Frank Weil said:
Matterwave said:
Mordred said:
It might be worth it-- if only to see that it is not necessary to "do anything to space or time" to get GR. It's really all just a question of prejudice-- do we think it's harder to monkey with matter, or with space? As for the constants, if expressed in non-dimensional form, which is the sensible form for constants anyway, nothing would need to happen to them.craigi said:
I hope you didn't think my "crank" comment applied to you. It only applied to my uneducated reaction to you! But the point I was making still holds-- if all GR does is tell us ratios of scales, we can never know what is "causing" the ratios to change, be it changes in spacetime, changes in matter, or even if there is any meaning in that distinction. Would you say the situation is different in twistor theory?Frank Weil said:
craigi said:
Ken G said:
craigi said:
I don't see why there are any more variables. You write all the constants in dimensionless form, and you write the EFE. That's it, there's your physics, that's everything you can test. The metric is dynamical, the CMB is redder than when it was emitted.Chronos said:
In GR, bound systems don't "expand with space." That answers your question-- what would need to shrink, in the other picture, is bound systems. The universe is not one of those, so no, it does not shrink, it stays put. Nothing happens to it at all, that's kind of the point.Matterwave said:
The same way you draw the line when you say "space expands"-- I could just as well ask you, "how about the space inside atoms?"
Matterwave said:
Chronos said:
Mordred said:
Indeed, the symmetries should be the centerpiece. The main symmetry in cosmology is translation symmetry, which is easier to support when nothing is going anywhere, i.e., when the matter is shrinking. If the space is expanding, it's of course the same thing, but a little harder to see that there is translation symmetry because the space is not just sitting there! If space is not said to be expanding, then clocks must be said to be speeding up, because it takes light more time to cross between galaxies. That is to say, the ratio of the time for light to cross between galaxies, and the time for an atom to oscillate, is increasing. The dimensionless speed of light stays the same (all dimensionless constants do), by which I mean, the ratio of the number of rulers crossed, to the number of atomic oscillations during the crossing, stays the same. That is all consistent with the rulers shrinking.craigi said:
Yes, I did just make it up, but then, so did whoever said space is expanding. How is that not just made up, can anyone cite evidence that is happening? Of course not, we have no model of space, we have only the predictions of GR. No part of the mathematics of GR says space is expanding, it's pure made-up language, accepted uncritically as if it was really saying something we could ever test, which then prompts people to ask "where does the space come from." That's my point here, the question emerges from a non-model, no part of GR asserts that space is actually expanding. Indeed, it seems to me one of the main points of all of relativity is noticing the difference between observations and coordinate systems!craigi said:
No, it wouldn't affect a single observable that GR predicts, that's obvious because all GR predicts is the dynamics of the metric, and all a metric gives you is the number of rulers that could lay end to end between two events, or the number of times some atom oscillates. That's it, that's all you ever measure, and that's all the mathematics of GR ever refers to. If you doubt that, then feel free to tell me something else it refers to.Mordred said:
None, the Planck mass does not vary, if expressed in dimensionless form (like how many protons is it).
I don't know quite what this "cosmon" metric is, but if it's what I'm talking about, it cannot be the least bit different from the standard metric. Perhaps they are talking about something different, or perhaps they are talking about the same thing and don't realize it. I'm sure that what I'm talking about is the exact same metric, for the reasons I just gave.
I would say they are equivalent, but can be distinguished. It's just semantics, if you'd prefer some other word for models that make all the same predictions, that's fine by me. It's actually not relevant to the picture I'm describing, as the steps for doing the math are precisely the same as for any GR solution. There is no difference at all until the mathematics, which leads to the predictions, is translated into words that say why the predictions come out the way they do. That's the part that isn't really physics at all, which is what I'm saying-- "space is expanding" is not physics, its social convention, much like a coordinate system. It can affect our cognitive resonance, but is ultimately subjective-- when we do an inclined plane problem, some people like to align their coordinate axes with gravity, others with the plane, but the physics is not different, and we do not say one picture is right and the other is wrong unless it yields an answer that does not predict the observations.abitslow said:
It sounds like you are saying you want the spacetime manifold to be a real entity. What I'm saying is, we don't want to attribute it any real properties than are more than we need for it to support GR, since GR is the only tested theory we have here. I'm not a mathematician, and they are the ones who keep careful track of what aspects you need to keep the same on the various possible manifolds and metrics you could have that would induce all the same observable physics. All I'm saying is that there is one example which pretty obviously induces the same physics, which is a universe that globally respects a cosmological principle, and either has space expanding with age, or bound systems contracting with age. The "entity" of the spacetime manifold might sound like it is doing two different things in those cases, but it could just be the same thing seen from a different perspective. The distinctions are superfluous, like looking at a painting from different angles. What do we usually do with superfluous elements? Ask the aether!guywithdoubts said:
guywithdoubts said:
Hello craigi,craigi said:
Mordred said:Universe geometry
The origins of the universe is unknown in cosmology. The hot big bang model only covers the history of the universe from 10-43 seconds forward. Prior to that is described as a singularity. However its important to note that the singularity is not a black hole style. Instead singularity in this case simply means a point in time where our mathematics can no longer accurately describe it. Numerous youtube videos and pop media articles would have you believe our universe exploded from some super particle. This was never predicted by the hot big bang model.
The observable universe which is the portion we can see is a finite, sphere with a radius of 46 Gly, which is equal to 46 billion light years. The 46 Gly particle horizon refers to the today's distance of objects, whose radiation emitted in the past we receive today. The overall size of the universe is not known, it could be infinite or finite. If its infinite now then it would be infinite in the past, a finite value can never become infinite. So why is geometry so important to cosmology if we know the size of the observable universe? The answer to that question lies in how geometry affects the following aspects, Light paths, rate of expansion or collapse and overall shape.
In regards to light paths and geometry a closed universe described as a sphere will have two beams of light emitted at different angles eventually converge. An open hyperbolic universe such as a saddlebag will have those same two light beams diverge. A flat universe will have parallel light paths (provided the beams at emission were parallel to begin with)
You will notice on each image there is a triangle, this triangle represents how the geometry affects our measurements. In a flat curvature the three angles of a equilateral triangle will add up to 1800. A positive curvature will add up to greater than 1800, a negative curvature will add up to less than 1800
Image from http://universeadventure.org
The topography of the universe is determined by a comparison of the actual density (total density) as compared to the critical density. The critical density is represented by the following formula
\rho_{crit} = \frac{3c^2H^2}{8\pi G}
\rho=energy/mass density
c=speed of light
G= gravitational constant.
density is represented by the Greek letter Omega \Omega so critical density is \Omega crit
total density is
\Omegatotal=\Omegadark matter+\Omegabaryonic+\Omegaradiation+\Omegarelativistic radiation+{\Omega_ \Lambda}
\Lambda or Lambda is the value of the cosmological constant often referred to as "dark energy" more accurately it is the vacuum pressure that attributes to expansion.
the subscript "0"for \Omega shown in the image above denotes time in the present.
Energy-density is the amount of energy stored per unit volume of space or region. Energy per unit volume has the same physical units as pressure, the energy or mass density to pressure relations are defined by the equations of state (Cosmology). see
http://en.wikipedia.org/wiki/Equation_of_state_(cosmology)
\Omega=\frac{P_{total}}{P_{crit}}
or alternately
\Omega=\frac{\Omega_{total}}{\Omega_{crit}}
Geometry in 2D
In developing a theory of space-time, where curvature is related to the mass-energy density, Scientists needed a way of mathematically describing curvature. Since picturing the curvature of a four-dimensional space-time is difficult to visualize. We will start by considering ways of describing the curvature of two-dimensional spaces and progress to 4 dimensional spaces.
The simplest of two-dimensional spaces is a plane, on which Euclidean geometry holds.
This is the geometry that we learned in high school: parallel lines will go off to infinity
without ever crossing; triangles have interior angles that add up to 180. Pythagoras’
theorem which relates the lengths of the sides of a right triangle also holds:
c2 = a2 + b2
where c is the length of the hypotenuse of the right triangle, and a and b are the
lengths of the other two sides. One can generalize the Pythagorean theorem to three dimensions as well:
c2= a2 + b2 + c2
see image 2.0 below
On a plane, a "geodesic" is a straight line(shortest distance between two points). If a triangle is constructed on a flat 2 dimensional plane by connecting three points with geodesics. The curvature can be represented in 2D, if you establish each angle of a equilateral triangle with
\alpha,\beta,\gamma for a flat geometry this follows the relation
\alpha+\beta+\gamma=\pi.
image 1.0
On a plane, (shown above) we can set up a cartesian coordinate system, and assign to every point a coordinate (x; y). On a plane, the distance ds between points (dx and dy) is given by the relation
d{s^2}=d{x^2}+d{y^2}
If a triangle is constructed on the surface of the sphere by connecting the angles will obey the relation
\alpha+\beta+\gamma=\pi+{AR^2}
image 1.1
where A is the area of the triangle, and R is the radius of the sphere. All spaces in which
\alpha+\beta+\gamma>\pi are called positively curved" spaces. It is a space where the curvature is homogeneous and isotropic; no matter where you draw a triangle on the surface of a sphere, or how you orient it, it must always satisfy the above equation.
"On the surface of a sphere, we can set up polar coordinates "north pole" and "south pole" and by picking a geodesic from the north to south pole to be the "prime meridian". If r is the distance from the north pole, and \theta is the azimuthal. angle measured relative to the prime meridian,"(1) then the distance ds between a point (r; \theta) and another nearby point (r+dr+\theta+d\theta) is given by the relation
{ds^2} = {dr^2} + {R^2} {sin^2}(r/R)d\theta^2
"An example of a negatively curved two-dimensional space is the hyperboloid, or saddle-shape. A surface of constant negative curvature. The saddle-shape has constant curvature only in the central region, near the "seat" of the saddle."(1). David Hilbert proved that a constant negative curvature cannot be constructed in a Euclidean 3D space. Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices \alpha
\beta,\gamma obey the relation \alpha+\beta+\gamma=\pi-{AR^2}.
{ds^2} = {dr^2} + {R^2} {sinH^2}(r/R)d\theta^2
image 1.2
A negative curvature is an open topography
If a two-dimensional space has curvature or flat which is homogeneous and isotropic, its geometry can
be specified by two quantities k, and R. The number k, called the curvature constant, R is the radius
k = 0 for a flat space,
k = +1 for a positively curved space,
k = -1 for a negatively curved space
Geometry in 3D
A two dimensional space can be extended to a three-dimensional space, if its curvature is homogeneous and isotropic, must be flat, or have uniform positive curvature, or have uniform negative curvature.
The 3 possible metrics for homogeneous and isotropic 3D geometries can be represented in the form ds2=dr2=Sk(r)2dΩ2
where
dΩ2=dθ2=sin2d\phi2
<br /> S\kappa(r)=<br /> \begin{cases}<br /> R sin(r/R &(k=+1)\\<br /> r &(k=0)\\<br /> R sin(r/R) &(k=-1)<br /> \end {cases}<br />
If a three-dimensional space is flat (k = 0), it
has the metric
ds2 = dx2 + dy2 + dz2 ;
expressed in cartesian coordinates or
{ds^2} = {dr^2} +{r^2}[d\theta^2 + {sin^2} d\phi^2]
If a three-dimensional space has uniform positive curvature (k = +1), its
metric is
{ds^2} = {dr^2} +{R^2}{sin^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]
A negative curvature in the uniform portion has the metric (k=-1)
{ds^2} = {dr^2} +{R^2}{sinH^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]
Geometry in 4D
Thus far we have discussed the 2 and 3 dimensional components. The Friedmann-Lemaitre-Robertson-Walker metric (FLRW) can be used to describe the 4D dimensions with the use of a(t). a(t) is the scale factor. See the redshift and expansion article for more information or the cosmocalc.
http://cosmology101.wikidot.com/redshift-and-expansion
http://cosmocalc.wikidot.com/start
Scale factor in a homogeneous and isotropic universe describes how the universe expands or contracts with time.
The FLRW metric can be written in the form
d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]
references
(1)"Introductory to Cosmology" Barbera Ryden"
images 1.0,1.1 and 1.2 (see (1))
(2)"Modern Cosmology" Scott Dodelson
(3)"lecture notes, Introductory to Cosmology" Dr. Ka Chan Lu
Frank Weil said:
craigi said:
Frank Weil said:
Except that it fails miserably to reproduce the Standard Model...Frank Weil said:
Frank Weil said:
Drakkith said:On a side note, please don't quote the entirety of very long posts. It just clutters up the thread.
Son of a... I didn't know that Mordred... thanks!Mordred said:
Drakkith said:
Mordred said:
