Saturday, December 15, 2007


Last time we explored Einstein’s special theory of relativity and saw how observers in relative motion with each other measured different times and distances between events. This created a problem for causality because there was no absolute “now”, so observers could disagree on the sequence of events and could possibly even observe an effect happen before its cause. The only way we were able to resolve this problem and preserve causality was to presume that information could not travel faster than light. We also saw how Einstein took a very positivistic approach to his new concept of space and time by only dealing with observable phenomena like the ticking of light clocks, the paths and timings of light beams, and the lengths of objects measured directly with yard sticks. Einstein made no reference to an absolute space or time, that we presume exists, but which we cannot directly measure.

People frequently sum this all up with the phrase that “everything is relative”, meaning that anything goes. But that is not the case at all. It turns out that my distances and times may disagree with yours, but I will always be able to convert them into your distances and times through the use of a set of handy conversion formulas known as the Lorentz transformations. For example, you might claim that a certain classic Led Zeppelin CD costs 10 euros, while I maintain that it goes for just under $15 U.S. dollars. That is because of the current currency transformation:

1 euro = 1.47139 U.S. dollars

We will always disagree on the exact numerical cost of items when you use euros and I use dollars, but we will always be able to convert one cost to the other with a transformation formula like the one above.

The Lorenz transformations all have the following term in them:
√1 - v²/c²

where v is the velocity of your friend’s spaceship speeding by, and c is the velocity of light. When v is 10% of the speed of light, then

v/c = 0.1


v²/c² = 0.01

√1 - v²/c²   =
√1 - 0.01   =
√0.99   =   0.99498743710...   ≈ 1

For relative velocities under 10% of the speed of light, the Lorenz transformations are similar to the situation where:

1 euro ≈ 1 U.S. dollar

and everybody agrees on the numerical cost of goods. In such a situation, everybody agrees on the distances and times between events to a very good approximation, and Newtonian mechanics makes very accurate predictions.

The Newtonian idea of a fixed and absolute space, providing a stage or background upon which the events of the Universe could unfold, was so appealing that it was very hard to part with. In Einstein’s original positivistic concept of relativity, there is no stage; there is only a dialog between the players in terms of their relative observations of each other. In 1907, Hermann Minkowski, a former university math professor of Einstein, realized that we could still have a stage for our players if we combined space and time into a 4-dimensional spacetime. By adding time as a 4th dimension to our traditional 3-dimensional world, we could unite space and time into spacetime. But how can you possibly combine the 3-dimensional world of x, y and z coordinates that are measured in feet, yards, miles, or meters with time that is measured in seconds? To understand this better, let us turn to an example from American football. The playing area of an official NFL football field is 100 yards in length and 160 feet in width. For some unknown reason, distances along the x-axis of a football field are measured in yards, while distances along the y-axis are measured in feet and inches from the sidelines. For example:

NFL Hash marks are the narrowest of all the levels of football play. They measure 70 feet, 9 inches from the sidelines and are 18 feet, 6 inches wide. (same as the width of the goal posts)

Now imagine that you are at a football game with a friend. The both of you watch a play where the quarterback finds that nobody is open and decides to run the ball himself (see top section of Figure 1). The quarterback receives the football at point A and proceeds to run in a straight line to point B where he is tackled. You observe that the quarterback made a gain of ∆x yards during the play, and that he managed to run ∆y feet across the field as well. Recall that ∆y gains across the field are meaningless in football, so long as you stay within the playing area defined by the sidelines. It turns out that your friend has started the celebration party a little too early with the help of Mr. Jack Daniels, and is in no condition to make observations, but does so just the same. Your friend sees the same play that you just witnessed, but is looking cockeyed at the field (see bottom section of Figure 1). Your friend observes that the quarterback made a gain of ∆x` yards during the play, and that he managed to run ∆y` feet across the field at the same time. When the two of you compare notes, you find that you disagree about both the X-dimensional gains in yards and the Y-dimensional gain in feet:

∆x ≠ ∆x`
∆y ≠ ∆y`

Being in a better mental condition than your friend, you remember the Pythagorean theorem from high school. You realize that if you both make your X-dimensional and Y-dimensional measurements in feet rather than in yards and feet, then you can use the Pythagorean theorem to find the total distance D that the quarterback ran. The total distance of the play should be the same for all observers no matter how they look at the playing field:

D² = ∆y² + ∆x²
D² = ∆y`² + ∆x`²

And sure enough, when all of the calculations are completed, you both agree that the quarterback ran a total of D feet before being tackled, even though you still disagree on the X-dimensional and Y-dimensional gains of the play.

Figure 1 (click to enlarge)

Let’s extend the above example, which was conducted in space, to a similar scenario conducted in spacetime. The top section of Figure 2 is a spacetime plot of the next football play. Because it is hard to plot more than 2 dimensions at a time on a piece of paper, in a traditional spacetime plot we usually plot one of the normal 3-dimensions X, Y, or Z of space vs. time T. The normal convention is to plot T vs. X, and we shall do the same. The X-axis in the spacetime plot represents the distance down the length of the football field, as it did in Figure 1 above, and T represents the time of the play. In the second play, the quarterback again decides to run the ball, but this time he runs straight down the field in the X-dimension from point A to point B in a straight line and does not run across the field at all. With the T-dimension we measure the time it takes for the quarterback to run the play. As before, we measure the progress of the play as motion through spacetime with ∆x and ∆t components.

Now imagine that you and your friend are on spaceships in relative motion with each other watching the second play. Even though the quarterback only runs straight down the field in the X-dimension, you and your friend will again disagree on the number of yards the ball was carried ∆x and you will also disagree on how long the play lasted ∆t because your friend’s relative motion makes him look cockeyed at your spacetime! From the bottom section of Figure 2 we see that:

∆x ≠ ∆x`
∆t ≠ ∆t`

Is there some way to come to a mutual agreement for the second play as we did for the first play? Yes, just as the distance D measured in space is the same for all observers, the interval Ι in spacetime is also the same for all observers. The interval Ι is like the distance between events in spacetime. You and your friend will both agree on the interval Ι of the play, so long as you both measure time and space in meters, just as you both agreed on the distance D traveled in the first play when you both measured ∆x and ∆y in feet. After all, there is nothing profound about measuring time in seconds. A second is just some arbitrary unit of time that has something to do with the current rate of spin of the planet that we happen to find ourselves on at the moment. The Universe really doesn’t know about, or care about, seconds. To convert seconds to meters, we need to use the speed of light. The speed of light is 3 x 108 meters/second, so to measure time in meters we need to use a conversion factor, just like we needed a conversion factor to change yards into feet:

1 yard = 3 feet

1 second = 3 x 108 meters of time

Once you convert your time measurements into meters of time, you can apply a modified Pythagorean theorem:

Ι² = ∆t² - ∆x²
Ι² = ∆t`² - ∆x`²

Notice that to measure the interval Ι between events in spacetime, you need to use a “-“ sign instead of a “+” sign in the equation.

Once again, when all of the calculations are completed, you and your friend will now both agree on the interval Ι of the play, even though you will still disagree on the X-dimensional and T-dimensional measurements.

Figure 2 (click to enlarge)

An interesting thing about intervals is that since light travels exactly one meter of distance in one meter of time, the interval between any two events defined by the motion of a light beam is always zero. For example, the interval between a photon leaving the surface of the Sun and arriving at the Earth 8 minutes later, after traveling 93 million miles is zero! If you examine Figure 1 and Figure 2 in my posting Is Information Real? , you will find a similar situation. In this situation, there are two events – the flash of the flash unit and the recording of the flash of light in the camera. Let us say that the mirror of the LightClock is 5 meters from the flash unit and camera, giving a round trip distance of 10 meters in the Y-dimension. Your friend will observe that there is no X-dimensional distance between the flash of light and the recording of the flash by the camera because the flash goes straight up and down for him. He will observe some T-dimensional distance between the events because it will take 10 meters of time for the flash to go straight up to the mirror and reflect back into the camera. Notice that in this problem, we have to deal with 2 dimensions of space, ∆x and ∆y and one of time ∆t, so we need to tack on another dimension to our formula for the interval Ι:

Ι² = ∆t² - ∆x² - ∆y²

Figure 1
∆y = 10 meters of space
∆x = 0 meters of space
∆t = 10 meters of time
Ι = 0

In Figure2 you will observe the same two events, but you will observe that there is an X-dimensional distance ∆x between the flash of light and the recording of the flash by the camera because the light beam has to do a bank shot off the mirror. You will also observe that the T-dimensional distance will be greater than 10 meters of time because the flash has to travel a longer distance to make the bank shot, and it will take more than 10 meters of time to do so.

Figure 2
∆y = 10 meters of space
∆x > 0 meters of space
∆t > 10 meters of time
Ι = 0

In both cases, you and your friend will disagree on the X-dimensional and T-dimensional differences ∆x and ∆t between the two events of the flash unit flashing and the camera recording the flash, but in both cases you will agree that the interval Ι is zero. The extra ∆x and ∆t you observe cancel each other out, so that you still obtain an interval Ι = 0 too, just like your friend.

In this analysis, I am using the convention for the interval Ι that Edwin Taylor and John Wheeler use in their excellent book on the special theory of relativity Spacetime Physics (1963) because it yields positive intervals for everyday experiences, which is more in keeping with our experiences of positive distances in space. However, most treatments of the interval reverse the signs on ∆x and ∆t and define the interval as:

s² = ∆x² - ∆t²

but that definition yields negative intervals between most day-to-day events in the Universe. The signs on the ∆x and ∆t are really just a meaningless human convention that the Universe is not concerned with. The important thing is which term is larger. For example, suppose you see a man walking past your home on the sidewalk and you want to calculate the interval between his footsteps. Let’s suppose that relative to you the man takes 1 meter footsteps every second in your reference frame. What interval do you measure? Going back to our formula:

Ι² = ∆t² - ∆x²

∆x = 1 meter of space
∆t = 3 x 108 meters of time = 300,000,000 meters of time

Ι² = 300,000,000² - 1²
Ι² = 90,000,000,000,000,000 – 1
Ι² ≈ 90,000,000,000,000,000

Taking the square root we find:

Ι ≈ 300,000,000 meters

So in our reference frame, the man is mainly moving through spacetime in time and hardly moving through space at all as he walks by! This would be true even if he were driving past your home in a racecar or flying past in a jet airliner. As you sit there quietly at rest in your own reference frame, watching the man slowly stroll by, naturally your ∆x = 0, so you will find yourself traveling through 300,000,000 meters of spacetime every second. Even when you are just sitting still looking out the window, you will find yourself speeding through spacetime at the speed of light, but only in the time dimension! This is the reason nobody stumbled upon the special theory of relativity until 1905. Because we are all speeding through spacetime at the speed of light, mainly in the time dimension, we do not experience relativistic effects. Only when ∆x becomes large, and of the same magnitude as ∆t, do we experience the effects of the special theory of relativity. For example, if the man we were observing was able to run past our house at 50% of the speed of light, then his ∆x between footsteps would become 150,000,000 meters and our calculation of the interval Ι between his footsteps would drop significantly. This would make us see his wristwatch run slow and his body shrink in width as he rapidly flew by. Stranger yet, if we were to observe photons speeding by our home, we would measure an interval of:

Ι² = 300,000,000² – 300,000,000² = 0

because we would observe a ∆x = 300,000,000 meters and a ∆t = 300,000,000 meters.

But since all observers must agree upon the same interval between events, and the photons think that they are at rest with a ∆x = 0, the photons must calculate their intervals of zero as:

Ι² = 0² – 0² = 0

So in the reference frame of photons, which we see speeding by at the speed of light, there is no change in time or space for them – the photons are frozen in their spacetime and do not experience time or age at all! So for many photons, the Big Bang just happened and they are just sitting around waiting for something to happen, which is hard to do when your clock does not move.

When ∆t is greater than ∆x, physicists say that events are timelike because time dominates the calculation of the interval, and when ∆x is greater than ∆t, physicists say that the events are spacelike because space dominates the calculation of the interval. When events A and B are timelike, it means that there is enough time between the events for a beam of light to reach from event A to event B, and consequently it is possible for event A to cause event B to happen. When events A and B are spacelike, it means that there is not enough time between the events for a beam of light to reach from event A to event B, and consequently it is not possible for event A to cause event B to happen.

The Stage Returns
In Minkowski’s reformulation of special relativity, we once again have a stage or background of spacetime upon which the events of the Universe can unfold. Einstein did not at first like this reformulation of his theory of relativity because he thought it was too close to the old Newtonian concept of absolute space. But after a while he warmed up to the idea, and found it of great use as he tried to generalize his theory of relativity for all observers. Recall that his original version of relativity, published in 1905, only covered the special case of observers moving at constant speeds in straight lines with no acceleration. However, if you go back to your spaceship and turn on the engine, Einstein’s first postulate that you cannot perform any experiment that reveals if you are moving relative to an absolute space literally flies out the window. When you accelerate, you can easily tell that you are moving relative to an absolute space because everything tumbles back towards the back end of your spaceship! For example, if you try pouring one of those little bottles of gin, while your airplane rapidly accelerates for takeoff, you will most likely end up with it all over your shirt!

