# If the local is defined as the 'Hubble radius'

1. Apr 15, 2015

### BTBlueSkies

I have great trouble with the concept of 'local' as it seems very generic in ways. Local can be a plank distance away, or a meter away.. or even a Hubble radius away (if your scale is in Hubble radii (sp) or so).

If distance in the direction of travel approaches zero as velocity approaches the speed of light... how does the concept of 'local' vary in real units? Is 'local' a function of speed? I often hear 'local this' and 'local that'... 'local' is a pain if you ask me... (but no one did so this I suppose is a monoloug) (i'll try to keep it clear and short.. and maybe smily.. spelling will not be checked).

I assert the word 'local' should not be so ambigouous.

I was reading the thread 'The rapidity of the FTL expansion of space' and the word 'local vicinity' showed up in the same paragraph as the word 'Hubble radius'. In another conversation a couple weeks ago, my personal first her on PF btw.. 'local' was expressed to me as 'two points 1mm apart are not 'local''... local is actually closer than 1mm apart.

hmmmmmm.....

So... 'Local' is somewhere between less than 1mm and the Hubble radius. ..

It seems to me that the word 'local' needs to be broken down into some formalized vobaculary.

I propose 'local' should be defined as ...

Given a point in space time.. Pn(x,y,z,t), that the idea of 'local' should be absolutely bounded... like |P1-P2| is in some range of local that can be named as like 'Local level 1' or 'Local level 2'

We should not just use the word 'local' and expect the other to 'automatically understand' the bounds of 'local in this situation'.

Back to the quote at the top..
If the local area is a 'hubble radius', or even a 1mm radius, ... then light outruns light in the sense that the photon in front of the photon behind it, with both photons going in the same direction, the first photon will be going faster than the photon behind it.... and the space between the photons will be ever increasing...

Which makes me ask... how can distance go to zero in the direction of travel when considering two photons going in the same direction... if the distance between the two photons is ever increasing reguardless how close together they originated.

'Photons travel at the speed of the space they occupy?'

I suppose, how we are taught is this... as we approach the speed of light, the distance in the direction of travel goes to zero. I interpret this to mean that all distances in front of me go to zero, meaning I am infinitely in front of myself... simultaneously completely as far as imaginable into the forward distance.

I am thinking, instead, I should interpret that distance goes to zero as meaning I travel zero distance through space... meaning I am at rest in space. I am moving at the same speed as space, but not relative to space. If I am not moving through space, then distance is zero. However, space is itself moving...

Interesting..
I

Last edited: Apr 16, 2015
2. Apr 16, 2015

### Orodruin

Staff Emeritus
How "local" is defined is not up to you or this forum to propose. It essentially means "in a sufficiently small neighbourhood".

3. Apr 16, 2015

### Ibix

There's usually a well-specified meaning when physicist types are talking about "local", although it may vary between applications and it may or may not have an expression and it may or may not include a scale in the sense you want to define. If I'm not mistaken about the context of PeterDonis' comment, "local vicinity" means "a patch of space-time small enough that space-time curvature is negligible". He's talking generally, and the actual size of the patch varies wildly depending on the space-time you are in. Near a black hole, "local" might be almost no size at all. In special relativity it is the whole of space-time. Most realistic cases are somewhere in between.

That is the basic problem with the approach you are proposing. "Local" is a well defined concept, but it may translate to a whole host of different scales, even in one application of the word.

4. Apr 16, 2015

### Ben Niehoff

This is correct. Keep in mind that "sufficiently small" depends on context. Sometimes it can mean a neighborhood that is well approximated by the tangent space; other times it might mean an entire coordinate chart, or any contractible region of spacetime.

5. Apr 16, 2015

### Staff: Mentor

This is true, and the responses to your post give the ways "local" is used in GR. However, I was actually using it in a much simpler sense: no object outruns a light beam that is passing right next to it.

"Local" in the usual GR sense (the sense other responders are talking about) means "local in spacetime", not "local in space". Changing frames does not change spacetime, so it doesn't affect what "local" means.

In the sense I was using "local", this doesn't really matter, because we are talking about an object and a light ray that are assumed to be spatially co-located at some instant of the object's proper time.

6. Apr 17, 2015

### BTBlueSkies

Thanks for the replies.... I appreciate the patience on this forum.