceleration (g) in equation 4) goes down, but equation 5's acceleration (g) increases--and vice versa!
To claim the g in the rocket is indistinguishable from the g on earth requires us to ignore the physical processes involved. Prior to any mass loss, if one drops a pen in a rocket accelerating at 9.8 meters per second per second, the pen's inertia keeps it in place and allows the floor of the rocket to accelerate up to it. A casual observer aboard the rocket will most likely get the impression the pen dropped to the floor. Is this really sufficient evidence to support the claim that inertia and gravity are one and the same? If so, one could also claim that gravity is not a genuine force; it's a byproduct of another force such as the rocket's thrust force.
If gravity and inertia are the same, it should be possible to fool the kidnapped scientists aboard the rocket. Our first attempt failed--the rocket lost some of its mass causing g to increase. Let's try a different approach: We smother the scientists with ether rags and place them in a rotating, donut-shaped space station. When they regain consciousness, it is hoped they will fail to distinguish between the angular acceleration (g) and the earth's gravity.
The advantage of this new arrangement is the space station stays in one location, making it easy for a maintenance ship to top off the station's fuel tank. We avoid a mass decrease, due to fuel consumption. This should provide an artificial gravity indistinguishable from earth' gravity, right? But then something very routine happens: the scientists place their garbage and waste in the disposal chute. The waste is jettisoned into space. The scientists are not aware of this. For all they know, the garbage goes to a dumpster here on earth. But once again, when they weigh themselves, their respective weights increase.
Curses! It's that bloody conservation-of-momentum rule! When overall mass decreases due to garbage disposal, the velocity increases, so does the angular acceleration. If the scientists were on earth, jettisoning waste would still decrease the space station's mass, but the overall mass of earth would remain virtually the same, since the garbage ends up in a dumpster on earth. As a consequence, earth's gravity maintains its value.
Then again, if the garbage stays on the space station, the perceived gravity there should be the same as earth's, and, we should be able to say with confidence, "Inertia and gravity are one and the same." But before we get too cocky, let's put this claim to another test. Imagine the earth and a red meteor coming together. We can interpret this event in one of two ways: the meteor is falling to earth, or, the meteor is at rest and the earth is moving toward it.
If inertia and gravity are the same, then it should be possible for a blue meteor to be at rest on the other side of the earth. The earth should move toward both meteors:
No force acts on the meteors--they are at rest, floating in space. The force, whatever it may be, is moving the earth in opposite directions to meet the meteors. Then again, maybe it doesn't really work that way. Could it be that the meteors are really falling to earth?
Assuming the meteors are falling to earth, a force must be causing them to accelerate. We call that force gravity. It seems clear at this point that gravity and inertia are not one and the same but are only similar to a casual observer who doesn't ask too many questions.
There is another key difference between inertia and gravity that involves time ... the time it takes a boson, say, a graviton to notify matter of a change in status in other matter. To properly distinguish between gravity and inertia, we need to make Newton's equation and Einstein's field equations time dependent. The same goes for the rocket-thrust equation.
Take a look at equation 15) above. If the sun's mass (m) were to drastically change in the current time (t), it would take about eight minutes (r/c) for us here on earth to feel the gravitational effects. The change will be felt by us at time t+r/c. Equation 16) concurs. Any change in spacetime curvature won't happen instantaneously. It takes time for gravitons to move through space from the sun to the earth.
Equation 17) tells a different story. If the thrust force changes in the rocket ship we discussed earlier, the acceleration (or perceived gravity) changes instantaneously--there is no need for gravitons to notify distant objects of the change in status, and to tell them to accelerate. For it is the rocket floor that is accelerating to objects at rest. Thus, when inertia mimics gravity, it takes less time.
The diagrams below illustrate the time difference between gravity and inertia:
Notwithstanding the case presented above, it is possible to mathematically prove the equivalence principle. Imagine a mass m. It is equal to itself (see equation 18 below). To get m moving we add some energy.
If we add its rest-mass energy to its momentum we have a total energy of m'c^2 (See equation 19). We perform a little algebra, but when we get to equation 21, there's a problem.
Nothing can go faster than light, so we take the momentum term (mv). We reduce the velocity a little and increase the mass a little (see 22). We do some more steps. At equation 26 we get a velocity that is less than light speed (c). We derive equation 28 which shows mass m increasing to m' when it is going velocity v.
We can conclude that when mass m is in motion, its mass increases. When it is at rest, its mass stays the same. Now let's do the same exercise; only this time mass m is always at rest and never increases. However, we place mass m near mass M which has a strong gravitational field:
Mass m appears to be falling or orbiting but it is really at rest. It's as if mass M is moving instead, but it's at rest also. So what's going on? What's moving are the coordinates of spacetime. Mass m appears to be accelerating because its point of location is accelerating. Anything located at that point, regardless of its mass, will fall at the same rate.
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. 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.
Suppose Alice and Bob can see outside, and, inside each other's ship. 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 Bob sees the whole universe whisking by him, and Alice only sees Bob's ship whisking by, 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 the time a photon takes to cross that distance is longer, but going right to left, distance is shorter and less time is needed. 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:
