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Spacetime and Velocity



Here is a thought on our universe. Our universe is set in a dimension we call space-time and define as having four dimensions. General relativity shows that the ‘shape’ of space-time is elastic from one frame to another, depending on relative velocity. You can think of one dimension stretching as the other contracts – more space, less time; more time less space. Nothing travels faster than the speed of light, and as you approach that speed ‘time dilation’ occurs, meaning time slows down, with the effect  that the speed of light remains constant with respect to you. 

In a universe with only energy and not mass, everything travels at the speed of light. As there is nothing to compare in this universe, let’s set the speed of light at zero. If energy in this universe converts to mass, then mass travels slower than the speed of light. Consider these sub-luminal velocities to be less than zero. If mass travels at increasingly slower sub-luminal velocities, then seen from the perspective of mass, the universe will seem to be expanding, as everything ‘accelerates’ away from it. Also, as time speeds up corresponding to the slowdown, time will seem to have a direction – the arrow of time. Well, that’s what it looks like from our perspective, so maybe that is the universe that we are in. And perhaps we are this side of the speed of light barrier, and ‘something else’ goes the other way, that we cannot perceive - dark energy? I shall leave this for someone else to prove, but in the meantime, let’s elaborate on the above thoughts. 

We can visualise space, with its three dimensions, but grapple with space-time and its fourth dimension of time. Let’s work on some visualisations. 

For the first part of the visualisation, take a cube with length, width (albeit the same) and height containing space. Imagine that when you compress the height of the cube it elastically expands in length and width. As you continue to collapse the height, you are finally left with just two dimensions of length and width. You have compressed three dimensions into two (a plane). Now pull the height back up, contracting length and width, until you are left with just height. You have compressed three dimensions into one (a line). 

If the cube was a vacuum, then you have pressed your vacuum first down into a plane and then upright into a line. Since it is a vacuum, and there is nothing there, that should not be a problem. If it is a line, with nothing there, maybe it is actually just a point (with zero dimensions), or in fact nothing at all. That could suggest that if we pulled the height back down (to get length and width), we would create something out of nothing. Hold this thought for when we move on to space-time, and treat it in a similar elastic manner, stretching and contracting dimensions to change ‘shape’. 

For the second part of the visualisation, we will construct a diagram of space-time as a light cone. This can be applied to any source of light or energy, or indeed to a big bank starting a universe. Take a source of light the size of a pea. It will radiate photons in all directions at the speed of light. If you count the photons off by the second, you can construct a series of shells around the pea. Cut each shell though its equator to create a disk and stack up the disks to form a cone with the smallest disk on the bottom. 

The outer surface of this cone, its wall, represents the speed of light. Each part of the surface comes from a spherical shell at a distance from the centre reached at the speed of light. The central spindle of the cone is at a state of rest.

According to the general theory of relativity, the speed of light is constant relative to you, whatever speed you are travelling at. This has the effect that time slows down for you relative to a slower object. You cannot exceed the speed of light, but if you reached it, time would stop for you. Just before you reach the speed of light time would run so slowly for you as to almost go on forever, In the light cone time is running fastest at the central spindle and slows towards the edges of the cone. For the photons at the cone’s wall time has stopped.

Imagine that this light cone represents a universe that has only energy and no mass. Its particles, photons, travel at the speed of light, so they exist on the wall of the cone, where time has stretched out to slow to a stop. It is similar to the cube above, where we have collapsed the dimension of height, but in this case we have collapsed the part of space-time that is space. If there is no space or time to measure how distance is travelled by the photons then the speed of light becomes moot. It could be anything you like, but for the moment, we will say that the speed of light is zero. Every particle in this mass-free universe obeys this constant of zero speed of light.

However, we have an equivalence of mass and energy (energy equals mass times the speed of light squared). If we introduce mass into this universe, then massive particles will move towards the centre of the light cone diagram, as they travel slower than the speed of light. We have set the speed of light at zero, and mass travels at subluminal speeds. From the perspective of the photon, it is as if the massive particles have a negative velocity. For the slower travelling particles time is speeding up. What is now happening is that the introduction of mass into the universe is stretching out the ‘space' portion of space-time. This is similar to pulling the cube’s height back up from the plane. The ‘shape’ of space-time is changing.

Continuing the process of converting energy to mass, the ‘shape’ of space-time would tend towards ‘all-time-and-no-space’ on the wall of the cone to ‘all-space-and-no-time’ near the spindle. At each of the extremes, the geometry of the light cone collapses – it is not a cone. In both cases, there seems to be either infinity or nothing.

With this ‘geometry’ in mind, let’s come back to the vacuum. Is a vacuum nothing? If it is nothing, does it take any space, or experience time? 

Quantum physics has a vacuum teaming with particles coming in and out of existence. In fact, this has to happen to allow the processes of physics to happen: it is integral to the theory. Conceptually, the changing ‘shape’ of space-time predicts that this should happen. It also suggests that energy and matter can emerge apparently spontaneously as the proportions of space-time change. It will also give rise to perspectives of expansion or collapse for the inhabitants of universes, depending on where they are on the continuum.

I have addressed this mathematically, but unfortunately the size of the file exceeds the capacity of what I can upload onto the web for the moment.

@ Fred Piechoczek 2015