paths

18:40

Learned the other day (or re-learned perhaps) that the time dilation (relative to a a comoving observer that’s essentially infinitely far away) for an object hanging out in a gravitational field is the same as if it had fell from infinity and was currently at that same radius from the center. In the former case “due to” being in the gravity well and in the latter “due to” the speed it accumulated by falling inwards.

So it’s essentially as if everything here on Earth just fell in from infinity and then stopped suddenly (and then the Earth’s rotation was added in, and of course there’s all the other relative motion but that’s not important here).

Then I realized – that’s because everything here on Earth did basically fall here from infinity. Or at least it took some convoluted path from the beginning of time to here. It might have been energy and is now matter, or some more complicated sequence therein. But nothing here got here by magic. 

The consistency of spacetime is really pretty cool when you see it.

nats

 April 12, 2021

13:51

Last night I realized that the BH entropy bound functions were using nats as the unit of entropy, not bits. So the area relationship for bits is A/(4*ln(2)). I need to read more about why entropy uses e as the base, but I was thinking if maybe this means there’s an adjusted Planck length to be used that removes that ln(2) – and possibly the 4 - so that the area, expressed in these units, is equal to the information bound in bits. The adjust value would be about 2.69E-35 meters.

Well of course I’m not the first to think of this:

https://www.lri.fr/~parrighi//popular/Planck.pdf

Was nice to see that at least I calculated the right value, lol. 

I hesitate about the 4 part because I don’t think it’s clear if the entire surface creates the limit or just some aspect of it (eg the shadow – I still like my idea of it being the shadow, to provide orientation of the state. However the ratio of 4 only applies when there are three spatial dimensions, and I’ve seen enough written up about the bound being valid in however-many dimensions to doubt that. Though, I also wonder – would Planck’s constant be the same in a higher-dimensional environment?).

[later: apparently the answer to 'why use natural log' is 'because it makes all of the calculus neater' and then you can switch units after you're done because it's just a constant adjustment]

Jerk et al

 April 11, 2021

15:01

Today’s crackpot thought: I’ve often wondered just why our intuitive grasp of “derivatives of position relative to time” stops at 2: position, velocity, acceleration. After that it’s just “stuff related to the change in acceleration”. We have names for it (jerk being the famous one), but there’s nothing intuitive, and no fundamental relationship between those derivatives and other things we feel/observe (like we get for F=ma).

I think I did come to the conclusion that circular motion (in flat space, eg not orbiting) involves an acceleration with constantly changing direction, so there’s that. But it’s not quite the same.

Anyway, today’s thought: maybe it’s because they don’t exist. If acceleration is fundamentally quantized, then there’d be no derivative. We can talk about the macroscopic change in a versus t, but in reality there’d be no rate of change, just change. Acceleration would be forced to jump from one level to another. 

Somehow I got to this while thinking about the “time dilation inside an accelerating spaceship” thing, where even a constant “gravitational” field leads to time dilation between to points at different “heights”. And then realizing that perhaps the force necessary to create acceleration is just because it takes something to maintain such a time gradient. But can’t reconcile that with SR.

Yay internet

 OK thanks to some helpful folks at math.stackexchange, now I know that I was coming up with something called a Volterra Integral (https://en.wikipedia.org/wiki/Product_integral). 

https://math.stackexchange.com/questions/4095179/e-to-an-integral-as-an-infinite-product-via-the-definition-of-the-integral

math!

 Oh, OK. So I'm pretty sure that works out to just

1/c^2 * g(x) * Lp

Where g(x) is the local gravity at position x, and Lp is the Planck length. It's unitless. 

If you have the product of N of (1+that) where N is h/Lp, it's reasonably approximated by exp(1/c^2 * integral of g() from x to x+h. Since h/Lp is going to be something pretty huge so it's not an infinite product, but it's like 10^40 products which I think is close enough.

And thanks interwebs for helping me find the full proper gravity in the context of GR: https://physics.stackexchange.com/questions/47379/what-is-the-weight-equation-through-general-relativity

Anyway, so the thing at the top is the "extra time penalty" at each step. I'm looking now at 1/that, which could be interpreted as the rate at which a layer of spacetime can "handle" the information it is being requested to convey outwards. Then factoring in terms of Rs, then in terms of As, then I = As/4, and then factoring r into A, to get some notion of the rate in terms of information and area. Then splitting A up into As+Ar again since things stop when As is full of bits, so it must really depend on the remainder. I've got a thing, it's not necessarily pretty yet though.

math (sigh)

 April 1, 2021

23:18

I think I basically am just looking for a particular part of the EFE. The time dilation from one spot to another spot slightly further “out” is going to be (1+k), and I want to know that k. The cumulative dilation over a longer distance will be the product of all those (1+k_i), and I think if each of those k_i is going down linearly? Or perhaps with the square of the distance, then you end up with a product series that leads to a power of e. Like – subdivide the k_i function into enough pieces, and it’s perhaps something like e^(integral of that function)? Which is what I see as one solution for a simple gravitational well:

T_{d}(h)=\exp \left[{\frac {1}{c^{2}}}\int _{0}^{h}g(h')dh'\right]

https://en.wikipedia.org/wiki/Gravitational_time_dilation

I’m just not up to speed enough to reverse back out of that and figure out what the k_i would be. But it’s probably just the spacetime gradient. I want to get that and swap in I (from I = As/4) and A to get something that looks like the “information flow function”. Then the question is, is that continuous or is it quantized to planck length “onion skin” segments (and then is it the radius or the area that’s quantized?).