# Are there upper and lower limits to the inverse square law?

• I
Gold Member

## Summary:

Does the inverse square law hold for luminosity distance at very, very large distances?
Does the inverse square law hold for gravity at very, very small distances?

## Main Question or Discussion Point

Quoting from Modern Cosmology by Andrew Liddle on pages 130 and 131: "Let me stress right away that the luminosity distance is not the actual distance to the object, because in the real Universe the inverse square law does not hold. It is broken because the geometry of the Universe need not be flat, and because the Universe is expanding."
In the book "Reality is not what it seems", in the chapter on "Quanta of Space", Carlo Rovelli, states that space must have a minimum dimension and posits that atoms of space must exist. This is part of the proposed Loop Quantum Gravity theory. In the book, it is explained that this assumption is necessary to prevent infinities from arising during normalizations from quantum physics to classical physics.

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stefan r
Gold Member
"Very very small distance" depends on what you mean. Quantum effects are not relevant if you are talking about an astronomy observation of some object. A cosmic ray hitting your retina might be close enough. Should not matter because a single event is not an astronomy observation. Everything beyond the lens of your eye is definitely far enough away for distance squared to be observed as expected.

I am skeptical about the idea that expansion of the universe effects the inverse square law. The light is still spreading out over an area as it travels. It's intensity decreases with square of distance traveled because the area illuminated is a surface. If space is expanding the light from an event will be spread out over the new larger surface area.

anorlunda
Mentor
Have you ever seen a fun house mirror. The image of a light in that mirror might be spread over a large area, or it might be concentrated in a bright spot. In that respect, a curved mirror appears to break the inverse square law.

Curved space can also distort images, and appear to break the inverse square law. But the law is not really broken because what we mean by distance must be altered in curved space, so distance squared must also be different.

Bandersnatch
Expansion causes any light emitted over interval $dt_e$ to be received over a longer interval: $dt_0=(1+z)dt_e$, since the emitter is receding as it is emitting;
AND
the received light has its frequency reduced by a factor of (1+z), i.e. the regular cosmological redshift effect.
So any time we measure bolometric flux from a standard candle source, the received flux is reduced not only by the inverse square law, but by and additional factor of $(1+z)^{-2}$:
$$F=\frac{L}{4\pi S_k(r)^2 (1+z)^2}$$
where ##S_k(r) is the curvature dependent part of the FLRW metric, and which for a flat universe is just the comoving distance (r) while being a function of r and the radius of curvature in the other two cases.

Gold Member