Discuss events which are simultaneous in one frame?

[JesseM]

Not really, if the universe is bigger (and curved) you just need longer rods for your triangle to make a noticeable difference in the sum of internal angles. I took an optimistic case, that the observable universe is all there is.

Fair enough, as long as you agree that the total universe could also be much larger than the observable universe.

With regard to the Schwartzschild radius argument, it is not just the radius that matters, it is the density. The argument goes a little like this:

1. The Copernican Principle states that wherever we are in the universe it looks pretty much the same (which means there is no big empty space around us into which the mass of the universe is expanding) and leads to the cosmological principle.

2. This means that our observable universe is not essentially different from the observable universe as observed from the most distant reaches of our observable universe (or the Andromeda galaxy, to use your example.)

OK, I agree that this is implied by the Copernican Principle.

4. If the radius of the observable universe and the mass/density of the observable universe matches that for an event horizon, then the universe being bigger and isotropic just means that the mass/density of the universe will be greater than that required to constitute an event horizon.

Only if you imagine the universe is not expanding. Since it is, you can't assume that the Schwarzschild calculation can tell you whether to expect an event horizon. And note that the larger the region of space you consider, the greater the rate at which points on opposite ends of this region are moving apart due to the expansion of space.

(Figures from http://en.wikipedia.org/wiki/Observable_universe.)

The observable universe has a radius of about 4.65 billion light years, or 4.65 billion times9,460,730,472,580.8 km = 4.3*10^25 m.

This gives a volume of 3.6*10^77 cubic metres.

The observable universe is calculated to have a mass of 3*10^52 kg - this is taken from the measured stellar density of 3*10^-27 kg/cubic metre (wikipedia contributers did this calculation, not me).

The Schwartzschild radius for the mass of the observable universe is:

r=2Gm/c^2=2*6.67*10^-11*3*10^52/(3*10^8)^2=4.4*10^25 m

Remember if the radius is greater, then the density of the universe has to be lower than has been measured and I have only ever heard arguments for the reverse, that the density of the universe is greater than measured because of "dark matter".

I know you have said that this equation does not apply to expanding matter, but equally, can you see that its a pretty nice match between figures here?

The wikipedia article gives the radius as 46.5 billion light years, not 4.65 (and I corrected the article to read 46 billion, since this is the number given in the reference they cite). So that would give a radius of 4.35 * 10^26 meters, and a volume of 3.45 * 10^80 meters^3. Multiply this by the density of 3*10^-27 kg/m^3--and this figure is based on the estimated total density of all forms of energy including dark matter and dark energy, not just the "stellar density" as you wrote--and we have a mass of about 1.0 * 10^54 kg. Multiply by 2G/c^2 to get the corresponding Schwarzschild radius and I find it works out to 1.5 * 10^27 meters, about 3.4 times larger than the actual radius--this is still fairly close, but not as close as the 1.023 difference that you got.

 

[JesseM]

I have no problem with spacetime being curved due to mass and energy, locally this will be very noticeable. But in a grander scale, curvature of the entire universe due to the mass and energy it contains will only be noticeable if the universe is closed and you are playing around with extraordinarily long rods. I have no problems with that. But elsewhere you said that the universe is not considered to be closed by most people.

If you're talking about using rods, then you're still dealing with the curvature of space, not of spacetime. The curvature of spacetime shows up in the proper time along worldlines, not in the spatial distance along rods.

And yes, the football visualization is just for a closed universe. I can't think of any decent way to picture a flat or open universe using a visualization where space is a curved 2D surface; if we could visualize curved 3D surfaces this would probably work better. Anyway, the point of the visualization is just to give some intuitive notion of what "spacetime curvature" would mean, not to accurately represent the shape or future of our universe. (doesn't your visualization also assume the universe has the shape of a hypersphere, which it might very well not?) Although I would point out that since our universe is very close to the critical density Omega which would give zero spatial curvature, it could be slightly above it by an amount that falls within the margin of error; because of the cosmological constant, even if our universe has positive spatial curvature it probably would expand forever rather than collapse. For such a universe you could picture something that starts out looking like a football on one end, but with the cross-sections just getting bigger and bigger forever as you move away from the pointed end.

 

[neopolitan]

The wikipedia article gives the radius as 46.5 billion light years, not 4.65 (and I corrected the article to read 46 billion, since this is the number given in the reference they cite). So that would give a radius of 4.35 * 10^26 meters, and a volume of 3.45 * 10^80 meters^3. Multiply this by the density of 3*10^-27 kg/m^3--and this figure is based on the estimated total density of all forms of energy including dark matter and dark energy, not just the "stellar density" as you wrote--and we have a mass of about 1.0 * 10^54 kg. Multiply by 2G/c^2 to get the corresponding Schwarzschild radius and I find it works out to 1.5 * 10^27 meters, about 3.4 times larger than the actual radius--this is still fairly close, but not as close as the 1.023 difference that you got.

You do realize that that puts us even more inside an event horizon? How much more mass can you have in an event horizon because the universe is expanding?

Could you reread this http://en.wikipedia.org/wiki/Observable_universe#Estimation_based_on_the_measured_stellar_density and fix it if is wrong or let me know if I misread it.

cheers,

neopolitan

 

[neopolitan]

If you're talking about using rods, then you're still dealing with the curvature of space, not of spacetime. The curvature of spacetime shows up in the proper time along worldlines, not in the spatial distance along rods.

Well, I didn't bring in the triangle initially, so I am happy to drop it, if it no longer applies. All I am getting at is that the larger the space or spacetime being considered the greater the overall curvature has to be to actually matter.

<snip> Although I would point out that since our universe is very close to the critical density Omega which would give zero spatial curvature, it could be slightly above it by an amount that falls within the margin of error; because of the cosmological constant, even if our universe has positive spatial curvature it probably would expand forever rather than collapse. For such a universe you could picture something that starts out looking like a football on one end, but with the cross-sections just getting bigger and bigger forever as you move away from the pointed end.

So, the one ended football is what you prefer? I will post my diagrams tomorrow.

 

[JesseM]

You do realize that that puts us even more inside an event horizon? How much more mass can you have in an event horizon because the universe is expanding?

Presumably it depends on the rate of expansion (and the bigger the volume you're considering, the more significant expansion becomes on that scale). Really, do you think it's likely that general relativity predicts the matter in the observable universe should form an event horizon and collapse, but no cosmologists have realized this because they haven't bothered to do the analysis?

Could you reread this http://en.wikipedia.org/wiki/Observable_universe#Estimation_based_on_the_measured_stellar_density and fix it if is wrong or let me know if I misread it.

Yes, there were some errors there, they were assuming the mass due to stars would be most of the mass of the observable universe, when in fact the study of the microwave background radiation in combination with the most popular cosmological model suggests visible matter makes up less than 5% of all the mass, the rest being composed of dark matter and dark energy. I added a sentence to this section to reflect that, and I also updated this section as well. So using the stellar density would also be the wrong way to calculate the mass of all parts of the observable universe, we should actually be using the "critical density" required for the universe to have nearly-flat curvature as suggested by the WMAP probe, which would be 9.9*10^-27 kg/meter^3. So with the radius at 4.35*10^26 meters, the total volume at about 3.45*10^80 cubic meters, the mass would be at 3.4*10^54 kg. The corresponding Schwarzschild radius would be 2.5*10^27 meters, a little under 6 times larger than the actual radius.

 

[JesseM]

Well, I didn't bring in the triangle initially, so I am happy to drop it, if it no longer applies. All I am getting at is that the larger the space or spacetime being considered the greater the overall curvature has to be to actually matter.

I don't know if I'd agree with that phrasing--the larger the scale, the easier it is to notice the effects of curvature, as is true in the spatial case where the angles a very small triangle would not differ noticeably from 180, but the angles of a triangle spanning a significant portion of a closed universe would.

So, the one ended football is what you prefer? I will post my diagrams tomorrow.

Both the two-ended football and the one-ended one are ways of visualizing valid solutions in GR, but the one-ended one represents a solution that's more likely to be true in our universe (though it's still quite possible that space is not positively curved). If the purpose of the visualization is just to help understand GR, it shouldn't really matter whether you're visualizing a GR solution that's likely to model the real world or not.

 

[neopolitan]

All I am getting at is that the larger the space or spacetime being considered the greater the overall curvature has to be to actually matter.

I don't know if I'd agree with that phrasing--the larger the scale, the easier it is to notice the effects of curvature, as is true in the spatial case where the angles a very small triangle would not differ noticeably from 180, but the angles of a triangle spanning a significant portion of a closed universe would.

Ok, clumsy phrasing. I do think it was clear in context (if you look at past posts for instance and see what how the triangle was being used - which was exactly in line with what you said about a very small triangle). It seems we agree anyway, but let's add a couple of words to see if it makes a difference to clarity:

All I am getting at is that the larger the totality of the space or spacetime being considered the greater the overall curvature has to be to actually matter locally.

cheers,

neopolitan

 

[neopolitan]

Presumably it depends on the rate of expansion (and the bigger the volume you're considering, the more significant expansion becomes on that scale). Really, do you think it's likely that general relativity predicts the matter in the observable universe should form an event horizon and collapse, but no cosmologists have realized this because they haven't bothered to do the analysis?

Your turn to appeal to authority? (Just joking, Jesse.)

No.

But I am not saying that the matter in the observable universe should form an event horizon and collapse.

I am saying that it is possible that we lie within an event horizon and that we always have. I realize that this is all counterintuitive so I think it is entirely possible that many cosmologists and other scientists have pondered it and dismissed it. But not all of them have (note that these mostly consider us to be "outside" a white hole, which I would argue is the same as being "inside" a black hole, but they may differ on that):

http://www.space.com/scienceastronomy/white_hole_030917.html
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BlackHoles/BlackHoles.html
http://space.newscientist.com/article/dn12853-black-holes-may-harbour-their-own-universes.html
http://space.newscientist.com/artic...khole-universe-might-explain-dark-energy.html
http://www.npr.org/templates/story/story.php?storyId=6545246
http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/universe.html (Your reference here, JesseM, which suggests the possibility of the universe as being "outside" a white hole.)
http://www.qsmithwmu.com/the_black_...f_speculative,_current_physical_cosmology.htm (Not sure how reliable this is, it seems to be more of a meta-research paper, analysing other papers, including Lee Smolin's for example. But the point is, other people have bothered to do the analysis.)

cheers,

neopolitan

 

[neopolitan]

Here are the diagrams I promised.

I use the description onion lightheartedly. It is not meant to be a prescriptive term.

The first diagram is "onion simultaneity" - this is equivalent to diagrams which appear earlier in this thread.

Then there is "onion at rest" - ie at rest in the 2+1D universe as onion.

Finally there is "onion motion" - ie in motion in the 2+1D universe as onion.

Each diagram is notated. Please work the maths through yourself to derive length contraction and time dilation equations.

cheers,

neopolitan

 

[JesseM]

Each diagram is notated. Please work the maths through yourself to derive length contraction and time dilation equations.

OK, no, the length contraction and time dilation equations of SR definitely aren't derived from an assumption of a hyperspherical expanding universe, and I think you know this. It looks like you are indeed trying to come up with new physics rather than just visualize widely-accepted facts about relativity, in which case this is not the forum to do it.

(anyway, I don't see how you could 'derive' the correct behavior from your pictures, because the pictures imply that length contraction at a certain velocity depends on being a certain distance apart on the sphere, whereas length contraction has no distance dependence--try drawing two rulers in motion relative to one another but at exactly the same position at some time, what would you 'model' say about their lengths at that time?)

 

[neopolitan]

OK, no, the length contraction and time dilation equations of SR definitely aren't derived from an assumption of a hyperspherical expanding universe, and I think you know this. It looks like you are indeed trying to come up with new physics rather than just visualize widely-accepted facts about relativity, in which case this is not the forum to do it.

(anyway, I don't see how you could 'derive' the correct behavior from your pictures, because the pictures imply that length contraction at a certain velocity depends on being a certain distance apart on the sphere, whereas length contraction has no distance dependence--try drawing two rulers in motion relative to one another but at exactly the same position at some time, what would you 'model' say about their lengths at that time?)

No, I am not trying to come up with a new physics at all. And yes, I know that the equations are not originally derived from an assumption of a hyperspherical expanding universe. I am saying that they can be and that this does not detract from SR in any way.

I did ask you to work through the maths, because it is so simple, but I can do it for you if it has to be that way. I can't do it tonight, since it is late, but I will try to find time to do it tomorrow.

A quick answer to your question, in terms of the model, those two lengths will be oriented such that their lengths are not in the same "plane". It is as if both see the other from an angle, such that their lengths are foreshortened. The magnitude of this foreshortening is related to their relative velocities - the equation given by length contraction.

There is no requirement for the lengths to be a distance apart from each other. There is no distance dependence.

cheers,

neopolitan

PS - Please bear in mind my question a few posts back "Is this inconsistent with SR?" My argument is that it is not inconsistent with SR. I do not argue that physics needs to be revised to be in accordance with my model.

Would it help to think of this: your one ended football model is viewed from the side so that the "arrow of time" goes from left to right, for example. If you swing your camera angle around so that the "arrow of time" is coming out of the paper and right at you, what does your model look like now?

 

[JesseM]

No, I am not trying to come up with a new physics at all. And yes, I know that the equations are not originally derived from an assumption of a hyperspherical expanding universe. I am saying that they can be and that this does not detract from SR in any way.

But your ideas clearly aren't compatible with SR, because as I said before, a universe with positive curvature cannot be one where SR is correct globally. GR does reduce to SR locally, but the whole point of this reduction is that you zoom in on a sufficiently small region of spacetime so that the curvature of spacetime, including the curvature of the universe as a whole, becomes negligible in this region. And GR reduces to SR locally in all allowable spacetimes, not just ones where space is a perfect hypersphere at all moments. So, any explanation which regards space being curved into hyperspheres as integral to understanding length contraction can't possibly agree with either SR or GR.

Feel free to email me your math if you want a specific critique, but I really don't think it belongs in this forum.

Would it help to think of this: your one ended football model is viewed from the side so that the "arrow of time" goes from left to right, for example. If you swing your camera angle around so that the "arrow of time" is coming out of the paper and right at you, what does your model look like now?

The time axis that goes through any point on the surface (the point representing a single event in spacetime) always lies along the spacetime surface, perpendicular to the surface of simultaneity at that point; it isn't even meaningful to talk about a time axis that jumps off the surface and into the higher-dimensional embedding space, since as I said before, the embedding space has no physical meaning. Also, although you can slice the spacetime into different possible surfaces of simultaneity and this will give you different time axes through a given point, you don't have arbitrary freedom to draw the time axis in any direction, since it always has to lie within the light cones of that point.

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

But your ideas clearly aren't compatible with SR, because as I said before, a universe with positive curvature cannot be one where SR is correct globally. GR does reduce to SR locally, but the whole point of this reduction is that you zoom in on a sufficiently small region of spacetime so that the curvature of spacetime, including the curvature of the universe as a whole, becomes negligible in this region. And GR reduces to SR locally in all allowable spacetimes, not just ones where space is a perfect hypersphere at all moments. So, any explanation which regards space being curved into hyperspheres as integral to understanding length contraction can't possibly agree with either SR or GR.

Feel free to email me your math if you want a specific critique, but I really don't think it belongs in this forum.

Since I am saying this is consistent with SR and would like to see where this is not consistent with SR, then I think it is the right forum. If it helps I can preface each post with something along the lines of "I am providing the following in an attempt to see whether a conceptual model is consistent with SR, please check the posts of advisors and mentors in reply to the following content." (Or what I have written above.)

What does disturb me is that you clearly do not understand what is in the model and rather than say that, you make assumptions about what I am saying which are not correct and then assign me with some odd ideas which I would have to have if your assumptions were correct.

Can you please make an attempt to understand what I am getting at and then provide a critique based on that?

The time axis that goes through any point on the surface (the point representing a single event in spacetime) always lies along the spacetime surface, perpendicular to the surface of simultaneity at that point; it isn't even meaningful to talk about a time axis that jumps off the surface and into the higher-dimensional embedding space, since as I said before, the embedding space has no physical meaning. Also, although you can slice the spacetime into different possible surfaces of simultaneity and this will give you different time axes through a given point, you don't have arbitrary freedom to draw the time axis in any direction, since it always has to lie within the light cones of that point.

Here seems to be a point of misunderstanding. Take another look at my model. At any point on the surface of simultaneity, the "arrow of time" is perpendicular.

I never said anything about having a time axis which jumped off the surface into a higher-dimensional embedding space. All I am talking about it changing your camera angle.

Try to take step back, think of the the football again. Look at the diagram attached.

Are we obliged to look at the model from only one viewing area? If not, think of the arrow of time as a handle, with chains down to the the model. Then twist the model around 90 degrees. What do you see then?

The reverse can be done with my model. If I think of what I have drawn as resembling a collapsed telescope, then I can also look at it from from the side. If I want to look at it with the instants spread out, rather than look at them all together, then I have draw out the "telescope" and, lo and behold! - I will see the football model.

Can you see this? Can you see that perhaps what we are saying is pretty much the same with a very slight difference in perspective?

cheers,

neopolitan

 

[JesseM]

Since I am saying this is consistent with SR and would like to see where this is not consistent with SR, then I think it is the right forum. If it helps I can preface each post with something along the lines of "I am providing the following in an attempt to see whether a conceptual model is consistent with SR, please check the posts of advisors and mentors in reply to the following content." (Or what I have written above.)

What does disturb me is that you clearly do not understand what is in the model and rather than say that, you make assumptions about what I am saying which are not correct and then assign me with some odd ideas which I would have to have if your assumptions were correct.

Can you please make an attempt to understand what I am getting at and then provide a critique based on that?

Sure, but here's my basic question--aren't you trying to "derive" Lorentz contraction in a way that depends on space being curved as in your onion diagram? Or can your idea about deriving the Lorentz contraction equation be generalized to a case where there is no spacetime curvature? If the former, the derivation can't be correct, for the reasons I said; if the latter, then please present your derivation for a situation where space is flat (i.e. instead of surfaces of simultaneity being concentric circles, they should just be a stack of straight lines).

Here seems to be a point of misunderstanding. Take another look at my model. At any point on the surface of simultaneity, the "arrow of time" is perpendicular.

I never said anything about having a time axis which jumped off the surface into a higher-dimensional embedding space. All I am talking about it changing your camera angle.

OK, I misunderstood that, I thought you meant keeping the orientation of the "football" constant while changing the orientation of the time axis so it was coming out at you, which would mean it was perpendicular to the surface. But sure, you can view the "football" (with or without the Big Crunch end) head-on, and if you ignore depth the surfaces of simultaneity would then look like concentric circles (although in the case where the universe contracts, you might have pairs of different surfaces of simultaneity, one from the expanding phase and one from the contracting phase, which seem to occupy the same points on the depth-less diagram).

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

Sure, but here's my basic question--aren't you trying to "derive" Lorentz contraction in a way that depends on space being curved as in your onion diagram? Or can your idea about deriving the Lorentz contraction equation be generalized to a case where there is no spacetime curvature? If the former, the derivation can't be correct, for the reasons I said; if the latter, then please present your derivation for a situation where space is flat (i.e. instead of surfaces of simultaneity being concentric circles, they should just be a stack of straight lines).

We might be able to bypass the need, if I can make something clear - in terms of space, only space, the surface which the sphere represents is flat. Yes, in terms of spacetime it is curved, but not in terms of space. So, a stack of straight lines as surfaces of simultaneity is basically what I have anyway.

I am not sure you have thought about it this way, but you have pretty much the same situation with concentric circles as you have with increasingly large stacks of lines. Think of 10 concentric circles made of elastic, with radii ranging from 1cm to 10 cm. The tricky thing here is that the elastic is such that, when relaxed, each circle is formed of the same length of material. Mark out an arc defined by 36 degrees on each of them a different colour (at the top of the circles to make it easier to visualise, but it doesn't really matter). Cut all the circles at the lowest point and flatten out the lines.

What were previously arcs are now segments, but the lengths don't change because you have them flat rather than circular. Your ruler of length L given by these arcs/segments expands, but not in a way that you can see. However, recall that I said that the "elastic is such that, when relaxed, each circle is formed of the same length of material". Once the elastic relaxes, each length marked will be the same.

I do believe this is the equivalent to what would happen in your flat model, you would just have the elastic stretched in a straight line, rather than into a circle.

If you can derive Lorentz contraction in your flat model, then the same method can be used to derive Lorentz contraction in my model (flat space wrapped around into curved spacetime).

Sadly I am heading off to Easter so I can't rework the diagrams that we have already exchanged in another discussion and post them today, but I think you know the ones I am talking about (ie in the "time dilation is not the temporal equivalent of length contraction" discussion - Lorentz contraction is derived in flat space there. All you need to do is to keep in mind that you must use consistent units and note that your rulers are expanding - so you have to account for that.)

Happy Easter,

neopolitan

 

[JesseM]

We might be able to bypass the need, if I can make something clear - in terms of space, only space, the surface which the sphere represents is flat. Yes, in terms of spacetime it is curved, but not in terms of space.

What do you mean "in terms of spacetime it is curved"--what observable implications is that supposed to have? Are the circles not supposed to represent curved spacelike surfaces of simultaneity, like the circles in my football diagram? Would not the angles of large triangles on such a surface fail to add up to 180? That would be spatial curvature, not spacetime curvature.

I am not sure you have thought about it this way, but you have pretty much the same situation with concentric circles as you have with increasingly large stacks of lines.

Well, this is another problem--what is the physical meaning of having either the circles or the lines grow "increasingly large", if you imagine rulers are growing along with them as in your diagram? In cosmology we do not assume that bound systems like rulers expand along with space, one of the links I posted earlier mentioned it can be shown using GR that things like a solar system would not be expected to expand along space, and in any case the whole notion of "expansion of space" would become physically meaningless if there was no empirical way of observing it since all our measuring-devices were expanding too. So is the expansion in your diagrams not supposed to be physically meaningful, but just some kind of weird coordinate system where the coordinate length of objects is continually increasing even though their physical length isn't changing in any meaningful sense? You're of course free to use weird coordinate systems in flat SR spacetime, so maybe we could come up with a coordinate transform that would give something like the diagram you'd draw with "increasingly large stacks of lines", but then you'd have to drop the notion that this has anything to do with the physical idea of the expansion of the universe in GR.

Think of 10 concentric circles made of elastic, with radii ranging from 1cm to 10 cm. The tricky thing here is that the elastic is such that, when relaxed, each circle is formed of the same length of material. Mark out an arc defined by 36 degrees on each of them a different colour (at the top of the circles to make it easier to visualise, but it doesn't really matter). Cut all the circles at the lowest point and flatten out the lines.

What were previously arcs are now segments, but the lengths don't change because you have them flat rather than circular. Your ruler of length L given by these arcs/segments expands, but not in a way that you can see. However, recall that I said that the "elastic is such that, when relaxed, each circle is formed of the same length of material". Once the elastic relaxes, each length marked will be the same.

I do believe this is the equivalent to what would happen in your flat model, you would just have the elastic stretched in a straight line, rather than into a circle.

If you can derive Lorentz contraction in your flat model, then the same method can be used to derive Lorentz contraction in my model (flat space wrapped around into curved spacetime).

I don't know what you mean by "if you can derive Lorentz contraction in your flat model". If by "my flat model" you mean something like the standard minkowski diagrams used to visualize spacetime in relativity, you don't really derive Lorentz contraction from those diagrams, although you can see how it looks on the diagrams. But remember that those minkowsi diagrams are just based on the Lorentz transformation, showing how the different coordinates of two of the inertial coordinate systems related by the Lorentz transformation would look when plotted together (so if you pick one coordinate system to draw in a cartesian manner with time and space axes at right angles, you can then plot the time and space axes of the other system in terms of what coordinates they cross through in the first system). Since Lorentz contraction can be derived from the Lorentz transformation, naturally it can be visually illustrated in such diagrams.

In contrast, you seem to be starting from a visual picture that isn't grounded in any well-defined coordinate systems which can be constructed in some physical way like inertial coordinate systems in SR, and then trying to "derive" Lorentz contraction from the way rulers are drawn in this physically ungrounded visual picture. This just seems like such a confused approach to how physical derivations work that I don't even know where to start explaining why it doesn't make sense.

All you need to do is to keep in mind that you must use consistent units and note that your rulers are expanding - so you have to account for that.)

Again, if rulers are expanding this would seem physically meaningless--or are your diagrams trying to say that rulers only expand at the same rate if they are at rest relative to one another, and that rulers in motion expand at different rates and that this explains Lorentz contraction? This would at least be some kind of physical-sounding hypothesis, although it wouldn't have any relation to the way the expansion of space works in GR.

 

[neopolitan]

It's clear that I will have to explain. But it won't be today.

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

What do you mean "in terms of spacetime it is curved"--what observable implications is that supposed to have? Are the circles not supposed to represent curved spacelike surfaces of simultaneity, like the circles in my football diagram? Would not the angles of large triangles on such a surface fail to add up to 180? That would be spatial curvature, not spacetime curvature.

Do you want the triangles or not (look about a dozen posts back)? What causes the curvature in your football diagram? Does it help to refer to space and spacetime as being asymptotically flat?

Well, this is another problem--what is the physical meaning of having either the circles or the lines grow "increasingly large", if you imagine rulers are growing along with them as in your diagram? In cosmology we do not assume that bound systems like rulers expand along with space, one of the links I posted earlier mentioned it can be shown using GR that things like a solar system would not be expected to expand along space, and in any case the whole notion of "expansion of space" would become physically meaningless if there was no empirical way of observing it since all our measuring-devices were expanding too. So is the expansion in your diagrams not supposed to be physically meaningful, but just some kind of weird coordinate system where the coordinate length of objects is continually increasing even though their physical length isn't changing in any meaningful sense? You're of course free to use weird coordinate systems in flat SR spacetime, so maybe we could come up with a coordinate transform that would give something like the diagram you'd draw with "increasingly large stacks of lines", but then you'd have to drop the notion that this has anything to do with the physical idea of the expansion of the universe in GR.

What is preventing the rulers from expanding?

Do you recall me stating:

Consequential curvature due to mass is not in the model I gave since it is an SR thing, not a GR thing. So far in this discussion (and hence in my model as shown) I haven't brought in mass to cause curvature.

Yes, all the measuring devices expand too. While I disagree with your dismissive terminology, on a certain level yes, "the expansion in (my) diagrams not supposed to be physically meaningful, but just some kind of weird coordinate system where the coordinate length of objects is continually increasing even though their physical length isn't changing in any meaningful sense".

Do you recall me asking this question:

Is my model inconsistent with SR?

I don't know what you mean by "if you can derive Lorentz contraction in your flat model". If by "my flat model" you mean something like the standard minkowski diagrams used to visualize spacetime in relativity, you don't really derive Lorentz contraction from those diagrams, although you can see how it looks on the diagrams. But remember that those minkowsi diagrams are just based on the Lorentz transformation, showing how the different coordinates of two of the inertial coordinate systems related by the Lorentz transformation would look when plotted together (so if you pick one coordinate system to draw in a cartesian manner with time and space axes at right angles, you can then plot the time and space axes of the other system in terms of what coordinates they cross through in the first system). Since Lorentz contraction can be derived from the Lorentz transformation, naturally it can be visually illustrated in such diagrams.

In contrast, you seem to be starting from a visual picture that isn't grounded in any well-defined coordinate systems which can be constructed in some physical way like inertial coordinate systems in SR, and then trying to "derive" Lorentz contraction from the way rulers are drawn in this physically ungrounded visual picture. This just seems like such a confused approach to how physical derivations work that I don't even know where to start explaining why it doesn't make sense.

This is where I need to make an attempt to explain the approach and what is, as far as I can see and only in my opinion, the one single benefit of my model over yours. I have been thinking about it, but I think I will have to craft the explanation carefully, so it is clear and concise and free from unnecessary distractions. And that takes time.

Again, if rulers are expanding this would seem physically meaningless--or are your diagrams trying to say that rulers only expand at the same rate if they are at rest relative to one another, and that rulers in motion expand at different rates and that this explains Lorentz contraction? This would at least be some kind of physical-sounding hypothesis, although it wouldn't have any relation to the way the expansion of space works in GR.

Sort of, but not quite. And we are not yet in GR. How about, when you think you have a full understanding of what I am saying in terms of SR, you knock down the model in terms of SR? Then we don't even need to bother with seeing if the model has any relevance to GR.

If it is ok in terms of SR, then we can move on to see if it is totally incompatible with GR? Does that sound fair?

cheers,

neopolitan

 

[JesseM]

Do you want the triangles or not (look about a dozen posts back)?

It's not an issue of whether I "want" them or not--my point in the earlier post was that the triangles were a representation of spatial curvature, not spacetime curvature. But in my more recent comment I was reacting to your statement "in terms of spacetime it is curved, but not in terms of space". If the angles of a triangle don't always add up to exactly 180, then it is curved in terms of space--do you disagree?

What causes the curvature in your football diagram?

The mass and energy which is assumed to fill all of space evenly in cosmological models of an evenly curved universe.

Does it help to refer to space and spacetime as being asymptotically flat?

Not if you are using that term incorrectly--a universe where space is curved into a hypersphere can never be asymptotically flat. Asymptotically flat means an infinite space where all the mass is concentrated in one region, and as you get farther and farther from that region the curvature goes to zero. In terms of a 2D analogy, it would look like an infinite plane that has been distorted with a small local depression (look at the diagram http://io.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/gravity-lens-space.gif and imagine extending the plane to infinity in all directions).

What is preventing the rulers from expanding?

See this page which I already linked to earlier, along with section 2.6.2 of this paper.

Yes, all the measuring devices expand too. While I disagree with your dismissive terminology, on a certain level yes, "the expansion in (my) diagrams not supposed to be physically meaningful, but just some kind of weird coordinate system where the coordinate length of objects is continually increasing even though their physical length isn't changing in any meaningful sense".

Well, that at least is helpful. So, your model has nothing to do with the expanding hypersphere picture in general relativity, correct? The curvature of surfaces of simultaneity in your diagrams is just some kind of coordinate feature, and actual physical triangles created out of rods would always have angles adding up to 180? If this is the case I have another question--are the circular surfaces of simultaneity supposed to be distorted pictures of the surfaces of simultaneity used by inertial frames SR, or are you using a different definition of simultaneity in flat SR spacetime? If the former, how do you deal with the fact that each surface of simultaneity for an inertial frame is supposed to represent an infinite space, but yours seems to be only finite?

I don't know what you mean by "if you can derive Lorentz contraction in your flat model". If by "my flat model" you mean something like the standard minkowski diagrams used to visualize spacetime in relativity, you don't really derive Lorentz contraction from those diagrams, although you can see how it looks on the diagrams. But remember that those minkowsi diagrams are just based on the Lorentz transformation, showing how the different coordinates of two of the inertial coordinate systems related by the Lorentz transformation would look when plotted together (so if you pick one coordinate system to draw in a cartesian manner with time and space axes at right angles, you can then plot the time and space axes of the other system in terms of what coordinates they cross through in the first system). Since Lorentz contraction can be derived from the Lorentz transformation, naturally it can be visually illustrated in such diagrams.

In contrast, you seem to be starting from a visual picture that isn't grounded in any well-defined coordinate systems which can be constructed in some physical way like inertial coordinate systems in SR, and then trying to "derive" Lorentz contraction from the way rulers are drawn in this physically ungrounded visual picture. This just seems like such a confused approach to how physical derivations work that I don't even know where to start explaining why it doesn't make sense.

This is where I need to make an attempt to explain the approach and what is, as far as I can see and only in my opinion, the one single benefit of my model over yours. I have been thinking about it, but I think I will have to craft the explanation carefully, so it is clear and concise and free from unnecessary distractions. And that takes time.

Well, take your time then. But please keep my questions above in mind when creating your response. In particular, I'd like to know if you are drawing a distorted picture of the coordinate systems of inertial frames in SR (so each circle is just a normal surface of simultaneity) or if you're defining a new coordinate system; if the latter, I'd like to know the physical basis for its construction (like the inertial rulers and clocks synchronized by the Einstein synchronization convention for inertial coordinate systems in SR), and how it relates to SR inertial frames (what would it look like to plot the axes of an inertial frame on the same diagram as your own new coordinate system)? If it's supposed the same coordinate system but drawn in a distorted way, I'd like to know what new insights the distorted picture is supposed to give, and what is the physical meaning of the features of your diagram that you think are interesting, like the fact that if you draw a straight horizontal line through the two pairs of worldlines representing two rulers, the length of the intersection regions are equal (if your drawings are distorted pictures of inertial frames, then a 'straight' line in your diagram wouldn't be straight in a Minkowski diagram, and the intersections probably wouldn't be equal either).

Sort of, but not quite. And we are not yet in GR. How about, when you think you have a full understanding of what I am saying in terms of SR, you knock down the model in terms of SR? Then we don't even need to bother with seeing if the model has any relevance to GR.

If it is ok in terms of SR, then we can move on to see if it is totally incompatible with GR? Does that sound fair?

Sure, but it was you who was connecting your model to the expanding hypersphere idea in GR, if you had said from the start it was supposed to be a new coordinate system in flat spacetime or a distorted drawing of existing SR coordinate systems, we could have avoided a lot of unnecessary discussion of GR. But then, it doesn't seem like you were clear on these points yourself, which is why I am suspicious of the claim that your diagrams are actually well-thought out enough for my questions above to have answers--again, it seems like you may have started from diagrams that have some features you find interesting, not started from any clear idea of whether the curvature was supposed to be physical or coordinate-based, or on what type of coordinate system the diagrams were supposed to represent. If this is the case then your ideas are "not even wrong", just too ill-defined to have clear physical meaning.

 

[neopolitan]

Sure, but it was you who was connecting your model to the expanding hypersphere idea in GR

I should be in bed, but I have to respond to this quickly.

I may have misled you somewhere, but I don't recall doing so. Where I did connect my model to the expanding hypersphere idea in GR? Could you point out the actual post where I said that.

thanks,

neopolitan

 

[JesseM]

I should be in bed, but I have to respond to this quickly.

I may have misled you somewhere, but I don't recall doing so. Where I did connect my model to the expanding hypersphere idea in GR? Could you point out the actual post where I said that.

Weren't you talking about "your model" when you made the comment about how the expansion of space is time in post #162? And likewise in post #164, wasn't this comment supposed to be based on your own model?

Think about the fact that 1) the universe is expanding and 2) the universe is not expanding uniformly. If it were expanding uniformly, we would never notice it because we would expand with it. What is expanding is the space between masses (masses being concentrations of energy).

And then skipping ahead to post #190 you seemed to be saying that you tossed aside your model but then later rethought that decision after reading some cosmology (you mention Hubble) and realizing that in the standard model the Big Bang did not have any specific center in space:

A very very long time ago (more than 20 years), when I first had it mind as a way of explaining to myself why the two spaceships/two flashlights scenario works (spaceships approach each other at ½c and shine lights at each other, etc etc). I am beyond that now of course - so please don't go into an explanation unnecessarily.

Anyway, I put this model aside because it would imply that the entire universe would be expanding in such a way that everything was moving apart from everything else and things that are further away would be moving away from us faster than things that were close. You can see that that is a problem, since at the time I had the concept of a big bang in which there was a defined centre to the universe. Then one day I had a bit of time at a library and looked things up (20 years ago remember, no internet). Hubble had something interesting for me. So I took another look at my model.

In post #193 you seemed to say that the curved surfaces of simultaneity in your diagrams represented the actual physical curvature of space:

Note that my "universe as onion" is intentionally 3+1 dimensional. The surface of the sphere represents curved 3D space.

And then again in post #198 you suggested the curvature was something physically real and measurable, not just a consequence of using a particular coordinate system in flat SR spacetime:

In any event, if there is curvature which is inherent rather than consequential to mass, effectively this will only manifest over large volumes of the universe - as I alluded to in a recent post. If the universe is infinite then it won't manifest. If it is bounded then my model seems more fitting than yours and the curvature will manifest, but only noticeably if you were to take readings which are ridiculously distant from each other.

In post #204 you responded to a question about whether the angles of a physical triangle would add up to 180, and said that they wouldn't in your model, again indicating the curvature was something physical:

And also including the law that if two straight rigid rods meet at right angles, and a third rod is laid out to meet the ends of each one, then the angles where they meet will always add up to 180? (aren't you allowing for the possibility of curved space in your model, or have I misunderstood?)

Well, if you have an absolutely huge set of rods, which span a substantial portion of the universe, no ...

So, none of this would indicate to me that your onion diagrams in post #219 were always just meant to represent some weird coordinate system (or a distorted drawing of standard SR inertial coordinate systems) in flat spacetime, rather than representing a successive moments in a universe that was actually physically curved into a hypersphere. Then in post #221 you even said that although the Lorentz contraction equation is not normally derived from the notion of an expanding hyperspherical universe, your point was that it "can be":

And yes, I know that the equations are not originally derived from an assumption of a hyperspherical expanding universe. I am saying that they can be and that this does not detract from SR in any way.

So if you are now saying that your model has nothing to do with the physical notion of a universe which is actually physically curved and actually physically expanding in a meaningful sense as in cosmology, this would reinforce my notion that you started with some geometric relationships seen in ill-defined visual diagrams, and are only trying to assign the diagrams a "meaning" in retrospect.

 

[neopolitan]

Checking the context of those comments and will get back to you.

I especially want to check the context of where I said "curved 3D space" because I think I tried to say that the 3D space was wrapped around a 4D shape, but not that 3D is inherently curved itself. So that does seem to be unclear.

But I don't have the time right now

cheers,

neopolitan

 

[JesseM]

Checking the context of those comments and will get back to you.

I especially want to check the context of where I said "curved 3D space" because I think I tried to say that the 3D space was wrapped around a 4D shape, but not that 3D is inherently curved itself. So that does seem to be unclear.

But I don't have the time right now

Sure, no rush--take your time.

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

Post https://www.physicsforums.com/showpost.php?p=1637336&postcount=162".

My comment in #162 does include this:

If it is also the case that the universe expands in such a way that that expansion can be interpreted as the passage of time, then I am also happy. - Note, I am not saying that the universe is expanding with time, or over time, but effectively that very expansion is time. If that is the generally accepted case, then I am very happy.

However, I don't say anything about a "expanding hypersphere idea in GR" and I don't claim the concepts are indistinguishable. I am, however, very interested to hear more about the "expanding hypersphere idea in GR", especially if this is a standard concept. If it is a standard concept, then I may be floundering on the border of proper understanding, ignorant of the fact that my ideas have already been fleshed out by someone else.

Post https://www.physicsforums.com/showpost.php?p=1648368&postcount=193" is poorly phrased. I apologise for the confusion. It is inherently confusing, I suppose, since I am thinking of flat space which has been wrapped around a hypersphere so the whole of it is curved, but only in terms of 4 dimensions, not in terms of 3dimensions. I have said that a few times.

In post https://www.physicsforums.com/showpost.php?p=1650460&postcount=198" I wrote:

In any event, if there is curvature which is inherent rather than consequential to mass, effectively this will only manifest over large volumes of the universe - as I alluded to in a recent post. If the universe is infinite then it won't manifest. If it is bounded then my model seems more fitting than yours and the curvature will manifest, but only noticeably if you were to take readings which are ridiculously distant from each other.

Ignoring the introductory clause "In any event", you may notice that all those sentences start with the word "if". That paragraph followed these paragraphs:

You referred to curvature as a consequence of mass, a gravitation effect. But you seemed to be saying that spacetime is inherently curved. Which is it?

Consequential curvature due to mass is not in the model I gave since it is an SR thing, not a GR thing. So far in this discussion (and hence in my model as shown) I haven't brought in mass to cause curvature.

Can you see that I was not making a statement here, but rather continuing a line of discussion sparked by Dr Greg in post #192 to which I was replying in posthttps://www.physicsforums.com/showpost.php?p=1648368&postcount=193" and also presenting an argument against any meaningful 3D curvature? Also, even if my phrasing in #193 was poor, in the grander scheme of things, the post should still be read in the context in which it was written. That said, while it makes it more unwieldy, I think for the sake of clarity I probably should have written the last sentence something like this:

If the universe is bounded and if there is curvature which is inherent rather than consequential to mass, then my model seems more fitting than yours and the curvature will manifest, but only noticeably if you were to take readings which are ridiculously distant from each other.

Regarding post https://www.physicsforums.com/showpost.php?p=1651603&postcount=204".

In post https://www.physicsforums.com/showpost.php?p=1648368&postcount=193", I posed a question to DrGreg:

I do think that the inherent curvature that you are discussing will similarly only come into noticeable effect when you are considering relatively large chunks of the universe. Do you agree?

DrGreg never replied. JesseM did though, in post https://www.physicsforums.com/showpost.php?p=1648420&postcount=195", where he wrote:

Again, you're talking about spatial curvature here. On human scales, space does appear pretty Euclidean. But spacetime curvature is a lot more obvious--for example, it's why balls travel on parabolas rather than in straight lines (take a look at the nice illustration here, from a book by Wheeler which I definitely recommend picking up a used copy of, showing how although the paths of balls thrown at different speeds trace different curves in space, they can be visualized as having the same curvature in spacetime)

(JesseM seemed to back away from this in post https://www.physicsforums.com/showpost.php?p=1648602&postcount=197" - I say this in case the quote above was made mistakenly, so as to give JesseM due credit for fixing it, not to point out the mistake on his part.)

Anyway, JesseM seems to have attributed a response to something DrGreg wrote as a statement on my part. Hopefully he can see the error there.

I can't do anything about JesseM's notion that I "started with some geometric relationships seen in ill-defined visual diagrams, and are only trying to assign the diagrams a "meaning" in retrospect". The best I can do is show how my ill-defined visual diagrams do actually work.

I would like to do that, but first I need to check what is understood and what is not understood about what has previously been said in this thread.

With respect to post #229, JesseM, if your football has mass and energy in it, then our models are not representing the same thing. I think you have already grasped that.

I won't use "asymptotically flat". I just wondered if it could help.

Your http://math.ucr.edu/home/baez/physics/Relativity/GR/expanding_universe.html" didn't really answer my question directly, but it did indirectly. The ruler I am thinking of is conceptual, not a bound system, not a structure of atoms and molecules. It is a "length" not a physical ruler.

You said:

each surface of simultaneity for an inertial frame is supposed to represent an infinite space

and mine seem to be finite.

According to who or what must a surface of simultaneity for an inertial frame represent an infinite space? My surface of simultaneity is not bound, but not infinite. This may confuse. In my model, 3D space is flat, but if you traveled long enough (and fast enough) you could end up traveling through the same part of the universe again.

Long enough seems clear enough, but why fast enough? Well the universe is expanding in such a way that to travel in one direction and come back to your start position, you would have to travel faster than the speed of light and you can't do that. Space between you and your destination would expand to prevent you getting there. (Anyone for a re-reading of Zeno of Elea's paradoxes?)

So, in effect, my hypersurfaces of simultaneity are infinite even if, strictly speaking, they are finite. (They would only be finite if you could climb out of the universe entirely and observe them from there, but you run into other, more immediate problems if you do that.)

So, I think I have answered everything now. Are we ready to move on?

cheers,

neopolitan

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

Here are the first three diagrams in another attempt to explain motion in the 2+1D universe as "onion".

When I look again at a diagram I posted before, I can see I was trying to explain too much too quickly, so I can understand that it was confusing. Sorry about that.

Try these diagrams.

Think about what Q might be, and then try to think about what happens if something has a velocity greater than Q. You can also think about three inertial observers, the first nominally at rest, the second with a velocity of 2Q/3 relative relative to the first and the third with a velocity of -2Q/3 relative to the first, and how the vectors between the second and third observers will work out.

cheers,

neopolitan

 

[neopolitan]

Two follow on diagrams, on length contraction, are missing. I did try to post them, not sure what happened, but I will try to post again tomorrow.

neopolitan

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

Another attempt to post the last two images. These should be viewed in concert with the images at https://www.physicsforums.com/showpost.php?p=1664296&postcount=235"

cheers,

neopolitan

 

[belliott4488]

Neopolitan and JesseM: I'm kind of intrigued by this discussion you've been having, but frankly, I'm not sure I want to read through all 237 posts to try and figure out what it's all about. Can you point me to a couple of key posts, especially where neopolitan explains this onion picture?

The particular questions I have are the basics: what do these represent? Space and time in a polar plot? What are the coordinates seen by one inertial observer? How would he see the world line of a second observer moving inertially relative to him? Can you do all this in flat space-time, or is this only relevant when the curvature is non-zero?

thanks

 

[neopolitan]

Neopolitan and JesseM: I'm kind of intrigued by this discussion you've been having, but frankly, I'm not sure I want to read through all 237 posts to try and figure out what it's all about. Can you point me to a couple of key posts, especially where neopolitan explains this onion picture?

I have tried to make the diagrams self explanatory, so that if you look at the first two of https://www.physicsforums.com/showpost.php?p=1654211&postcount=219" and then all five in #235 and #237 (they are just below, scroll down to see them) it should be possible to work out what I am saying.

The particular questions I have are the basics: what do these represent? Space and time in a polar plot? What are the coordinates seen by one inertial observer? How would he see the world line of a second observer moving inertially relative to him? Can you do all this in flat space-time, or is this only relevant when the curvature is non-zero?

They represent flat 3D space mapped onto a hypersphere which expands. Coordinates shown are those for an observer nominally at rest (Observer C). You don't see world-lines, but the result would be identical to that you conceptually could "see" with standard SR. The clock held by an observer in motion relative to you ticks slower (and relative to him, your clock ticks slower too - the very first diagram in post #219 attempts to show this, but it may be a bit busy).

Not sure what you mean by "Can you do all this in flat space-time, or is this only relevant when the curvature is non-zero?" Nor do I understand where the question leads.

I could have sworn JesseM wrote something about how curvature in spacetime only shows up in your worldlines rather than odd behaviour of triangles. I can't for the life of me find it though (a problem with so many posts). If I could find it, if it existed, then that might be a step towards answering your question ... possibly.

thanks

cheers,

neopolitan

 

[belliott4488]

I have tried to make the diagrams self explanatory, so that if you look at the first two of https://www.physicsforums.com/showpost.php?p=1654211&postcount=219" and then all five in #235 and #237 (they are just below, scroll down to see them) it should be possible to work out what I am saying.

Sorry, it wasn't obvious to me how to relate these to traditional pictures of Minkowski Space.

They represent flat 3D space mapped onto a hypersphere which expands. Coordinates shown are those for an observer nominally at rest (Observer C). You don't see world-lines, but the result would be identical to that you conceptually could "see" with standard SR. The clock held by an observer in motion relative to you ticks slower (and relative to him, your clock ticks slower too - the very first diagram in post #219 attempts to show this, but it may be a bit busy).

Okay - step 1.: Can you explain how this mapping works? Suppose I have a traditional M. Space diagram, with an observer at the origin and some event at a point (t,x,y,z). How would this observer and event be represented in your diagram, i.e. what would their coordinates be?

Now, when you speak of "the surface of a sphere which is an instant in time, or a hypersurface of simultaneity," I gather you've mapped the t-axis to a radial coordinate. What is the sigificance of the point r=0? Is it the beginning of time? If not, then do times in the past get mapped to r in a way that assymptotically approaches r=0 at t-> -infinity?

One more: your two observers have different hyperspheres of simultaneity, which are evidently not concentric, so they have different r=0 points. What does this mean? Every observer has his own r=0 point, so is that a separate "beginning of time" event for every observer?

Not sure what you mean by "Can you do all this in flat space-time, or is this only relevant when the curvature is non-zero?" Nor do I understand where the question leads.

Not to worry. I saw a reference to "curvature" and thought maybe you were discussing something that applied only to the curvature of space-time produced by gravity in GR. If that's not the case, then let's not even go there.