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World Lines

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World lines are convenient ways of drawing out the history of an entity through space as well as time. Imagine a standard graph with one quadrant where the horizontal (x) axis represents space or distance and the vertical (y) axis represents time. We can draw a line on the graph based on how an entity moves through space and time. Let's take a simple example.

The horizontal axis represents the present1 so we will start drawing from the origin. Let's draw the world line of a hypothetical hamster called Geraldine. At the beginning of the day, Geraldine is asleep amongst some comfortable pink bedding on the top level of a multi-levelled cage. Since she is not moving through space2, her world line is a straight vertical line from the origin, meaning that she is only moving through time.

When Geraldine wakes up she strolls from one side of the top level of her cage to the other side. Now she is moving in both space and time, so we have to draw a diagonal line going to the right of the diagram. Since Geraldine can't move very fast, the line has a steep gradient because she won't get very far in a given period of time. To get down to the next level of her cage, Geraldine must climb down a vertical 'ladder'. This is still movement through space, so the only thing that might change the line on our diagram is her speed. When climbing, she is probably going slower, so the line is an even steeper gradient.

Suddenly Geraldine falls from the ladder straight down to the level below. The world line's gradient now becomes shallower because during descent, the speed of movement through space is greater. From this simple diagram we can now draw a very significant conclusion, and it is not the conclusion that hamsters should not climb ladders too early in the morning. On the part of the line that represents climbing, a certain distance along the time axis (we'll say 2cm) represents a slightly longer distance on the real world line. When Geraldine moves faster in the 'falling' part of our world line, the same 2cm worth of time represents a longer length of the world line than the climbing stage.

In other words, Geraldine's movement through time slowed down when her movement in space sped up. Does this sound familiar? Yes, it's Albert Einstein's theory of relativity. It was Einstein himself that pioneered the world line way of drawing things. There is a variation of the world line diagram where we have two axes of space against one axis of time. Such a diagram is more accurate than Geraldine's world line, and they are usually called 'light cones'. 'Light cones?' you think; why are they called 'light cones' when they have nothing to do with light and should never necessarily resemble cones? Rest assured, there is a reason.

It turns out that for Geraldine's world line to go at a 45-degree angle from the vertical, she would have to be travelling at the speed of light. However, according to Einstein, nothing can actually pass this barrier. Entities that travel slower than light speed are called 'tardyons'. To move, tardyons need energy, which is quite obvious. Geraldine obtained her energy from the food she ate, for example. To move faster, you need more energy. The trouble is, according to Einstein, energy is the same thing as matter. So the faster you go, the more mass you acquire. It is common sense that heavier objects need more energy to move them, so the faster you go, the more energy you need to go even faster. All this is subject to the well-known relationship between matter and energy, where the energy, E in joules is equal to the mass, m in kilograms, multiplied by the speed of light squared (c2) (where c is in metres per second). The mass of the object is therefore equal to E/c2.

The standard or traditional scale for a world line is that the length that represents about three hundred million metres along the x-axis is the same length as the length that represents one second up the y-axis. This is because light travels at three hundred million metres per second. Photons, the particles that represent light, aren't the only things that travel at light speed. Many other fundamental particles, including neutrinos, travel at the same speed. The collective term for such particles is a 'luxon'. If you plot the world line of a luxon, the line will be at 45 degrees from the vertical.

If you could accelerate to the speed of light, then your mass would increase to infinity, and would hence need an infinite amount of energy to move it. This is the reason that faster-than-light travel is regarded as forbidden to tardyons, and this also means that world lines that go beyond 45 degrees from the vertical are also forbidden.

Now consider a graph for world lines that has four quadrants. If you plot the lines that represent the light barrier you get a cross through your grid. It is good practice to shade in all areas that are beyond the light barriers on the diagram, because these represent forbidden zones to tardyons. Physicists refer to these zones as 'elsewhere'. All places in space-time that would need faster-than-light travel to reach them are 'elsewhere'. For example, the sun as it was one minute ago, as seen from Earth, is elsewhere, because it takes information of the sun approximately eight minutes and 20 seconds to reach Earth.

The interior of black holes are also classed as being elsewhere. This is because anything that goes into a black hole would need to travel faster than light speed to escape, due to the massive gravitational pull. Technically speaking, we say the escape velocity of a black hole exceeds light speed. Now, because the light speed barrier only applies to tardyons, physicists have theorised the existence of particles that naturally travel faster than light - they are called tachyons. Tachyons would be the only type of matter that could easily escape from black holes, but because they constantly exist elsewhere, it would be very difficult if not impossible to prove their existence or interact with them.

In a world line diagram that has two dimensions of space and one of time, the world lines of tardyons exist in cones, and this is why they are called light cones. The top light cone is the future of the item in question, the bottom light cone is its past. The origin of the graph is called the 'here-now' and represents the present. So far we have only dealt with straight world lines - they are called 'geodesics' - but curved world lines are easily created by entities that are accelerating.

A geodesic that is the world line of a tardyon follows a world line that is described as 'timelike' because they are nearer the time axis. Tachyons would have world lines that are 'spacelike' because their world lines are closer to the space axis.

Einstein stated that world lines could not be cut, and that they are not created from nothing. This would violate the Law of the Conservation of Mass and Energy. With this restriction, changing time as in the film Back to the Future is not possible. So, you might ask, if world lines cannot be created or destroyed, what happens to our world line when we die?

The World Lines of Humans

When we are born, our world line branches off from our mother's world line. When we die, the world lines of the atoms that make up our body go on. As you can see, world lines at no point cut off or appear suddenly.

You might think of your world line when you are alive to be a single thing, but we can still think of it as a collection of atoms. With this revised picture, drawing the activities of humans on a world line diagram is drawing millions of world lines that are all next to each other. But this is not quite accurate. Our cells are always duplicating themselves and then destroying themselves. Dead cells on the top layer of our skin are being shed at the estimated continuous rate of 10 billion per day. Your skin has been totally replaced after a month. The lining of the stomach replaces itself in just five days. Other people then end up breathing in the shed cells, and we are constantly inhaling the air that has already been exhaled by billions of other people in the history of the planet. The world lines of our atoms are never the same.

Humans are therefore not single entities. Your body now probably has nothing in common with the body you had a few years ago, except for the general arrangement in which the atoms are placed. The only thing that makes up you as a human is the pattern of your world lines, but the world lines themselves are constantly changing. It is very likely that we all contain atoms from Albert Einstein himself. So, strictly speaking, it isn't possible to draw the world line of a single human; we have to trace the world lines of trillions of different atoms that at some point come to exist in your body before going off to be part of someone else.

If we drew the world lines of all the atoms on the surface of the Earth, we would see that the human race looks very interconnected. Here is a relevant observation of the mathematician Rudy Rucker on this subject:

The simple process of eating and breathing weave all of us together into a vast four-dimensional array. No matter how isolated you may sometimes feel, no matter how lonely, you are never really cut off from the whole.
1Or is this an illusion? Follow this link for more on the nature of time.2Strictly speaking, she is moving through space because the Earth is hurtling around the sun at almost 30km per second, but this fact is usually ignored since it applies to all terrestrial objects.

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