MeJennifer said:
The past light "cone" is a 4d object and is like an expanding sphere deformed by spacetime curvature.
Jennifer we are talking at the level of schematics, not actual 4D objects
. I mentioned the "teardrop silhouette" which you get for example in Lineweaver's Figure 1. Also, IIRC, in Hawking and Ellis book. It is a good schematic description using a space time diagram. 1D for space, 1D for time.
StandardsGuy said:
Thanks. It wasn't just a mental picture, there was a drawing, but not by a cosmologist. I am looking for how it is drawn correctly. The teardrop was brought up as an alternative presentation.
Guy, I looked at Jorrie's balloon thread at CR4 and I now understand what you are talking about and how the heartshape schematic illusion comes about. Personally I don't like it because it identifies time with the radius of curvature of the hyperspheres and I don't know any reason in nature to do that.
But it is a catchy schematic. Everything I'm saying assumes the finite hypersphere case. We will eventually find out which is right, the infinite or the finite hypersphere picture, but the data is not good enough yet to make the call. Let's assume that we know finite. So then the radius of curvature at a certain moment of universe time can be mathematically defined and measured based on observations.
In the schematic, you don't have a unique time
axis strictly speaking. You have representations of space at given moments of universe time which are
nested.
Like a Russian doll. Earlier versions of space are inside the later ones. Or like an onion.
It is catchy, even seductive, but it doesn't correspond to the time on anybody's clock. The radius of the hypersphere is called the "radius of curvature" to make sure everybody realizes that it doesn't actually have to exist in nature. Just something one can compute from observations of spatial curvature. A hypersphere does not have to be embedded in a larger 4D spatial surround. So it is not necessarily a real objective physical radius.
Furthermore the radius of curvature does not correspond linearly proportionally to any reasonable idea of time, because radius increases at an uneven rate!
In the past the RoC increased more rapidly on a per second or per year basis. The Roc is basically proportional to the scalefactor a(t). In the past, like in year 380,000 when the CMB was released, the scalefactor was increasing faster than today. And now it is increasing pretty slow but the rate is increasing! If you use * for time deriv, then a* is small but a** is positive. Friedman universe time is the unspoken time coordinate, unless otherwise specified.
So if you nest the hyperspheres and "connect the dots" you get an appealing schematic and under a very special unrealistic assumption you get a heartshape.
The heartshape comes only in the case that the observable corresponds to the total, which observations have ruled out by better than 95 percent. A cosmologist wouldn't draw that schematic and imply "this is how it is". But the heartshape is cute. And if the universe were smaller and more curved then it could be.
Anyway the heartshape in CR4 does
NOT correspond to Lineweaver's Figure 1 teardrop. Lineweaver is consistent with current observations and the heartshape is not.
For observations see Komatsu et al 2009. Google "komatsu WMAP cosmology" They indicate that IF the finite hypersphere case were in fact true (which we don't know yet) then the circumference is at least 630 billion lightyears, with 95 percent certainty, based on curvature measurement.
So the halfcircumf of our hypersphere is at least 314 call it 300 billion lightyears.
But the radius of the observable is much less! It is more like 45 billion!
To get the heartshape you need them equal, you need 45 to equal 300. Then the antipode is visible to us, part of our observable. But 45 is not equal 300 and the antipode is not part of our observable. And as far as we know (using LCDM) it never will be. Alas.
The upshot is that nesting does not give a good spacetime representation in which time is presented proportional or linear, but it's appealing.
And if you go ahead and go with that representation regardless of the unrealism, then you do not get a heartshape anyway, for the observable or the past lightcone. Because in actual fact the observable is not big enough to go all the way round (even if you take the absolute minimum possible size allowed by the observations.) Alas.
