How Entropy, Temperature and Chemical Reactions Impact Time Dilation

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:

Entropy Succeeds Where the Ideal Gas Law Fails

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.