The Equivalence Principle
In 1907, Einstein had another revelation that solved the above problem. He realized that if a man fell out of a window and was in free-fall, then all effects of gravity would simply disappear. For example, we frequently watch astronauts in free-fall on the Space Shuttle become “weightless”. With this inspiration/revelation, Einstein concluded that he could reinstate his first postulate of relativity by postulating that gravity and acceleration were equivalent. This has become known as the Equivalence Principle:

"we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system."

With the Equivalence Principle, it once again became possible to conclude that no experiment could be performed to determine if you were moving or standing still relative to an absolute space. For example, suppose you are on a spaceship and let go of one of those little cans of Spam, and you observe that it immediately falls to the floor. According to the Equivalence Principle, you will not be able to tell if you are sitting on a launch pad on Earth in a gravitational field, or if you are out in outer space accelerating upwards at 32 ft/sec². Similarly, if the Spam just sits there and floats like it does on the Space Shuttle when it is in orbit, you will not be able to tell if you are in free-fall in a gravitational field or far out in interstellar space just coasting along.

Spacetime Curvature
Einstein’s new postulate of the Equivalence Principle had even more far reaching consequences than Einstein’s original two postulates of special relativity, but at a high intellectual cost. The original special theory of relativity, published in 1905, used little more than high school algebra for the most part. On the other hand, Einstein’s new general theory of relativity, published in 1915, used tensor calculus and non-Euclidean geometry which taxed even the most math proficient physicists of the day. Einstein’s new general theory of relativity relied heavily on Minkowski’s concept of spacetime. Not only did we have our stage back, but now we found that the stage could be distorted, like watching a play in a curvy fun-house mirror.

Imagine a large merry-go-round that can spin very quickly. Start out with the merry-go-round not spinning and measure the radius R of the merry-go-round with a standard foot ruler. Next measure the circumference C of the merry-go-round using the same standard foot ruler. If you divide the circumference by the radius, you will get:

C / R = 2π ≈ 2 * 3.141592…..

This should not come as a surprise because that is what they taught you in high school geometry, also known as Euclidean geometry, where the three angles of a triangle add up to 1800 and π is determined using a circle and the above formula.

Now spin up the merry-go-round to a high rate of rotational speed and measure the radius R and circumference C again. You will find that the radius R will not change because the rotational motion is perpendicular to the radius R and there is no relativistic shortening in that dimension. However, when you measure the circumference C, you will find that you get a larger number than before because the standard foot ruler you use will shorten in the direction of the rotational motion because of relativistic effects:

C / R > 2π

This is clearly not what you learned in high school geometry. According to the general theory of relativity, the reason that you no longer measure a value of 2π is that the merry-go-round is accelerating, and its spacetime is no longer flat, but is curved and deformed. When the merry-go-round spins, the direction of its motion on the circumference is constantly changing, and that is an acceleration. That’s why you feel a tug trying to pull you off the merry-go-round when it is spinning. If you put clocks along the circumference of the merry-go-round while it is spinning, you will also notice that they run slower than a clock placed at the center of the merry-go-round. Again the rotational motion along the circumference of the merry-go-round is causing time to slow down because of relativistic effects, and this is further evidence of the distortion of spacetime caused by the acceleration of the merry-go-round. If accelerations can curve spacetime, then using the Equivalence Principle, we can predict that gravitational fields must also curve spacetime too. In fact, the general theory of relativity proposes that gravity really is just a curvature of spacetime itself.

It’s easier to see how this works for the curvature of a 2-dimensional surface than for a 4-dimensional spacetime, so let’s go back to our football field. Imagine our football field again, but this time it is situated on the Equator of the Earth with its X-dimensional length aligned in an east-west direction along the Equator. If you add up the angles in a triangle on the football field by stretching a rope between three players, you will get 1800, just as you learned in geometry class in high school. Now let the size of the football field expand many times over, until the 50 yard line reaches the North and South Poles of the Earth. Don’t worry about what happens to the length of the football field during this example. If you now add up the angles in a triangle formed by a player on the Equator in South America, a player on the Equator in Africa, and a player at the North Pole, you will get more than 1800, in fact, the angles will add up to about 2700 because each angle in the triangle will be close to 900. This is a sure sign that the surface of the Earth is curved and not flat. Euclidean geometry, the kind of geometry you learned in high school, only works for flat surfaces. For curved surfaces, you have to use the non-Euclidean geometry developed by Bernhard Riemann in 1854.

