Relativity and Cosmology

We have seen how cosmic redshift can be understood as photons conserving angular momentum by 'shedding energy' in terms of frequency as the cosmic balloon expands. Is it reasonable to think that particles simply shed speed in order to also maintain a constant angular momentum? So it seems.

Intro

Fig. 1 (right) shows a single massive particle (pink arrow) being shot along the balloon surface at a time when R=25 units. As the balloon expands, the particle maintains its angular momentum relative to the balloon center by traveling more slowly along the balloon surface. This is visible as the shrinking of the angle between the arrow and the radial lines, representing static observers on the surface. Instead of continuously spiraling out, as a photon would have done, the particle's angle to a local radial line eventually approaches zero. This means that over time the particle's surface velocity (momentum) decays to zero.

This effect is qualitatively independent from the expansion profile, be it decelerating, constant or accelerating expansion, as long as there is expansion and not contraction. This is a rather surprising result of cosmic expansion, because it looks like a 'drag' or 'friction' that the cosmic balloon surface (space) exerts on particle movement. However, when there is no expansion (a static universe), the effect disappears, ruling out that it is a friction. What is more, should the cosmic balloon shrink, the particle will gain speed relative to balloon surface, as it has to in order to maintain a constant angular momentum.

Analytics

In Galilean dynamics, it is the kinetic energy (½mv²) and the momentum (mv) of a particle that decay due to an expanding universe. The velocity v is also known as the 'peculiar velocity' in order to distinguish it from the 'Hubble velocity', which is the apparent recession velocity (relative to us) of a distant object that is static relative to the balloon surface.

As the radius R of the balloon increases, the v of the particle decreases in order to keep angular momentum L=mRv constant - at least in Galilean (low speed) dynamics. In relativistic (high speed) dynamics, things are not much different, just with the special relativistic Lorentz factor gamma (γ) entering the equation.^[1]

L = m R v γ = constant -------- (1)

where γ = (1-v²/c²)^-0.5. It is relatively easy to plot the curve of Fig. 1 from this equation, together with the expansion equations of Friedman, of course. Depending on the speed of the particle and the expansion profile, the particle path may be virtually straight, as in Fig. 1, or it may be curved. In the end, it always ends up going more and more radially, meaning with less and less momentum along the surface.

Figure 3.5 above is from a doctoral thesis by Tamara M. Davis, under supervision of prof. Charles Lineweaver, titled "Fundamental Aspects of the Expansion of the Universe and Cosmic Horizons".^[2] It shows the decay of the velocity of particles for various initial particle speeds, with light (v=c) the top curve. It is clear that the larger the kinetic (momentum) part of the total energy of a particle (see Eq. 2 below), the more it behaves like a photon.

Particle Redshift?

It is possible to analyze this case in terms of redshift of de Broglie waves.^[3] Following the thoughts in the previous Blog entry ("The linear momentum of a photon is given by p = h/λ, where h is the Planck constant") and Roger's suggestions on de Broglie waves, the following interpretation is very interesting.

The relativistic energy of a moving particle is given by

E = √[p² + m²] c² = √[(h/λ)² + m²] c² -------- (2)

where p is the momentum, m is the rest mass and λ the de Broglie wavelength of the particle. It seems like a "de Broglie momentum" (h/λ) is added vector-wise to the rest mass (m), giving the particle its wave-particle duality. Particle mass is presumed constant, so in order to keep angular momentum relative to the balloon center constant, the de Broglie wavelength of the moving particle must redshift more or less like for a photon.

Angular momentum of a particle moving on the balloon is given in special relativity by eq. (1) above as: L = m R v γ = constant.

But, according to ref. note [3]:

γ = h / (λ m v) -------- (3)

so we can also write:

L = R h / λ = constant -------- (4)

exactly as for a photon. If R is increasing, λ must be increasing by the same ratio in order to keep L constant. This is same as a velocity decrease in order to keep the relativistic angular momentum of the particle constant on the expanding balloon. The velocity equivalent to a specific λ can be extracted from Eq. 3, taking into account that γ is also a function of v.

Caution

One must always remember that the balloon analogy is just that, an analogy. It may help to get 'handles' on some of the puzzles of the cosmos, but it does not represent the real thing. As an example: the cosmos might be precisely flat, with no curvature, yet the behavior discussed here is apparently still present. How and why, I don't know.

Jorrie

Notes

[1] This is essentially the special relativistic form of Kepler's second law of planetary motion: "a planetary orbit sweeps out equal areas around the Sun in equal time intervals", where the equal time intervals are now measured in proper time of the moving particle. We do not need the general relativistic form here, because particles on the balloon feel zero net gravitational acceleration (the whole surface is at the same gravitational potential).

[2] http://arxiv.org/abs/astro-ph/0402278, a doctoral thesis by Tamara Davis, under supervision of Charles Lineweaver, page 50.

[3] http://en.wikipedia.org/wiki/De_Broglie_Wave

-J

Despite having apparently simple explanations, cosmological redshift remains one of the mysteries of the universe. Neither 'expanding space', nor Doppler shift offer a completely satisfactory answer. Here is a better one (perhaps).

What is cosmological redshift?

It is the observed phenomenon that distant galaxies glow at a redder and redder frequencies the farther out they are. Astronomers measure galactic redshift by comparing the absorption lines of certain elements in the light spectrum of the galaxy to the same light spectrum in the laboratory (or to the spectrum of our Sun). The redshift is then defined as the change in wavelength (Δλ) divided by the laboratory wavelength (λ₀) of the absorption line in question, i.e.,

z = Δλ/λ₀ = (λ-λ₀)/λ₀ = λ/λ₀ - 1 ----------- (Eq. 1)

where λ is the observed wavelength.

In the expanding cosmic balloon, this is very easily pictured in terms of the ratio of the radius of the balloon now (R₀) to its radius (R) at a certain time in the past, i.e.,

z = R₀/R - 1 ----------- (Eq. 2)

as shown in Figure 1 (right). Here R₀=100 and the red circles represent earlier values of R. It comes directly from Eqs. 1 and 2. It is not difficult to see why wavelength is inversely proportional to the radius. Or is it?

Figure 1 shows the balloon size from around last scattering of photons (the red dot at the origin, R=R₀/1089, representing the CMB) up to today (R=R₀=100). The red rings are not time based, but size based, at 25%, 50%, 75% of the radius (or circumference) of today. The picture is valid for any expansion profile, provided that there is at least some expansion (also called a perpetual expansion scenario).

Explanations

In the time that the CMB photons were in flight, the balloon expanded by a factor 1089 and the photon wavelengths were 'stretched' by a factor 1089, giving their redshift as: z = λ/λ₀ - 1 = 1088. Quite reasonable, it seems at first sight,^[1] but how can a photon's wavelength be stretched? A single photon does not even have a defined size, so stretching it is not conceptually very palatable!

Another reasonable explanation may be that it is just different frames of reference between transmission and reception of the photon and that the redshift is a coordinate transformation issue, resulting in Doppler shift. However, the two frames of reference may be moving away from each other at greater than the speed of light in vacuum (c), yet we still measure a real, finite redshift. How do we reconcile this fact with the Doppler shift equations of Einstein, which do not work for recession speeds equal to or larger than c?

So, where does cosmological redshift come from?

The balloon analogy offers a neat 'crutch' that makes the phenomenon a little more palatable. It appears as if photons conserve angular momentum as they travel along the surface of the expanding balloon, almost as if they go into a larger orbit around the center of the balloon.^[2] Since photons cannot shed speed in order to keep angular momentum constant in the larger 'orbit', they shed linear momentum in another way - by reducing frequency.

The linear momentum of a photon is given by p = h/λ, where h is the Planck constant. Angular momentum magnitude of a photon relative to the balloon center is given by: R p = R h/λ. If R is increasing, λ must be increasing by the same ratio in order to keep angular momentum constant. Increased λ is the same as cosmological redshift.

I suppose in the end it is not too important which 'crutch' you use - stretching of wavelengths, conservation of angular momentum, or even Doppler shift, as long as it is accepted that the received to emitted photon wavelength ratio (λ/λ₀) is the same as the expansion ratio (R₀/R) since emission.

Jorrie

[1] See the animation on the Webb Space Telescope site.

[2] This is equivalent to Kepler's second law of planetary motion, stating that a planetary orbit sweeps out equal areas around the Sun in equal time intervals. This is the same as the conservation of orbital angular momentum. The same considerations cause any (massive) particle with a velocity relative to the skin of the balloon to conserve angular momentum and hence slow down, provided that the balloon is expanding. The opposite (speed up) happens if the balloon is shrinking. More about that in a follow-on Blog post.

-J

Now that we have a 'design' and an algorithm for the cosmic balloon, it is easy to apply it some cosmological problems. The first application is about the discussion on the 'cosmic teardrop' or 'cosmic heart'^[1] between GK and Physicist during the design phase of the cosmic balloon. Here are some numerically correct illustrations of the scenarios discussed, taken out of a spreadsheet.

The surface of the expanding balloon represents the entire cosmos at any particular time. Figure 1 (right) represents just a slice here, with the circumference of the circle representing 3D space. The interior and exterior of the balloon is a hyper-dimension into which the balloon expands or deflates.

When we look back in time (to distant sources), we essentially look 'inside' the balloon - not really inside it, because the balloon was just smaller at earlier times and we are still looking along the surface of the smaller and smaller balloon. The red and blue curves are photons coming from (say) our east and west sides, with the colors just to distinguish between them when they overlap in some scenarios.

This is easily simulated on the cosmic balloon. We just trace the paths of photons backwards along the surface of the shrinking balloon until we find the place where they originated (or until we find the BB!). For every time step, photons move a distance cΔt towards us along the surface of the balloon. Whether they approach or recede from us depends on the expansion rate and their distance from us.

For our present cosmos, even taking its minimum size, the 'heart' formed by the rays is not very remarkable - it is stretched very thin. The photons originated at the time of transparency of the cosmos (the CMB epoch), around 400,000 years after the BB, which is 'at the origin', for all practical purposes.

The balloon radius is 100 Gly, (shown on a linear scale) and the time of photon travel is 13.7 Gy. The 'two-gigayear-rings' show that the expansion was originally very fast, then slowed down somewhat and is lately picking up speed again. This is understood in terms of the effect of matter domination at earlier epochs and vacuum energy domination at later epochs. The two dotted radial lines represent where the regions of the photon emissions are today - a proper distance of 46 Gly from us. When those photons were emitted (or rather were released to move freely), the source regions were only 42 million light years way from us, just over 1000 times closer than today.

The only way to get a fuller 'cosmic heart' (as discussed in the previous thread) is to slow the expansion down to a crawl. This will happen with a flat Einstein-de Sitter universe, but it takes an extremely long time. Interestingly, there is a matter+vacuum solution to the Friedman equations that does exactly that within a 'reasonable' time (Figure 2, left). With matter energy making up 50% of the critical density and vacuum energy 200% of critical density (case 0.5,0,2.0), the expansion essentially 'stops' at 50 Gly radius - it actually approaches the 50 Gly radius asymptotically. For a period roughly the age of our present universe, the expansion would continue, but then the rate drops off towards zero. This would have made the CMB photons to travel almost the radius of the total cosmos, arriving after some 150 Gy. The half-circumference of the circle is ~165 Gly.

If we let the time continue for long enough (and if nothing disturbs the near-equilibrium, almost static situation), the 'heart' spreads around the balloon and eventually the origins overlap as shown in Figure 3 (right). For this scenario, it happens after roughly 200 Gy. This scenario means that we could have observed the same region by looking into opposite directions in the sky.

Apart from the first (present day) scenario, these may be hypothetical cases, but it illustrates the power of the cosmic balloon model rather well. It is quite simple to model and results like these flows very easily from it.

As an aside, the requirement for this situation is not so far from the observed case. There is no theoretical reason why the cosmological constant could not have been some three times higher and the matter density around double what we deduce today. However, there are observations that seem to rule out such a set of parameters.

Jorrie

[1] The 'cosmic teardrop' of observable space appears when the expanding universe is presented on a flat spacetime diagram, as shown in Figure 13.2 from Relativity-4-Engineers (shown in Figure 4, right). The parabola is equivalent to the surface of the hyper-sphere above.

The main reason for not seeing the 'comic heart' in publications is the fact that one is bound to the hypersphere and cannot show time properly on it. The 'teardrop' view does not suffer from that drawback. However, there are also certain drawbacks in the flat spacetime representation, like how a closed universe would be pictured.

My 'two-gigayear-ring' trick is a way around the time problem. The 'cosmic heart' originally earned its name from taking a flat LCDM cosmos, where the scale of R₀ is arbitrary and setting R₀ to a convenient value like the Hubble radius (13.7 Gly). It gives this pretty heart shape trace for the CMB photons (Figure 5, right). Much nicer than the 'streamlined bomb' shape when the proper (minimum) R₀ is used, as in the main figure above.

The 'teardrop' is essentially just this 'heart', with the circular coordinates flattened out.

-J

I invite all interested readers to help design the 'perfect cosmic balloon', compatible with all of cosmology - if not, then at least with most it! The recent debate on this Blog, pitching the kinematical against the hyper-spherical (balloon) cosmic models, made it clear that there are lots of uncertainties about what the balloon model can and cannot represent.^[1]

First, let's do the system engineer's thing and write a brief high level specification for the perfect cosmic balloon, trying to keep it "as simple as possible, but not simpler" (A. Einstein).

1 Specification

1.1 General

The two-dimensional balloon surface (black) shall represent all of 3D-space. This means we disregard the third spatial dimension. The extra inside/outside dimension (pink) shall represent hyperspace. It is not accessible to us - just a visualization aid. It simply provides a direction for the surface to expand and curve into.

Formally, the hyper-radius (R) may tend to infinity, or even have an imaginary value (iR), but for this exercise we will stick to real, finite values of R. Our observable universe (blue) is a limited circular patch on the surface with a radius (the Hubble radius R_H), determined by how far light could have traveled since the balloon was created.

1.2 Specifics

1.2.1 Mass

The surface of the balloon shall be able to hold massive particles (green) in a frictionless manner, i.e., they shall be able to move freely across the surface, but never be able to 'fly' off the surface, even if the radius should suddenly shrink. These particles shall represent the ordinary and dark matter of the cosmos.

1.2.2 Radiation

The surface shall also be able to hold photons (red), always traveling at the speed of light along the local surface and hence have energy that depends on wavelength. These photons shall represent cosmic radiation energy and hence also the cosmic microwave background (CMB).

1.2.3 Vacuum (Change note: reply #243)

Vacuum energy (cosmological constant) shall be included in any inflating/deflating mechanism of the balloon.

1.2.4 Total Energy

It may be assumed that the energy of expansion is included in the algorithm that inflates/deflates the balloon.

1.2.5 Momentum

It may be assumed that the momenta of the skin and all energy on it are included in the algorithm that inflates/deflates the balloon.

2 Design

Assume a balloon material that retains its elasticity over a reasonable range of balloon radii (at least enough range to illustrate the principles). A sensor system measures the balloon radius (R) directly and also determines the rate of change (ΔR/dt). A pump/reservoir/valve system supplies or withdraws gas to/from the balloon at a rate that will keep ΔR/dt = R H, where H is a function of R and the energy density makeup of the cosmos to be simulated. H, the time variable Hubble parameter, is obtained from:

H² = H₀² [(1-Ω₀)/a² + Ω_m/a³ + Ω_r/a⁴ + Ω_Λ]

where H₀ is the (present) Hubble constant, Ω₀ = Ω_m + Ω_r + Ω_Λ is the present total energy density parameter, a=R/R₀ the expansion factor, Ω_m the present matter energy density parameter, Ω_r the present radiation energy density parameter and Ω_Λ the present vacuum energy density parameter.

The operating equation is then:

(ΔR/dt)² = R²H² = R²H₀² [(1-Ω₀)/a² + Ω_m/a³ + Ω_r/a⁴ + Ω_Λ]

'Guest' has proposed the following neat little high level algorithm for this system in reply #242:

1. Set gas flow direction valve for inflating the balloon.

2. Continuously measure the radius R of the balloon and calculate (ΔR/dt)² = R²H² = R²Ho² ((1-Ω)/a² + Ω_m/a³ +Ω_r/a⁴ + Ω_Λ), with all the constants given and a = R/Ro, where Ro is a value that corresponds to the R for which the parameters are given.

3. Calculate switch = (1-Ω)/a² + Ω_m/a³ +Ω_r/a⁴ + Ω_Λ. If switch goes negative, even temporarily, change the gas flow direction direction valve for permanently deflating the balloon.

4. Measure ΔR/dt, square the result and compare it with the calculation of (ΔR/dt)².

5. If squared result is smaller than the calculation, increase gas flow rate.

6. If squared result is larger than the calculation, decrease gas flow rate.

7. Repeat from 2 until exit condition is reached.

3 Tests

First a general discussion is given and then specific simulations and 'tests' (to follow).

3.1 General

Since the balloon is ensured to follow the Friedman equations (at least theoretically), paper 'tests' would not be very meaningful. It is proposed that the 3 cases (de Sitter, Einstein-de Sitter and Lambda-cold-dark-matter (LCDM)) be simulated and the results shown here for comparison to other simulations.

3.2 The de Sitter model

The de Sitter expansion curve for a flat (Ω_m = Ω_r = 0, Ω_Λ = 1) universe is obtained by integration of the following simplified form of the above expansion equation:

ΔR/dt = RH₀/978

where the factor 1/978 is a conversion of Ho from km/s/Mpc to 1/Gy.

This is also called case (0,0,1), as on tthe figure below. It is clearly an exponential expansion curve. A Hubble constant of 72 km/s/Mpc was used for the curve.

The age of such a universe would have been about 105 Gy, read off where the curve intersects the 100 Gly radius line (which is taken as a=1). De Sitter did not intend this to be a model of the real universe, but as a tool to investigate expansion dynamics.

3.3 The Einstein-de Sitter model

The first workable attempt to model the real universe came when Einstein and de Sitter made the assumption that the universe is flat and for all practical purposes contains only matter, i.e., Ω_m = 1, Ω_r = Ω_Λ = 0. This means that the Friedman equation reduces to: (ΔR/dt)² = (RHo/978)² Ω_m/a³, giving the parabolic curve below, again for Ho = 72 km/s/Mpc.

The curve intersects the 100 Gly radius line at around 9 Gy age, making such a universe uncomfortably young! It used to be no problem when Ho was still believed to be around 50 km/s/Mpc, but not any more.

3.4 The Lambda-cold-dark-matter (LCDM) model

This is the 'standard' model at the present time, comprising about 26% matter (ordinary plus dark matter), a tiny amount of radiation energy, with the bulk of the energy (74%) made up of vacuum energy (the cosmological constant). The full equation must be used here: (ΔR/dt)² = (RHo/978)² ((1-Ω)/a² + Ω_m/a³ +Ω_r/a⁴ + Ω_Λ)

This caused an expansion curve that started out with a decreasing rate, slowly turning over to an increasing expansion rate at around 7 Gy.

This model universe has a present age of around 13.7 Gy, which is quite comfortable.

4 Conclusion

While it may be impossible to 'design and build' a laboratory sized cosmic balloon that will 'automatically' have the properties of the real cosmos, it is definitely possible to construct one that can follow the Friedman equations, at least for a short period of time. Section 2 (Design) describes such a device and its method of operation.

Such a balloon can serve as a 'crutch' to lean on in discussions of cosmological principles. At least it is a little more 'tangible' than the presumed dark matter and dark energy of the real cosmos. In a next Blog entry I will attempt to use it to show how some cosmological issues can be explained.

Jorrie

After some 'slogging', Jonmtkisco and myself almost came to an agreement that the 4D hypersphere^[1] and the 3D kinematic^[2] cosmic models are quite equivalent. Then, in post 84 of that thread (and what followed), we again left it hanging in the air, at least to some degree. I have now worked an example that shows them to give equivalent results, at least for a matter-only, flat universe (the 1,0 case, referring to 100% matter and 0% dark energy in the literature).^[3]

On the left are the plots for hypothetical cosmic travelers (immortal beings!) that fly far away, turn around and come home again (the black curve), almost 'twin-paradox' style.^[4] The only differences here are the vast cosmological distances and timescales and that the speeds are not very relativistic.

At time zero, the cosmic travelers fire their super-propulsion system for a rapid, but modest Δv of 0.0234c relative to their home galaxy, which sits at the origin with zero peculiar velocity. The acceleration lasts for a time that is negligible compared to the gigayear (Gy) timescales involved and needs only modest g-forces. They then coast inertially for exactly 15 Gy of cosmological time, while their proper speed gradually decreases, as shown by the red curve. This is due to the decelerating expansion of the (1,0) universe that couples to their proper velocity.

They turn around at 15 Gy by means of a negative Δv of double the original magnitude (Δv = -0.0468c). Their speed now slowly increases (getting more negative) until they arrive home after another 13 Gy, for a total round trip time of 28 Gy. Due to relativistic time dilation, their onboard clock will read some 78 years less, but that is of no importance here (and utterly negligible compared to the Gy timescales).

The decelerating expansion of the universe creates an interesting negative acceleration profile for the travelers, as indicated by the blueish curve. So far the plots used the hypersphere model with the Friedman equations for the expansion rate. A comparison plot of the acceleration curve was then done for a kinematic model, using a Newtonian shell-theorem calculation of the traveler's acceleration relative to the origin, together with the Friedman equations for density. The acceleration curves proved to be identical - the pink 3D kinematic curve sits on top of the blue curve from the 4D hypersphere model.

The result shows that for at least the matter-only universe, one is free to use the 4D hypersphere or the 3D kinematic model. This equivalence is a hot topic in cosmology today. Some authors use the matter-only hyper-spherical model to advocate that "space is expanding", while others use the 3D kinematic model to advocate that "space is not expanding". If they are genuinely equivalent, it is a meaningless debate, not so? What do you think?

-J

Notes:

[1] The 4D hypersphere model (also loosely termed the balloon analogy) is broadly described in this post of the previous Blog topic. Some more details are available on my website and in the eBook (http://www.relativity-4-engineers.com). It is effectively a five dimensional spacetime (four space and one time) and uses the FLRW solution to general relativity as a basis. One can usually ignore most of the dimensions: a simple expanding two-dimensional (one space dimension and one hyperspace dimension) wire ring with frictionless beads strung uniformly along it does the trick very nicely.

[2] The 3D kinematic model is based on the low field limit of general relativity and strives to do all the dynamics only in the usual 4 dimensional spacetime. It also uses FLRW solution to general relativity as a basis, e.g. to calculate the changing energy density of the universe. Using the Shell Theorem for a spherical, uniform density body of frictionless gas, the acceleration of a test particle at any distance from the center can be calculated.

[3] I'm not sure if the LCDM universe (with dark energy) is also compatible with both approaches, but my guess is that it is. Some more about that in a future post (unless Jon has solved that one already).

[4] I've been prompted to these calculations by a draft paper: 'Cosmological Radar Ranging in an Expanding Universe' by Geraint F. Lewis et al. (arXiv:0805.2197v1 [astro-ph] 15 May 2008). I disregarded their accelerating rockets and just used an 'instantaneous Δv', which is OK due to the immense timescales involved. Modeling of the acceleration of the rocket burns just complicates the issue for no real benefit or insight.

The authors use the fact that their inbound trip takes a shorter time than the outbound trip to suggest that expanding space must be a flawed concept, e.g. they wrote: "However, it is the presence of matter that necessitates the inclusion of gravitational forces upon the motion of the rocketeers and it is this - the changing gravitational influence of matter in the universe on the rocketeers - that causes the increasing asymmetry moving down the panels in Figure 2, not that space physically expands."

The expanding 4D hypersphere seems to suggest otherwise. E.g., it sports expanding space and it gives equally valid results; in certain cases even more so. What's more, it models standard cosmology, easily calculated and easily visualized, provided one ignores some dimensions.

Some of the problems I have with the 3D kinematic model:

• It explains the fact that clocks tick at equal rates everywhere in a homogeneous universe by the relativistic time dilation of comoving particles moving through space relative to the origin of an arbitrary coordinate system.
• It runs into trouble when distances are large enough to give apparent recession velocities exceeding c.
• It may have problems incorporating the various forms of hypothetical dark energy.
  

AFAIK, the 4D hypersphere model does not suffer from any of these issues. (It probably has others, though...)