According to the theory of general relativity, mass causes spacetime to curve, and spacetime tells objects how to move. If you look at the equation below, it should be obvious why this is the case.
Well ... OK ... it is not obvious. Perhaps we can derive some field equations that are equivalent but more intuitive? Let's start with diagrams A and B below (r=radius; ct=spacetime portion of the radius; ct'=warped spacetime):
Diagram A shows an arbitrary sphere of space with no mass present. The field lines are straight (or flat) and connect the center with the outer edge. The field lines in diagram B curl like waves and pull all the space inward toward the center, shortening the spacetime wavelengths and creating a smaller sphere with more curvature. (To see more details on how this works, click here.)
The Lorentz equation above is pretty straight forward (G=Newton's constant). It shows how ct' is a function of mass (m). Add mass (m) and ct' shortens. We can use this equation as a model for our new field equations. Let's see what we can come up with:
The last equation above is kind of interesting. All the stuff on the left side must equal one. Let's multiply both sides by 8(pi)r, (the derivative of a sphere area) and do a few more steps (E=energy; T=energy density):
Add some Tensor indices:
We now have something equivalent to Einstein's field equations. Notice how each term contains an 8pi factor. We can do away with it.
What was not obvious before is now more obvious. If the stress-energy tensor (Tuv) changes, the spacetime variable (ct') also changes. Any particle in the vicinity will be affected by the changing ct', and move along a geodesic curve.
Notice there are a couple of 1/r^2's that can be factored. Time to dress this puppy up a little bit more:
We've come full circle. We now have a tensor version of the Lorentz factor we started with. To change things up a bit more, let's bring ct inside the parentheses.
Finally, we can name the ct and ct' tensors S and S', respectively:
The universe is getting bigger and colder. We are told that one day it could suffer a heat death. By contrast, the early universe was smaller and hotter. All this implies an inverse correlation between the universe's average temperature and its size.
We might be tempted to model it with the Ideal Gas equation. We could think of the universe as a big balloon filled with gas. As the volume of the balloon increases, the pressure decreases along with the temperature. Below is the equation (P=pressure; T=temperature; V=volume; R=universal gas constant; n=kilomoles of gas):
Right away the equation bursts our balloon. When volume (V) increases, pressure (P) decreases but the temperature can remain constant. So why isn't our universe the same temperature as it was in its early years? Its volume is increasing; its pressure is decreasing; yet, the temperature isn't remaining constant.
So what's the problem with the Ideal Gas Law? We know that pressure is equivalent to energy density, so let's plug energy density (E/V) into the equation in lieu of pressure (P):
The second equation above reveals the problem: The volume (V) in the denominator cancels the volume in the numerator. Change in volume in this case has no effect on the temperature.
We know that entropy increases as the universe expands. Perhaps lower temperature is a function of increased entropy? Let's take a look at the entropy equations (E=heat or energy; s=change in entropy; T=temperature; k=Boltzmann's constant; omega=number of different arrangements of particles):
If we equate the two equations above, we can derive a temperature equation:
Temperature appears to be a function of energy (E) over the number of ways particles can arrange themselves (omega). Once again, temperature is not a function of volume, but it seems like it should be. We know from experience that a candle can heat up and maintain the temperature of pickle jar, but a candle in an aircraft hanger can be a much colder space.
Let's explore the omega variable and see if we can make it a function of volume. Imagine a single particle with one state in a single space (see first diagram below). In that case, omega equals one. Now double the space. The particle will have two spaces it can occupy (see second diagram below). Omega now equals two.
Entropy has increased due to increased space. We can define omega in the following way:
Below are some examples that include the above diagrams, a coin toss, and dice:
Consider the coin-toss example above. Normally we think of a coin as having two states: heads or tails, so omega equals two, right? Well that's only true if the coin lands on the same spot on your floor because there is no other spot it can land. But suppose you have a big floor and you divide it into a grid of 20 spaces where the coin can land. Omega would then be 2^2 * 20 = 40 states.
Now suppose you have the same floor space, but you now have two coins. Omega becomes 1520 states! We are now ready to put together a temperature equation:
Notice the temperature is now a function of volume because entropy is now a function of volume. If the volume increases, the temperature decreases and so does the pressure (Boyle's Law) if the other omega variables and E are held constant. Remember pressure is E/V. The difference here is that E/V is no longer canceled by V*E/V.
Taking the temperature of the Universe or any other system can be done by this system of equations. The first equation is for systems that have more space than particles. The second is for singularities, high pressure systems, where the energy density is high. The third equation allows you to make a quick-and-dirty calculation. It is similar to the Ideal Gas equation albeit it includes entropy.