Now have the player in South America and the player in Africa both start walking due north towards the North Pole. As the two players proceed due north, they will find that they are slowly approaching each other and that they will finally meet at the North Pole. Even though both players were traveling due north in straight lines, it will seem as if there was some “force” pulling them closer together. Of course, there really was no force at all. The two players were merely walking in straight lines on a curved 2-dimensional surface defined by the surface of the Earth. It was the curvature of the 2-dimensional surface that brought them together, not some “force”. The general theory of relativity says the same thing about gravity. The apparent curved orbit of the Earth about the Sun is caused by the Earth moving in a straight line through a curved spacetime. The general theory of relativity proposes that matter, energy, and pressure cause spacetime to deform. However, it is mainly the great mass of the Sun that causes spacetime near the Sun to deform, but the thermal energy of the Sun and the great pressure at its core, also provide lesser amounts of deformation too. When an object moves through a deformed spacetime in a straight line, it seems to move in a curved line through space. We interpret the deflected motion of the object to be the result of the “force” of gravity, when really it is just a peculiarity of the distorted spacetime that the object is moving through. That is why all objects fall with the same acceleration, as Galileo observed. They are all just moving through the same deformed spacetime in straight lines.

All of the above can be expressed in terms of Einstein’s famous field equations for general relativity, which in their most concise form are:

Gμν = 8 π Tμν

The expression on the left represents the curvature of spacetime and the expression on the right represents the matter and energy content of spacetime. The simplicity of the above formula is a bit misleading. It really represents a large matrix of nonlinear partial differential equations. In Software Chaos, we will learn that we generally cannot solve nonlinear differential equations and must instead use computers to calculate numerical solutions for them. This is just the case for Einstein’s field equations. Exact solutions can only be calculated for the simplest of problems, while high-speed supercomputers are required for more complicated problems. However, approximate solutions to some problems can also be obtained by making some simplifying approximations to certain problems.

To illustrate this process, suppose you have a Mercator projection map of the world and wish to determine the distance between Panama City near the Equator and London. The Mercator projection is a projection of the spherical Earth onto a cylinder surrounding the Earth, which produces a map with straight lines of both latitude and longitude forming a grid. The advantage of the Mercator projection is that courses of constant bearing appear as straight lines on the Mercator projection, so if you set your ship on a Northwest course, it will follow a straight line on your map. The problem with the Mercator projection is that it stretches both the north-south and east-west scales near the poles so that Greenland appears to have the same land area as Africa even though it is really 14 times smaller in area. Now if you try to find the distance between Panama City and London on the map using the Pythagorean theorem by measuring the distance along the Equator from Panama City to the longitude of London and then measuring the distance due north from the Equator to London you will find that the distance D that you obtain will not be correct:

D² ≠ ∆y² + ∆x²

However, you will find that by applying some correction factors to the above equation, you can once again find an equation that approximately yields the distance D between Panama City and London:

D² = A∆y² + B∆x²

The correction factors A and B are a reflection of the fact that space on a Mercator projection is distorted.

Similarly, for gravitational fields that are not too strong, like the gravitational field around the Sun, the interval Ι can be approximately written for objects moving towards and away from the Sun as:

Ι² = (1 - 2GM/c² x) ∆t² - (1 + 2GM/c² x) ∆x²

G = Newton’s gravitational constant
M = the mass of the Sun
x = the distance from the Sun
c = the speed of light
∆x = a small change in the distance from the Sun
∆t = a small change in time

The above correction factors to the interval are again the result of spacetime being distorted by the mass of the Sun. So the special theory of relativity is simply the general theory of relativity applied to a flat spacetime in which our original equation for the interval has no correction terms and is like the Pythagorean theorem for flat space. It turns out that for objects moving at orbital speeds about the Sun, such as the Sun’s planets, nearly all of this distortion to spacetime is due to the distortion of time, so Newtonian gravity is really just an illusion resulting from the distortion of time! For more on the general theory of relativity see Gravity from the ground up (2003) by Bernard Schutz, which is an amazing mathematical treatment of the general theory of relativity using only high school algebra!

The general theory of relativity is an incredibly accurate effective theory that takes over when Newtonian mechanics and Newtonian gravity break down at high velocities and large masses. It works over a very large range of conditions and predicts things like the slowing down of time in gravitational fields, the curvature of light beams near massive objects, the physics of black holes, and the cosmological expansion of spacetime. For example, we have been able to observe stellar systems composed of two neutron stars orbiting each other. A neutron star is a star that has reached the end of its life and has collapsed into a mass of neutrons so dense that a teaspoonful has a mass of 5 trillion tons, and its surface gravity is a trillion times stronger than Earth’s. As the two neutron stars orbit each other, they create gravity waves, ripples in spacetime, that radiate energy away causing an orbital decay of the pair. And the observed orbital decay agrees with Einstein’s predictions to one part in 1014. Even your GPS unit relies on Einstein’s general theory of relativity. There are currently 31 GPS satellites orbiting at an altitude of 12,600 miles above the Earth, and each contains a very accurate atomic clock. The signals from the GPS satellites travel to your GPS unit at the speed of light, so by knowing the travel time of the signals from at least 4 of the GPS satellites, it is possible to determine your position on Earth very accurately. In order to do that, it is very important to have very accurate timing measurements. Because the GPS satellites are moving relative to your GPS unit on Earth, their onboard atomic clocks run slower and lose about 7.2 microseconds per day due to their high velocities. But because the satellites are also further from the center of the Earth and in a weaker gravitational field, where spacetime is less deformed than on the surface of the Earth, their atomic clocks also run faster and gain 45.9 microseconds per day due to the weaker gravity out there. The net effect is a gain of 38.7 microseconds per day, so the GPS atomic clocks have to be purposefully built to run slow by 38.7 microseconds per day before they are launched, so that they will keep in sync with clocks on the surface of the Earth. If this correction were not made, an error in your position of 100 yards/day would accrue. But despite the great success of the general theory of relativity, it is still only an effective theory that does not work for very small things where quantum effects predominate, but that is a story for another time.

Spacetime and Cosmology
The general theory of relativity had a huge impact on the new science of cosmology early in the 20th century because it could provide a description of the overall curvature of spacetime for the entire Universe as a whole. There will be a future posting devoted entirely to cybercosmology, but here is a brief thumbnail sketch of events as we currently understand them for the physical Universe. Our bubble Universe formed about 14 billion years ago in a Big Bang, perhaps from a rapidly expanding false vacuum (Andrei Linde 1986), or from a black hole in a previous bubble Universe (Lee Smolin 1992). The radius of the visible Universe, that portion of our bubble that we can see with telescopes, is currently only 14 billion light years because that is the furthest distance that light could have traveled towards us in the 14 billion years since the Big Bang. But the visible Universe is just a very tiny speck in our bubble which is more than 1026 times larger, or perhaps, even infinitely large. Our bubble is so large because it went through a very rapid cosmic expansion called Inflation during the first 10-32 seconds after the Big Bang that stretched the spacetime of our bubble until it was nearly flat (Alan Guth 1981). The spacetime of the Universe has continued to expand since the Inflation, but at a much lower rate of expansion. The rapid expansion of spacetime during the Inflation might sound like a violation of the restriction, imposed by the special theory of relativity, that nothing can move faster than the speed of light, but that is not the case. It is true that matter, energy, and information cannot travel faster than the speed of light through spacetime, but spacetime itself can expand much faster than the speed of light, and that is what happened during the Inflation. The end result is that we can only see a very tiny portion of our bubble Universe out to a distance of 14 billion light years, but our bubble continues on to an infinite, or nearly infinite, size filled with galaxies we cannot yet see. At the instant of the Big Bang, a huge quantity of positive energy in the form of photons, and a matching amount of negative energy in the form of gravitational potential energy was released. The net amount of energy in our bubble Universe seems to be zero, which certainly is the case if spacetime is truly flat. So it appears that our bubble Universe was essentially made from “nothing”. It seems that today, most cosmologists think that there probably are an infinite number of other bubble universes, forming an infinite multiverse.

In the general theory of relativity, the physical Universe is a 4-dimensional spacetime. The Software Universe, on the other hand, is a 2-dimensional cyberspacetime, consisting of a time dimension and a cyberspace dimension. The world currently produces about 500 billion microprocessors per year, so there should be about 1 – 10 trillion active microprocessors at the present time. Deep down at the hardware level, cyberspacetime is not a continuum - both dimensions are quantized. The cyberspace dimension is defined by the 1 – 10 trillion currently active discrete microprocessors, wherever they might be, and the individual system clocks of each microprocessor quantize the time dimension. Microprocessors currently run with a clock speed of about 109 Hz, so the time dimension of cyberspacetime is quantized into quantum chunks of time of about 10-9 second each.

As with spacetime in the physical Universe, cyberspacetime is the stage upon which the events of the Software Universe and IT unfold. The Software Universe began in a Big Bang about 2.1 billion seconds ago as a few bytes of machine code on Konrad Zuse’s Z3 computer in the spring of 1941, and has expanded and evolved into the complex Software Universe we see today, consisting of millions of terabytes of software residing on trillions of microprocessors. Cyberspacetime began on Earth, but has expanded to every planet in our solar system and is currently leaving our solar system on board the Pioneer 1 & 2 and Voyager 1 & 2 probes.

At the logical level, most familiar to IT professionals, the graininess caused by the quantization of cyberspacetime at the hardware level vanishes and so most IT professionals just view it as a large cyberspacetime continuum. Like the spacetime of the physical Universe, cyberspacetime is nearly flat. For example, I frequently work from home in a suburb of Chicago in the middle of the night (for me) with some of my fellow Middleware Operations team members who are located in India and with our Command Center which is located 26 miles from my home. Our Unix server farms and mainframe datacenters are scattered throughout the U.S. in different cities. But as far as I can tell, it is all just one large flat cyberspacetime to me. Programmers only have to worry about warped cyberspacetime if they write code for one of the 300 geosynchronous satellites orbiting at an altitude of 22,300 miles, one of the 31 GPS satellites orbiting at 12,600 miles, or distant space probes, because cyberspacetime is more warped here on Earth than way out there. But even we Earthbound programmers still have to contend with the limitation imposed by the special theory of relativity that requires that information cannot travel faster than the speed of light. This becomes a factor for telecommunication over satellite links and is why the software on the Spirit and Opportunity probes on Mars must be located on board the probes and not back on Earth in a JPL control room, since it can take up to 21 minutes for commands from JPL to reach Mars. When I worked for Amoco in the credit card processing section of IT, we had to worry about the 240 milliseconds that we lost in sending credit card transactions up from the gas stations to a geosynchronous satellite and back down to our mother Earth station in Tulsa. When customers swiped their cards in a CRIND (Card Reader in Dispenser) at the pump, we lost 240 milliseconds because our transactions could not travel faster than the speed of light, and those 240 milliseconds could not be regained by tuning the network no matter how hard we tried.

The Software Universe is not made of things; it consists only of processes and flows of information between discrete events in the two dimensions of cyberspacetime. There is the illusion that the Software Universe is filled with real things, such as files and databases with purchase orders and inventory levels on them, but that is an illusion. For example, when you edit a file with notepad, you are not interacting with a file; you are interacting with a process. Just press CTRL-ALT-DEL to open Windows Task Manager, click on the Processes tab and look for the notepad.exe. The PID is the process ID for the notepad process you are interacting with. The CPU Time is the distance the notepad process has traveled along the time dimension of cyberspacetime measured in dedicated CPU cycles to the notepad process. Nothing ever interacts with "things" in cyberspacetime because the "things" have to first be read into the memory of a computer and placed under the control of a process. Only the processes interact with each other in cyberspacetime. Therefore, files and databases on disk drives, tapes, and CDs are not part of the Software Universe; they are part of the physical Universe. In the Software Universe, we have two processes called "read" and "write" which allow us to pass information into and out of cyberspacetime. So at the logical level, above the hardware level, the Software Universe is simply made up of information and was created out of “nothing”, just like the physical Universe. We shall see in a future posting on cybercosmology, that a Universe simply made out of information and composed only of interacting processes, might be a good model for the physical Universe too.

Next time we will drill down even deeper by turning to another effective theory of physics called quantum mechanics and explore the nature of the microstates of statistical mechanics, the structure of atoms, and software at the level of individual characters in source code.

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Steve Johnston

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