Olbers's paradox tells us that in a static and infinite universe the night sky should be completely brilliant free regions devoid of light or dark. Out the terms of the paradox precisely help us understand better.
Specifically, the mathematical formulation of the paradox is to calculate the flow of light we receive to be located in the origin of a spherical coordinate system. To begin with we establish two hypotheses. The first hypothesis is that light sources are distributed homogeneously in space. The second is that space is static. first consider the flow of light that reaches us emitted from a source at a distance r. In a static space this flow is proportional to the inverse square of distance, f ~ 1 / r ². Then consider the amount of light sources in a spherical corona. This number n increases with the radius squared, n ~ r ². Therefore, the sum of the flow of light from all light sources located in a spherical shell at any radius is F ~ f n. This value is a constant as r dependence cancels. How many layers
spherical want to add? We start with r = 0 and we move to increasingly larger radii. Now we establish two hypotheses. First that the universe is infinite in extent. Two light sources that are eternal. Thus we see that the sum of spherical layers is infinite and that each layer contributes to a constant value (not infinitesimal) the total flux of light. The result is therefore infinite. This is the paradox of Olbers. 1. The light sources are distributed homogeneously in space
2. The space is static
3. The universe is infinite in extent
4. The light sources are timeless We are going
indicating these assumptions made at a time.
First, the homogeneity of the light sources. If the source distribution is not homogeneous paradox need not be. Benoit Mandelbrot, the creator of the fractal, proposed a cosmology that solved the paradox of Olbers precisely that. Specifically, in a universe in which the light source distribution of fractal dimension is less than two, the paradox does not occur even if you have infinite sources in an infinite universe whose existence is eternal.
The reason is that it no longer n ~ r ² and there will be no cancellation f ~ 1 / r ². What is true in general that for a fractal dimension D, n ~ r ^ {D-1}, making the paradox is resolved if D
second hypothesis, the static space. If space is not static and is not satisfied that f ~ 1 / r ², but in general for an expanding universe flow is more diluted. In some models this resolves the paradox. This was the solution of the stationary model of cosmology of Hoyle and Burbridge was based on a space-time de-Sitter.
third hypothesis, the infinite expanse of space. If space is not infinite and has an edge (or say that the source distribution Light has an edge), then the sum over spherical crowns ends at a certain radius. Thus the sum of the flow is finite.
But beware, if space is finite but unbounded (and the rest of the following assumptions remain valid, namely the old infinite light sources), then the light emitted in more remote past makes one or several times around the universe to reach us. Since the age of sources is infinite the number of crowns to consider again is infinite and the paradox is not solved. Fourth
hypothesis, age infinite light sources. This is obvious: If the light sources do not exist from the infinite past, but from a time T, then because the speed of light c is finite, the number of crowns to consider is finite, in particular to a radius R = c T. < 2. La justificación de un modelo cosmológico así viene del hecho que a cierta escala las fuentes de luz parecen agruparse en distribuciones fractales en el universo - aunque según me consta a mí los estudios no son concluyentes y no son extrapolables a muy grandes escalas.
is wonderful to see how this simple, everyday paradox confronts us with an unprecedented cosmological mystery and becomes a cornerstone of cosmology, as it shows the depth of the possible solutions to the paradox. Any model of the universe must necessarily deal with it.
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