Can anyone explain the Planck area?

Question

First of all, I am not an expert. I was reading about the holographic principle and came across the Planck area. It says that Planck area is the square of Planck length and there were some pictures like this: source

I know that this Planck area is used for black holes. But doesn't matter if this Planck area is for a triangle or a square?

And there was a clip that Leonard Susskind was saying that Planck area is $10^{-33}$ cm on a side. source 39:44.

But isn't it that the Planck length is $10^{-33}$ cm? If so, what is the "on a side"? Does it mean a face? Please keep it simple so I can understand it.

Answer 1

The Planck length is usually defined as

$$l_P=\sqrt\frac{\hbar G}{c^3}\approx 1.6\times 10^{-35}\,\text{m},$$

where $\hbar$ is the reduced Planck constant, $G$ is Newton’s gravitational constant, and $c$ is the speed of light.

The Planck area is usually defined as the square of the Planck length,

$$A_P=l_P^2=\frac{\hbar G}{c^3}\approx 2.6\times 10^{-70}\,\text{m}^2.$$

There are various quantum gravity theories, none yet accepted. The role of the Planck length and Planck area in physics is therefore still being researched. In particular, bits of information in black holes should not be pictured as being stored in nice little triangles or nice little squares on the event horizon. One bit of information probably “requires” about one Planck area, in some sense, but we don’t yet understand just how the information is stored so we can’t talk about it being in a particular place on the horizon or having a particular shape like a triangle or a square. That diagram is not to be taken literally.

As for the phrase “on a side”, it can he used to talk about the sizes of squares or triangles. A square has four sides and a triangle has three sides, so instead of talking about their area you can talk about the length of their sides.

Answer 2

The Planck length is approximately $1.616255\times 10^{-35}$ m. You get the Planck area by squaring that, just like you get the square metre by squaring a metre. So the Planck area is approximately $2.61228\times 10^{-70}\,\mathrm{m}^2$.

As Wikipedia says, we expect geometry to get weird near that scale, so despite your diagram it's probably not appropriate to think about the Planck area in terms of Euclidean squares and triangles.

The reason I say "expect" and "probably" is that we don't yet know what really happens at that scale, we need a proper theory of Quantum Gravity for that. And even when such a theory is developed we may never be able to verify its predictions at that tiny scale: the energy required to do that is vast. It's essentially equivalent to the energy density in the first few Planck time units after the Big Bang started.

Answer 3

The entropic argument cited in a previous answer is probably the best account because it is the one that generalizes to $n$ dimensions.

For n dimensions, the corresponding quantity would be expressed in terms of the $n$-dimensional version of Newton's constant, $G_n$, as the $n-2$ Planck "area": $$\frac{hG_n}{c^3} = \text{Planck } n-2 \text{ "area"}.$$

There are several observations that should be made about this that are almost universally neglected.

First, if the $4$-dimensional theory is an effective or dimensionally-reduced version of an $n$-dimensional theory, for $n > 4$, then it is actually the higher-dimensional Planck units which are fundamental, rather than the $4$ dimensional ones! The Planck length that comes out of the $n$-Planck area may be very different and on an entirely different scale than the Planck length that comes out of the $4$-Planck area. For instance, we may very well be on the threshold of new physics, if that quantity happens to be orders of magnitude larger than the Planck length!

Second, a simple dimensional analysis of the action integral for the Einstein-Hilbert action $$ S = \int{\frac{R}{2κ}\sqrt{|g|}d^4x}, \\ [h] = [S] = \frac{[R] \left[\sqrt{|g|} d^4x\right]}{[κ]}, \\ [hκ] = [R] \left[\sqrt{|g|} d^4x\right] = \frac{1}{L^2} L^4 = L^2 = \left[\frac{hG}{c^3}\right], \\ [κ] = \left[\frac{G}{c^3}\right], $$ shows (first of all) that the correct power of $c$ for the coupling coefficient $κ$ is $c^3$ not $c^4$ or $c^2$. All three appear in the literature, and $c^4$ appears in many on-line sites. These are wrong.

In 4 dimensions $κ = 8πG/c^3$, which generalizes in $n$ dimensions to $$ κ_n = \frac{(n-1)(n-2)}{n-3} \frac{π^{½(n-1)}}{(½(n-1))!} \frac{G_n}{c^3} = 2π\frac{n-2}{n-3} \frac{π^{½(n-3)}}{(½(n-3))!} \frac{G_n}{c^3}. $$ adopting the convention $(-½)! = \sqrt{π}$.

Second, and more importantly, it shows that the Planck area, when expressed as a function of $(h,κ,c)$ $$\frac{hG}{c^3} = ¼ħκ$$ or for the $n$-dimensional case in terms of $(h,κ_n,c)$ $$\frac{hG_n}{c^3} = \frac{n-3}{n-2} \frac{(½(n-3))!}{π^{½(n-3)}} ħκ_n$$ is independent of $c$!

Now, what does this mean?

Whatever theory is devised to integrate gravity with quantum theory will be based on an additional axiom that is independent of any notion of light speed and independent of the paradigm split between non-relativistic versus relativistic - and will apply equally well to Newtonian Gravity as it does to General Relativity.

Thus, such a theory can be found by combining Newtonian Gravity within the setting of Newton-Cartan chrono-geometries (or in the setting of a more comprehensive chrono-geometry that contains Newton-Cartan spacetimes) with quantum theory and proceeding from there.

The theory that combines general relativity with quantum theory must have this as its non-relativistic correspondence limit and must be derivable from it as a deformation in "theory space"; i.e. as a category-theoretic deformation of the category of quantized non-relativistic theories over either Newton-Cartan chrono-geometries or more comprehensive chrono-geometries that contains Newton-Cartan chrono-geometries into the category quantized general-relativistic theories over Lorentzian chrono-geometries.

The strongest hint as to what this theory should be, and what the additional axiom ought to be, lies with the fact that the Planck unit that comes directly out of combining the coupling $κ_n$ with $h$ is an $n-2$-area - the additional assumption should be something that involves $n-2$ areas and is something that may be posed generically for Newton-Cartan chrono-geometries, Lorentzian chrono-geometries (or even Riemannian, locally n-Euclidean, geometries or Carrollian chrono-geometries) and may, in fact, be an additional assumption that is generic to $n$-manifolds and to natural and gauge-natural bundles over $n$-manifolds. It is an additional axiom that may provide a quantum correction to, say, the classical diffeomorphism group over natural bundles and the classical gauge group over gauge-natural bundles.

My personal bias is that the additional axiom makes no adjustments to classical geometry at all, but only to the diffeomorphism and gauge groups: an axiom for "quantum" diffeomorphism/gauge transformations, i.e. a quantum correction to transformations that move space-time points.

Answer 4

When considering things like this I try to find a visualisation which helps with understanding.

Firstly I think it is important to recognise, as the graphic shows, that the formula for storing one bit of information requires 4 Planck areas. If you consider the Planck length to be, in some sense, a significant physical limit (and not just a random distance produced from a dimensional analysis of fundamental constants) then I think you have to take the "4" seriously and hence have to take issue with the graphic using collections of triangles to illustrate its point.

Let's consider squares. If we build a surface of Planck length squares and then overlay the bits of information on it I think the correct way of visualising this is not to put the bit of information on the Planck area because there is nothing to put it on. The Planck area is not a physical surface onto which something can be put (any more than a Planck volume is something into which something can be put). The physical reality is that there is only a network of connected points and hence the information is stored by points (junctions) and by their connectivity.

Considering a static snapshot, just for simplicity of visualisation, the unit cell is 4 Planck areas; the bit of information resides on the centre point with it's four Planck lengths radiating from it; basically a 2x2 chess board. The normal rules for sharing edges and vertices in unit cells apply. The perimeter Planck lengths (of the 2x2 array) contribute halves (becoming wholes with the next unit cell), the corners are a quarter of a point (junction) etc. etc.

Obviously this static image is not the physical reality which I view as entirely dynamic and which, depending on how you wish to use language to construct the dynamic physical reality, is either information flow along the Planck length connections between junctions in the network or the appearance/disappearance of the bits of information at the junction due to quantum fluctuations. In either case the dynamics are concerted so as to ensure that the one bit per four Planck area unit cell condition is met.

I would make one final point about black holes and this visualisation. The unit cell as described contains one bit of information. The unit cell is fine as a visualisation for an (essentially) infinitely extended surface but how can this one bit/4 area/ unit cell be rolled up / folded so as to make a closed volume/shape. I think this may be of interest in the context of "1 bit black holes". Probably, there are many options depending on the topology but one interesting one produces a torus. I think it works for all three types of torus, (ring, horn and spindle). If anyone (a lot cleverer than me) could consider this in the context of black hole geometry I would be very interested to hear some views.

Answer 5

I’m no physicist, but I’ve always liked to think that the square of Planck’s length is a real-life pixel in a simulated universe. The entire universe as we know it is a huge, pixelated depository of information. Where that information came from, I haven’t the slightest clue. I am not a big fan of the big bang theory as it simply does not make sense to me from a logical standpoint.

I assume there there’s a lot more involved with this universe that I can never hope to understand, but from what I have read and from whom I have listened to, it seems that our world and entire universe are nothing but a simulation, and Planck’s length is simply a small part of that.