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Space-Time

 

To locate an event we need to use 3 dimensions of space and 1 of time. We can, for instance, define the position of an object in a room stating that it is placed x metres from a wall, y metres from the adjoining wall and z metres from the floor. On the other hand, it can be said that this object (which may be a ball thrown into the air) is standing in that position, but only in an extremely tiny moment: we now use the time dimension to locate the object.

Time and space present 2 big differences between them: unlike space, time is unidimensional and seems to flow in one single direction. Nevertheless, the time dimension is also, in some way, comparable to the space dimensions, as it is possible to establish an equivalence between a given time lapse (let's say, 8 minutes) and a determined space interval (the distance separating the Earth from the Sun, which is precisely 8 light-minutes).

 

Properties

The empty space is recognized as holding 2 fundamental properties:

Homogeneity, or the equivalence between all the points placed at an infinite straight line (invariability in the spatial translation);

Isotropy, or the equivalence between all the directions that this straight line may assume (invariability in the rotation).

The empty time (without events), for being unidimensional, is only given one single property:

Homogeneity, or the equivalence between all the time instants (invariability in the time translation).

Each of these properties is corresponded by one conservation law. An object endowed with some quantity of energy (E) moves across the space conserving a quantity of movement (linear momentum P), whatever the direction it takes (angular momentum L). To make the reasoning more easy, we reduce here E to the energy implied in the movement (kinetic energy).

 

Space and Time in Classical Physics

The equations that, in classical physics, define the energy e and linear momentum are the following ones:

E         = Ms2 * ½

P         = Ms

 

Where:

M       = mass

s          = speed

Therefore, we reach the conclusion that in classical physics space and time are entirely independent entities, opposing to what happens in relativistic physics.

 

Linear momentum measured in horsepowers (MoonRunner Design UK)

 

Space and Time in Relativistic Physics

We can easily perceive the intimate relationship between space and time implied by relativistic physics if we imagine someone moving very fast in regard to a given reference vector. The amount of space traveled by the individual will be larger, but this movement implies that the time traveled by him will be smaller, or in other words, the pointers of a high speed moving clock tick in a slower pace than the pointers of a clock that is stationary in regard to that reference vector. So, if we add space we need to subtract time.

 

Symmetry and the Arrow of Time

Why does the time look to flow from the past to the future instead of the opposite? As scientific laws show to us, there doesn't seem to be any difference between past and future. The laws of physics are invariable under the combination of the symmetries known as C, P and T:

    1. C is the exchange of electric charge: particles vs anti-particles;
    2. P is the exchange of parities: left side vs right side;
    3. T is the exchange of the time direction: past vs future.

 

Symmetry P (MoonRunner Design UK)

 

If there is symmetry P, that means that the behaviour of the particle is the same, no matter if it moves to the left (in the opposite direction of the clock pointers) or to the right (in the direction of the clock pointers). If there is symmetry CP, that means that this behaviour equivalence is kept, but only if we replace the particle by its anti-particle. If there is symmetry CPT, that means that, besides that, we still have to change the direction of time.

With few exceptions (as in the case of the kaons and anti-kaons), the behaviour of the particles doesn't break the symmetry CP and therefore, for most of them, there doesn't look to be any distinction between past and future. However, according to the 2nd law of the thermodynamics, the disorder or entropy always grows as we make head towards the future. On the other hand, psychology tells us that we shall remember the past and not the future, as cosmology shows that the Universe is expanding and not contracting as we advance in time. Summarizing, there seems to be a disruption concerning to symmetry T.

The second law of the thermodynamics results from the existence of more possible disordered states than possible ordered states. For instance, on a snooker table, the balls are initially distributed in an ordered way (they form a triangle) but, as the match goes on, it is virtually impossible that they take again their initial kind of distribution because it is a highly unlikely result of a play. So, if something begins in a highly ordered condition it is almost unavoidable that, as time goes by, it will end up in a condition where the disorder is much higher than the initial one.

 

From geometry to chaos (MoonRunner Design UK)

 

Why does the Entropy increase?

Why does the Universe begin in an ordered condition? After all, by the already mentioned reasons, it would be much more probable that it started in a disordered state, so that the entropy would have no chance to significantly increase.

Stephen Hawking defends that the beginning of time is located in a regular point of the space-time, not in a singularity (infinitely tiny point where the laws of physics, as we know them, are not valid anymore). This theory is called the "Universe with no borders". Hawking also argues that his theory implies that the Universe must have started in a relatively regular and ordered state.

Though, according to some theorists, the regular state of the primordial Universe may also well be only an outcome of the strong version of the Anthropic Principle, according to which this Universe is one among many universes, a rarity where very special conditions allowed that life could begin and thrive. If this hypothesis is true, we may live in a Universe that began in a very ordered state for the simple reason that the growth of the entropy is essential for the development of life, as we shall see later.

 

The Psychological Arrow

And why do we perceive the increase of the disorder and not the opposite? Because a psychological arrow necessarily results from the thermodynamic arrow. In other words, when any information is registered in our memory, the order inside our brain grows, but that registration implies the use of a big amount of energy that is dissipated as heat, which is a form of energy in a highly disordered state. So, after the registration of the information in the memory, the global disorder has increased. In a statement, this means that the psychological past necessarily corresponds to the most ordered state.

 

The Cosmological Arrow

Concerning to the cosmological arrow (the expansion of the Universe), it goes along with the psychological arrow for 2 possible causes:

    1. The Universe will expand forever;
    2. The Universe expands in a first phase and contracts in a second phase. Under this situation, the increase of the entropy is bigger during the 1st phase than during the 2nd, because in the beginning of the latter the Universe shall have already reached a very high level of entropy (disorder), so it can't grow much more from then on. As we know, intelligent life demands a high level consumption of food, which unavoidably is converted into heat. In other words, it demands a high transformation rate of ordered energy into disordered energy. As a consequence, life can't be developed during the contraction period of the Universe, because then the transformation rate is very low, as we have already seen. 

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