We also need Heisenberg's Uncertainty Principle (p=momentum; x=position; h-bar=Planck's constant). (To find out how to derive the Uncertainty Principle, click here.) For our convenience we convert momentum (p) to energy (E) by multiplying both sides by light speed (c):
Divide both sides by E:
Multiply the above result by the Schwarzschild radius, then do some algebra to get the Planck length (x with a sub p):
To get the Planck time (t with a sub p), take the Schwarzschild radius and divide it by c:
Shall we call the above result the Schwarzschild time? We also need the other Heisenberg Uncertainty Principle, the one with energy (E) and time (t). Now do some more algebra to get the Planck time:
To get the Planck mass, set the Schwarzschild time equal to the Heisenberg time, then solve for E-squared:
Let's assume E is kinetic energy. Square it, make a substitution, do some more algebra, and get the Planck mass (m with a sub p):
Maybe you've read somewhere that particles can escape a black hole via quantum tunneling, Hawking radiation or pair production; however, according to classical physics, there is no way light can escape--or is there?
Light escapes a gravitational potential if it has sufficient escape velocity. In terms of energy, a photon's kinetic energy must exceed the gravitational potential energy. A black hole's event horizon, whose distance from the center is the Schwarzschild radius (r), is the place where kinetic energy equals potential energy.
Theoretically, if a photon passes through the event horizon toward the black hole's center, it can't escape. You will note that the second equation above assumes a photon has mass (bar-m). This should be converted to a mass equivalent: fh/c^2 (f=frequency; h=Planck's constant; c=light speed; G=Newton's constant; m=black hole mass). We can now derive the Schwarzschild radius:
Houston, we have a problem! The second equation above assumes that a photon's kinetic energy is (1/2)fh. Electromagnetism has kinetic energy (photons) and potential energy (which involves charge). If photons are kinetic energy, then it only makes sense to use a photon's total energy (fh) when deriving the radius (r).
If (1/2)fh equals the gravitational potential energy, then fh must be greater than the gravitational potential energy--and the photon should be able to escape.
The math above indeed shows that fh is greater than the gravitational potential. There is another consideration: does the photon have more than zero energy? The last equation above shows that fh would equal (1/2)fh if fh is zero. When a photon enters a gravitational field it gains energy or frequency (f). When it tries to leave, it loses energy or frequency (f). The equations below demonstrate this (d=distance the photon travels; g=gravitational acceleration):
The total frequency (ft) of the photon is the sum of the initial frequency (fo) plus or minus the change in frequency (triangle f). How much energy the photon gains or loses depends on its angle of entry or exit relative to the unit normal line (n). If the photon is parallel to the unit normal (n), cos(0) equals 1, the photon will lose a maximum amount of energy when exiting a black hole. If the photon leaves at an angle, it will lose less energy.
So, if a photon enters the event horizon it will gain energy. When it tries to leave, it will lose energy. If it has more than zero energy, will it escape? Let's compare the escape velocity (c) to the gravitational equivalent. As you recall, we derived this inequality:
Let's multiply both sides by C^2/fh, then simplify:
Since c is greater than .707c, the photon will escape. In fact it will escape from any point that is outside the radius of Gm/c^2--one half the Schwarzschild radius.
If we want to measure, say, the total energy of a field, it seems logical to measure the energy of each and every point in that field and add them all up to get the total. Unfortunately the field has an infinite number of points within its space, each with a finite amount of energy. The total energy is infinite according to our calculation--but when we actually measure the total energy, we get a finite value.
So let's forget about points in space. Instead, let's measure each wave frequency, then add them all up. Unfortunately, we get infinity again. We get infinity if we treat each particle-wave as though it had positive energy. Wouldn't it be great if a big chunk of the energy were negative? It would cancel the infinity and we would be left with the energy we actually measure.
Let's take a closer look at particle-waves. Below are a couple of waves. They each have a different frequency and amplitude. Note how each half cycle is either plus or minus, but not both; i.e., the plus and minus do not cancel each other. If we added the energies of these waves together, the total would be be the sum of their energies--or would it?
We know that each wave has its own wave function:
Let's do an experiment: Take a bunch of waves (up to an infinite number) with varied amplitudes and wavelengths, and run them all together:
When we looked at individual waves, we noticed that the plus half of the cycle never cancelled the minus half. The diagram above shows destructive interference when a bunch of waves are mixed together--the pluses cancel the minuses. Thus positive infinity is cancelled by negative infinity. Hopefully, we have something finite left over that agrees with actual measurements. Let's check and see.
We will now calculate the total energy in a vacuum. We will add up each integer (n) from minus infinity to plus infinity, and multiply the total by 1/2 the frequency (f) times Planck's constant (h).
Let's calculate the integral:
The constants (c) cancel. Next, we divide this definite integral into two parts: negative infinity to zero, and, zero to infinity.
This amounts to negative infinity squared plus positive infinity squared, which is equivalent to adding infinity to minus infinity--and the following equation where the epsilon limit is zero:
Substitute the Taylor series of each exponent and simplify:
As you can see, we end up with no infinities--just a finite energy.
According to Phys.org "[t]he universe may have existed forever, according to a new model that applies quantum correction terms to complement Einstein's theory of general relativity."
(Click here to read the article by phys.org.)
The universe-existed-forever hypothesis has been thoroughly debunked. I am amazed someone dusted it off and is trying to resurrect it. If the universe existed forever, then anything that is probable has happened already, since an infinite amount of time has passed, but wait! Infinity is not a finite number and it is improper to treat it as such. By its very nature, infinity never ends, so infinite time cannot pass, and the universe has not existed forever.
Even if we were to entertain the notion that the past could somehow stretch back to infinity, then it took an infinite amount of time to get to this moment, but infinity never ends, so this moment isn't here yet. But wait! It is here--so much for the universe-existed-forever hypothesis.
Here is another amusing gem from the article: "These terms keep the universe at a finite size, and therefore give it an infinite age. " Uh oh! I guess there's no Hubble constant or red shift. But wait! There is. The evidence shows the universe is expanding. The "terms" don't trump the evidence.
"Anti-matter looks like matter going backwards in time," is a quote I've been hearing lately. Here's a question that popped into my head: What does matter going back in time look like? The typical response is it looks like a rewinding video. However, if matter truly goes back in time it would simply vanish or would exist in the past, not the present or future. It would be unobservable. All we can see is it going in the opposite direction in forward time.
Case and point: positrons have been experimentally trapped for as long as 16 minutes. How is that possible if they go back in time? To exist for 16 minutes, they have to go forward in time for 16 minutes. If you ask me, anti-matter looks like matter with an opposite charge--FULL STOP.
Anti-time does not take you to the past, but rather, it makes the past possible. If there was no anti-time there would be nothing to cancel the current moment in time. The moments would pile up. You would not only be living in the present moment, but all your past moments as well. All your memories would be all too real. Hopefully, for your sake, they are good memories.
To demonstrate how anti-time works, let's start with a time-moment represented by an arrow:
The plus sign indicates that it is a positive, forward time-moment. The time-line arrow to the right is how we normally think of time: just a straight line going up in this instance. Now let's add a second time-moment and see what happens:
The first time-moment is canceled by an anti-time-moment (arrow pointing down). That leaves us with the present moment. We get a similar result when another time-moment is added:
But why isn't there an anti-time arrow to cancel the current time-moment? Well, at the beginning of time, there was no past, so time could only go forward, but once it went forward a little, there was some room to go back and still room to go forward--so we get a forward arrow followed by a backward arrow and another forward arrow, etc. As a result, we get time that has a forward bias.
This is a pretty bizarre theory! Can it be tested? Sure. Ask yourself, "Is history history or is it still happening?" If history is history, obviously something is cancelling those time-moments that would otherwise pile up. In mathematics we use a minus sign to cancel a plus sign, so it stands to reason that -time cancels +time leaving us only with the present.
Well ... OK ... it is not obvious. Perhaps we can derive some field equations that are equivalent but more intuitive? Let's start with diagrams A and B below (r=radius; ct=spacetime portion of the radius; ct'=warped spacetime):
Diagram A shows an arbitrary sphere of space with no mass present. The field lines are straight (or flat) and connect the center with the outer edge. The field lines in diagram B curl like waves and pull all the space inward toward the center, shortening the spacetime wavelengths and creating a smaller sphere with more curvature. (To see more details on how this works, click here.)
The Lorentz equation above is pretty straight forward (G=Newton's constant). It shows how ct' is a function of mass (m). Add mass (m) and ct' shortens. We can use this equation as a model for our new field equations. Let's see what we can come up with:
The last equation above is kind of interesting. All the stuff on the left side must equal one. Let's multiply both sides by 8(pi)r, (the derivative of a sphere area) and do a few more steps (E=energy; T=energy density):
Add some Tensor indices:
We now have something equivalent to Einstein's field equations. Notice how each term contains an 8pi factor. We can do away with it.
What was not obvious before is now more obvious. If the stress-energy tensor (Tuv) changes, the spacetime variable (ct') also changes. Any particle in the vicinity will be affected by the changing ct', and move along a geodesic curve.
Notice there are a couple of 1/r^2's that can be factored. Time to dress this puppy up a little bit more:
We've come full circle. We now have a tensor version of the Lorentz factor we started with. To change things up a bit more, let's bring ct inside the parentheses.
Finally, we can name the ct and ct' tensors S and S', respectively:
