Our expanding universe is not thought to be relativistic. Technically, the galaxies receding away from us are at rest. The space between them is expanding and creating the impression that the galaxies are in motion. However, each galaxy can be treated as a clock, and, it is not likely that any of these clocks tell time at the same rate.
Using the diagram below as a reference, consider Alice and Bob living in a static universe. Alice lives in galaxy A and Bob lives in galaxy B. Each has a photon gun and shoots one photon per second at the other. The vertical red arrows represent the photons fired from one galaxy to the other. When the first photon from Alice (after traveling light years) finally arrives at galaxy B, Bob intercepts it and records it as a unit of time. Since Alice fired photons at regular intervals, Bob only has to wait one second for the next photon and another second for the next one after that and so forth. He receives one photon per second from Alice, and, for similar reasons, Alice receives one photon per second from Bob. This is how Alice and Bob keep track of each other's time rate.
You will note there are some horizontal red arrows at the top of the above diagram. Also located at galaxy A is Gertrude. She also has a photon gun. She fires one photon per second at Alice. Alice therefore intercepts and records one photon per second from Gertrude. Bob also receives photons in the same manner from Norbert who is also lacated at galaxy B (see horizontal red arrows at the diagram's bottom). This is how Alice and Bob keep track of their own time rates.
When the universe is static, both Alice and Bob receive fired photons from all directions at a rate of one photon per second. All clocks seem to agree and equations 1 and 2 provide the relevant math. But suppose the galaxies move away from each other at velocity v due to expanding space. The next diagram represents this scenario:
The vertical red arrows show that the photons take longer to reach their respective destinations and fewer photons are received per second compared to photons represented by the horizontal arrows. Since each photon is counted as a unit of time, Alice and Bob have the impression that the other's time rate is slower. Since Gertrude is located in the same galaxy as Alice, she isn't receding from Alice the way Bob is, so Alice sees no change in her time rate, since she receives the same number of photons (time units) from Gertrude. Ditto for Bob and Norbert at galaxy B.
If we do a little algebra we can derive the Lorentz equation from equation 3 above. The final result is equation 10:
Equation 10 shows each galaxy's proper time (t') shrinking as they accelerate further apart. It does not matter if the galaxies are technically at rest, since the space inbetween and the photons are not at rest. If the space is at rest, here's the result after firing a photon gun for five seconds:
Notice the steady stream of photons. This is where t' = t. The next diagram shows what happens if space expands at a steady rate of v. The photons still have a steady rate but the interval between them has increased, so they are not counted as frequently. Time t' is less than t.
Finally, the next diagram is the most realistic, since the expansion of space is accelerating. Here the the interval between photons continues to grow and they are counted less and less frequently. Time t' is shrinking.
When Alice and Bob are far enough apart, neither will receive any photons from the other. Time t' will be zero. Thus the expanding universe is relativistic if one keeps track of the various time rates at different distances.
"Reality is merely an illusion, albeit a very persistent one."--Albert Einstein
It is apparent from the above quote that reality distinguishes itself from ordinary illusions by its persistent nature. Reality is true even if you choose not to believe in it. Thus, if we are trying to settle the question whether time is real, we should examine time to see, if like reality, it too is a persistent illusion.
The time variable is very persistent and ubiquitous in so many physics equations. On that basis we can claim it's real, but just how real is it compared to things like matter, energy, mass, distance, force, your neighbor's barking dog? It's not like we can grab time out of the air, hold in hand and look at it like a hunk of clay. However, like clay, time can be stretched or compressed depending on how close to the speed of light you are traveling. How is that possible if time is just a product of human imagination? Surely any relative differences in time would also be limited to the human imagination and not an empirical reality.
When examining time, one has to make the distinction between how we measure time and time itself. One popular argument claims that if all particles in the universe stopped changing their states and came to rest, time would stop and cease to exist. This seems reasonable. If nothing happens, how would we experience the "flow of time"?
But what if the "flow of time" is just our experience when we measure time? If your watch stops, you don't assume that time has stopped. You only assume your ability to measure and experience "the flow of time" has stopped. So it seems reasonable to assume that time continues even if every particle comes to a grinding halt. Think of a stalled universe as one big watch that stopped.
So what exactly is time if not a flowing, evolving, ever-changing environment of entropy? The following equation, for me, is a real eye-opener and has forced me to rethink time:
E is energy and psi is the wave function, tp is the Planck time, G, c, and h-bar are the gravitational constant, light speed and Planck's constant, respectively. The above equation shows that it doesn't matter how much or little energy there is, or whether states change frequently or not at all, whether they go forward or backward. No matter what values you plug in for E and psi, you get forward time, specifically, the Planck time. Imagine having zero energy, zero change and still having a Planck time. How is that possible? Thought experiment time:
Imagine a universe with no energy, no distance or space, no charges, no masses, no momentum, no oscillators--just a single zero-dimensional point, a singularity. According to the above equation, time still exists. Why? Because the singularity is persistent--it is real. What exactly is this singularity? It's literally nothing ... except time at a single reference frame, at a single point. No clocks, no observers, just pure time.
Time is so essential to reality, that no "persistent illusion" can persist without it. Time can persist without anything else we would deem real, but nothing we deem real can persist without time. The words "reality," "existence," "persistence," "presence" all imply the passage of time. At this juncture, one could argue that time is not only real, but reality's most essential component. And, when we perform the above thought experiment, we witness time in its purest form.
So if you ever encounter a skeptic who believes time isn't real, that particles exist without time, ask the following question (but don't hold your breath):
How do we measure time? Normally we take some event that happens over and over again, and, we assign it a time unit. For example, the complete rotation of the earth we call a day. We can take that time unit (or any fraction thereof) and assign it to other events--that may or may not be periodic--such as a chemical reaction, for instance. We say the chemical reaction happened in time t.
What if that chemical reaction were to slow down? Can we say its rate of time has slowed? If we define time as the rate of change, then the change in the rate of chemical reactions, entropy, or the earth's rotation would indeed impact the rate of time.
Consider the famous twin paradox, where one twin boards a rocket that hurls him into outer space close to light speed. According to Einstein, his time slows, he ages more slowly than his twin back on earth. His body's biochemical reactions are slower, entropy is reduced--at least that is the implication.
When he arrives back on earth, he will be younger than his twin brother. But his twin has a plan: while he's out in space, his twin cryogenically freezes himself. His twin will be thawed out by the time he gets back to earth. If his twin's plan works, they will both be the same age. Now, did the twin on earth reduce his time rate? Can the rate of time be reduced by means other than high velocities and mass density? This post shall address these questions, but first, let's define the variables we will use:
Let's start with entropy. If we somehow reduce the rate of entropy, will the time rate also be reduced?
If we take the Boltzmann constant (which has entropy units) and multiply it by the unit-less Lorentz factor's squared reciprocal, we derive equation 2 below:
Equation 2 shows that entropy is reduced when velocity (v) is increased. The time rate is also reduced. So far it appears we have a correlation between entropy and time. If equation 2 is valid, we should be able to use it to derive a more standard entropy equation:
Equation 6 above confirms the validity of equation 2. So we can say the twin in outer space, traveling near light speed, has reduced his entropy. Now the twin on earth wants to freeze himself, i.e., lower his body's temperature. Will this reduce his entropy? Equation 8 below confirms that it will. From 8 we derive equation 12 which shows lowering the temperature (T) reduces the time rate.
According to equation 12, the twin on earth will age more slowly. Does this imply the rate of his body's biochemical reactions will slow down?
Equation 13 is the Arrhenius equation, where k is the rate of a chemical reaction. When temperature T is reduced, so is the rate of the chemical reaction. From 13 we derive a new entropy equation at 19:
Equation 19 tells us that when the rate of a chemical reaction is reduced, so is entropy. And the reduced chemical-reaction rate reduces the rate of time:
So the age difference between the twins could be nil when the space twin arrives back on earth. If the earth twin cryogenically freezes himself, he may be younger than the space twin. Both twins have found a way to reduce their respective time rates, they have found ways to time travel into the future. Here are four ways to reduce the time rate:
Let's compare the point particle and the string. A point particle's dimensions all have a zero limit. By contrast, a string has a zero limit for its cross-section dimensions and a Planck-length limit on just one space dimension. So one could ask, if the Planck length is the shortest length, then how is it that a string has shorter dimensions along its cross section? Does space follow different rules along, say, the x-axis and y-axis than it does along the z-axis? It doesn't seem likely that it would--and we will mathematically prove it, i.e., we will disprove the string. First, let's define the variables we need:
Below we we define the Lorentz factor at equation 1; the de Broglie wavelength at equation 2; The Planck length at 3; and the Planck mass at 4. At equation 5 we express the Planck length in terms of the Planck mass and de Broglie function. The Planck mass is multiplied by c to give the maximum momentum theoretically possible: the Planck momentum.
If the momentum were larger than the Planck momentum, The Planck length would not be the shortest length possible. If equation 5 is true, then equation 10 below must also be true, but is it really?
If we check how much energy is involved in bringing, say, an electron up to the Planck momentum, we discover the amount falls far short of infinity. In fact, the the energy amounts to only around 300 lbs. of TNT. If we could somehow squeeze more energy into the particle, surely we could increase the momentum. The result would be equation 15 where the wavelength is less than the Planck length.
Unfortunately the particle's momentum is not only restricted by the light-speed barrier, but also by the Planck temperature, which we will discuss later. Right now, let's see if the Planck units are valid. To build the Planck units, the following assumptions are made (see equations 16 & 17):
But then there's the Heisenberg uncertainty principle. At equation 19 we see that h-bar (the reduced Planck's constant) isn't really the smallest value possible. H-bar/2 is smaller. If we use this smaller value, we can derive new and smaller units (see 23 and 24):
Let's plug in these new units into the uncertainty relation (see 25). Now, if we assume the Planck mass is valid, then the shortest length is half the Planck length (see 26). OK, we broke the Planck length barrier. Can we do better? Yes! According to special relativity, a particle approaching light speed has an upper mass limit of infinity (see 27). If we could somehow harness the energy of the whole universe, then surely the minimum possible length would be far shorter than half the Planck length (see 28 to 30).
OK, now is a good time to address the Planck temperature. The Planck temperature is allegedly the hottest temperature that can be achieved. If such is the case, then our particle maxes out at 300 pounds of TNT worth of energy and corresponding momentum, i.e., the Planck momentum.
The Planck temperature is defined at 31 below. Notice how equation 31 does not take relativity into account. Let's factor in relativity (see 32). At equation 33 we end up with a temperature that has an upper limit of infinity.
Some physicists have postulated that the Planck mass represents the mass of the smallest black hole possible. If such a black hole is at rest, we can certainly use equation 31 to find the Planck temperature. But what if the black hole is not at rest? Then its energy (E) would be equation 35 below. Once again we end up with a temperature greater than the Planck temperature (see 36,37). Even if we limited the energy to kinetic energy only, it seems highly probable such energy would exceed the Planck energy (Planck mass X c^2). Thus it seems probable the highest temperature would exceed the Planck temperature (see 38 & 39).
OK, we've laid the groundwork needed to disprove the strings of string theories. We begin our disproof with Gay-Lussac's Law. Using the Planck length and Boltzmann's for the constant and doing a bit of algebra we derive a temperature equation at 44:
Both a string and a point particle have zero volume due to one being one-dimensional and the other zero-dimensional. Zero volume yields infinite temperature (see 45 to 47). If we assume the Planck temperature is the highest possible temperature, then one-dimensional strings (and point particles) don't exist. If the string (or point particle) is at rest, there's infinite uncertainty re: the temperature, thus possible temperatures that exceed the Planck temperature (see 48, 49).
Now let's attempt to save string theory. If we assume there is a minimum distance greater than zero, then that minimum distance should exist in all directions. Given that assumption, is it possible to build a string? For illustrative purposes, let's assume the Planck length is the shortest distance possible. Here's what we get:
At 50 above we have the temperature equation. At 51 we shrink the dimensions down to the Planck length. We end up with a small volume of matter (circled in red) rather than a string. We use this object to derive equations 52 to 54. Of course, given what we have covered in this blog post, the Planck length could be cut by at least one half. However short the actual shortest distance, it is abundantly clear it doesn't make a one-dimensional string.
Is faster-than-light speed or warp drive possible? If Scotty were here to explain how warp drive works, he'd probably say, "Sorry, Capt'n, we don't 'ave enough power!" But assuming we have enough power, warp drive shrinks the spacetime in front of the Enterprise while it stretches the spacetime behind it. That way the Enterprise is closer to its destination and further from its starting coordinates without having moved through space. According to Scotty, this prevents time dilation, so the time on the Enterprise stays in sync with the time on earth.
You may be familiar with the twin paradox where one twin stays on earth and the other travels through space at high speed. According to special relativity, time on the spacecraft passes more slowly than the time on earth. When the space-twin returns home, he finds his brother has aged considerably while he has aged hardly at all. Warp drive allegedly solves this problem by not requiring the Enterprise to move through space. However, notice spacetime must be altered for warp drive to happen. This implies time as well as space is being altered, since, according Einstein, space and time are intrinsically connected.
Einstein also insisted that nothing goes faster than light, but as we shall see, the light-speed barrier can be beaten under the right circumstances. First, let's define the variables we are going to use:
Equation 1 below contains the famous Lorentz factor. At equation 2 we can see why Einstein believed nothing goes faster than light. Any particle with mass would be infinitely massive at light speed, which implies that an infinite amount of energy is needed to get the particle up to that speed. Since energy is limited, the logical conclusion is light speed for anything with mass is out of the question.
What Einstein insisted, is true if velocity (v) is not a function of mass. But suppose velocity is a function of mass, like in a gravitational field, for instance. What happens then? We know our earth orbits our sun. We know each is affected by the other's gravity. The equations below assume the sun's mass (M) and the earth's mass (m) increase as the Lorentz factor says they should.
So now the velocities are a function of mass, but this creates a runaway positive feedback loop!
The red arrows above show that an increase in the earth's mass causes an increase in the sun's mass which in turn causes another increase in the earth's mass and so on. According to NASA's database, the masses of the earth and sun do not need to be continually updated. Those masses stay reasonably constant. The same is true for other celestial bodies within our solar system. Also, at equation 8, notice how the sun and earth start out with less energy than they end up with. This violates energy conservation. That can only mean one thing: equations 6 and 7 are wrong.
We know equation 1 is right, however. So let's sort out this conundrum. We can derive equation 1 if we assume momentum is conserved. At equation 9, mc is constant and m' increases when velocity u decreases. Velocity u is defined at equation 10:
Using a little elementary algebra we derive equation 1 and restate it at 13 below:
So why doesn't equation 13 work for a gravitational field? Gravity is truly something special. Momentum is not conserved (see inequality at 13b). Different masses fall at the same rate. When a falling mass accelerates, the instant velocity (v) at a given point along the path is the same no matter how big or small the mass (m). Using this fact, we derive equations 20 and 21 below:
At equation 20 and 21, notice how the masses of the earth and sun don't change. No more positive- feedback loop and energy is now conserved.
Now, let's see what happens when a mass (m) moves at light speed in a gravitational field:
Mass m does not change! It remains the same no matter how fast it moves! This implies that infinite energy is no longer needed to achieve light speed. Higher speeds are possible when you consider that mass M has no upper limit and radius r has a zero-limit ( see equation 20). The only energy needed is GMm/r.
We know that time is slower where gravity is stronger and faster where gravity is weaker. If we replace mass (M) with time (t) at equation 21 we can derive equation 26:
Equation 26 confirms that time dilates in a gravitational field. This makes total sense when you consider gravity is a function of warped spacetime. This also explains why the Enterprise's warp drive must somehow place the ship in a protective spacetime bubble where the spacetime within is not warped, so time can be the same on the ship as it is on earth. Only the spacetime surrounding the bubble is warped.
Now, if mass m goes no faster than light speed, energy conservation is not a problem, but faster than light speed suggests a problem. If there are speeds faster than light then E does not necessarily equal mc^2. You would start out with that amount of energy but end up with more energy. Faster-than-light velocity might yield E = m(greater than c)^2. But if we begin with Einstein's complete energy equation (equation 27), we can derive equation 34 below:
At equation 34 we see that energy E is conserved no matter how fast mass m moves. At equation 37 we see why. Any increase in gravitational velocity is offset by a decrease in time velocity (u). When you add the square of those velocities you get c^2. NICE!
OK, another question: does time run backwards during faster-than-light spacetime travel? We work out the math below:
The variable u is time velocity. If it is negative, then proper time (t') is running backwards (see equation 41). However it appears earth time (t) is still running forward. At equation 44 we see that negative u cancels itself.
Imagine a spaceship (not protected by a warp-drive bubble) moving faster than light. Let's plug in some specific numbers and see what happens:
At equation 47 above, we see that two solutions are possible: positive and negative proper time (t') and imaginary numbers to boot. We factored out i to make the number real, but even then we have two possible solutions: proper time is running forward or backward. Forward time is more probable. To see why, click here. In either case, equation 53 confirms that earth time is not altered. It is still moving forward at a rate of t:
Apparently Hollywood and sci-fi novels failed to consider that a person going back in time is not exempt. If you went back in time, you would grow younger and younger. You would only be able to go back as far as your conception. Before that, you didn't exist. This might explain why we don't get tourists from the future.
Additionally, you are only going back in time in your reference frame. The rest of the universe is moving forward in time. So, if you are a space-twin, when you return to earth, your brother will still be years older than when you left, and, you will be much younger than when you left--perhaps just an embryo! The math below confirms this:
Thus the warp-drive bubble that protects the star ship Enterprise is an absolute necessity! We need to control time. Once we have this technology, we need the finite energy of a black hole to hurl across the universe faster than light!
Update: If a spaceship can go faster than light in a gravitational field, does light also go faster than light? Let's see what the math says. Here are the variables we will need:
Equation 58 is velocity according to special relativity (see derivation below). As the velocity v grows, so does the mass, creating an upper speed limit of c or light speed.
Equation 59 is velocity in a gravitational field. As we demonstrated above, mass m does not grow as velocity increases. There doesn't seem to be any upper limit to this type of velocity.
Equation 60 is light speed. If energy is added, the frequency increases, but the wavelength shortens. No matter how much energy is added, the increase in frequency is offset by the decrease in wavelength. As a result, a photon can't go faster than c.
At 61 below, we compare the momentum of a falling body to a photon's. The falling body's velocity will increase in a gravitational field with no apparent upper limit. The photon, on the other hand, will experience a frequency increase, but no apparent velocity increase.
"Captain," Scotty might say, "we have a problem!" If fermions in a system go faster than light, then the system's photons must also go faster than light or be left behind. If they are left behind, there's an energy loss that might reduce the fermions' speed, keeping it within the light-speed limit, and/or, the fermions could lose mass. No doubt Einstein is smiling in his grave. Anyway, as promised, here is the derivation of equation 58:
Update: What about galaxies beyond the cosmological horizon? Aren't they hurling away from us faster than light? Perhaps, but our galaxy's photons can't reach them and their's can't reach us, but said photons can reach galaxies that are closer. Locally, each galaxy is moving well under light speed. The relative velocity (v) of a galaxy is determined by Hubble's constant (H) and distance (D). Where v = HD. If HD < c, a system's photons can come, go, or stay. Within any galaxy, HD is less than c.
Update: Below is further proof that c is the top speed in our universe:
