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Relativity and Cosmology

This is a Blog on relativity and cosmology for engineers and the like. You are welcome to comment upon or question anything said on my website (relativity-4-engineers), in the eBook or in the snippets I post here.

Comments/questions of a general nature should preferably be posted to the FAQ section of this Blog (http://cr4.globalspec.com/blogentry/316/Relativity-Cosmology-FAQ).

A complete index to the Relativity and Cosmology Blog can be viewed here: http://cr4.globalspec.com/blog/browse/22/Relativity-and-Cosmology"

Regards, Jorrie

Another Relativity "Paradox"

Posted April 11, 2016 7:00 AM by Jorrie
Pathfinder Tags: acceleration relativity

A futuristic spaceship is stationed at its base, which is static at 10 light years (ly) from Earth in free space. The ship and base clocks are rate adjusted to run on Earth time.

The ship now departs from base and accelerates outward along the Earth radial at 1g on its accelerometer, for one year on its clock. Relativistic dynamics tells us that it achieves a speed of 0.76c relative to base and that the acceleration lasted for 1.175 base years. I also says that the ship was then 0.54 ly from base and 10.54 ly from Earth, all in base inertial coordinates.[1]

However, as observed in the ship's (new) inertial frame, that 10.54 ly Earth distance will be Lorentz contracted to 10.54(1-0.762)0.5 = 6.83 ly. This means that in ship coordinates, despite accelerating away from Earth for one year, Earth is 10-6.83 = 3.17 ly closer to the ship. What is more, this decrease in distance occurred at an average rate of 3.17 ly per year, which is effectively 3.17c.[2]

Paradoxical? Maybe. What do you think?

Regards, Jorrie

Notes (for the relativistically minded):

[1] Acceleration of 1g is very close to 1 ly per y2. Note that c=1 in these units. For 1g, the relative speed change is given by a simple hyperbolic tan function: Δv/c = tanh(Δt) = tanh(1) = 0.76, where Δt is the acceleration time as measured by the ship's clock.

In the base frame (X,T), time and distance traveled during the ship's acceleration is: ΔT = sinh(Δt) and ΔX = cosh(Δt)-1 respectively. It is obvious that we are here dealing with hyperbolic spacetime movement of the ship in the base frame. It is commonly known as Rindler coordinates. See https://en.wikipedia.org/wiki/Rindler_coordinates

[2] While we know that it isn't strange to find 'funny' speeds during acceleration, the initial and final distances were at least measured when the ship was inertial. It is true that after the acceleration, the ship could not have instantly measured the distance to Earth. But there could have been a handy inertial frame around, one that also moves at 0.76c away from Earth. The ship's captain could simply have asked a nearby stationary observer: "how far is Earth now?" The answer would undoubtedly have been 6.85 ly.

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Time Dilation and Lorentz Contraction Revisited

Posted March 27, 2016 11:00 PM by Jorrie

Intro

There are many fundamental relativity discussions that are buried deep inside related threads and hence very difficult to access at a later stage. I will try to elevate some of those issues to the level of a separate Blog entry.

Ralfcis' problems

To start with, deep inside a thread on "Conventionality of Relativity", Ralfcis wrote:

"One can derive the length contraction formula from the time dilation formula so why even bring in the redundant concept of length contraction, they're the same thing. However, you said time dilation is also not real (because both frames see time dilation in the other) but using the criteria of the twin paradox, one can measure an age difference consistent with time dilation that remains once the relative velocity ends."

Jorrie replied:

"This is the crux of the matter. There is coordinate dependent time dilation and then there is proper time dilation. There is coordinate dependent Lorentz contraction, but there is no proper Lorentz contraction. In this sense, the two are different, not 'the same thing'.

However, Lorentz contraction is real in the sense that when you make a real measurement of the length of a passing spaceship, you get a Lorentz contracted value. In this sense Lorentz contraction is 'real', but then, 'real' means different things to different people - a debate that I have no intention of entering.

As long as you have standard synchrony, i.e. Einstein synchronization of clocks, you have Lorentz contraction and the limiting one-way speed is c. The limiting 2-way speed is c, irrespective of the clock sync convention used."

To which Ralfcis replied:

"I'm lost here, totally blank. What you said may be beyond my ability to understand. I can't even formulate a question."

I think the problem is that Ralf has developed his own (evidently flawed) verbal framework for understanding relativity and he attempts to fit every bit of related information acquired into this framework. Some bits just do not fit into his framework and he gets stumped by it, or worse, he contorts the information to fit in.

Since I have spend thousands of words in an attempt to turn Ralf's framework into the mainstream direction and evidently failed, I recommended that he opens other threads to get more input and direction. This has apparently also failed to give any satisfaction.

Lately, Ralf has come around to a more conventional framework, but since I have mostly completed this Blog entry, I will post it anyway. It may be useful for other members of this forum - readers that have long since unsubscribed from that lengthy prior discussion.

Let us restrict the discussion to special relativity (SR) only, since without a solid SR foundation, it is useless to discuss general relativity (GR). Also, let us leave quantum physics and philosophical considerations out of it. SR does not answer "why" or "how" questions, only "what" (observable) questions.

The main unanswered questions seem to be about reciprocal time dilation, relative elapsed time, Lorentz contraction and the isotropy of the speed of light. In the latter, it is more specifically the one-way speed of light being the same in every inertial frame that trips up many a student of relativity.

a) The one-way speed of light

It seems appropriate to get this one out of the way first. There is no magic about the one-way speed of light being the same in all inertial frames. Einstein has simply declared it to be so as a convention[1] and then based the whole of his amazing theory of relativity on this assumption. The fact that relativity works flawlessly within its applicability, justifies Einstein's assumption without any shadow of a doubt.

Importantly, this also seems to be what nature prefers. Physics would have been very 'ugly' if any other clock sync scheme was used, e.g. the GPS system would have been all but useless, because the speed of light would have been different in different directions.

This assumption determines the method for synchronizing clocks throughout every inertial frame. In its simplest form, if we know the distance d between two clocks that are permanently at rest relative to each other and we send a time stamped signal, the receiving clock simply adds a propagation delay Δt = d/c to the time stamp and sets its time accordingly.

Because this is such a simple and universal scheme, many present day scientists simply accept that the one-way speed of light is c in every inertial frame and never give it a second thought. This sometimes leads to heated debate between scientists and "the rest", who are attempting to understand the reasoning behind the principle.

b) Reciprocal time dilation

The fact that when A and B are in uniform relative (inertial) motion, A observes B's clock to 'lose time' and B observes A's clock to 'lose time' is directly related to the above convention about the one-way speed of light. It comes about due to the way clocks are synchronized, using the convention.

It does not determine who ages slower or faster, but just how the one observer observes the others clock. Time dilation can be viewed as simply a change in 'spacetime observation angle' - each views the others time vector at an angle in spacetime, which depends on their relative speed, which is reciprocal.

This does not make time dilation "an illusion" or "not real". When proper scientific measurements of time are made between two inertial observers in relative motion, the results are as real as any measurement can be; but, it is reciprocal and hence coordinate dependent and not absolute.

c) Relative elapsed time

This is where "aging slower or faster" comes in. Every inertial object follows a trajectory through spacetime, called a 'worldline'. When two inertial objects in free space are at rest relative to each other, they follow equivalent (not necessarily identical) worldlines, so they age identically.

If they are not at rest relative to each other, they are following non-equivalent worldlines and they may age differently. They can synchronize their clocks when they move past each other and after that the one that experience the largest change of inertial frame will age less. This is why the traditional "away-twin" always ends up younger than the "home-twin".[2] If neither of them experiences any change of inertial frame, we cannot tell who ages more or less than the other.

It just so happens that in mostly quasi-inertial cases (where the acceleration phase is short relative to the inertial phases), the aging difference is approximately the same as that given by the SR time dilation formula. This fact has led to a lot of confusion in the popular literature. There is no difference in elapsed times unless there has been a difference in the change of inertial frames, which requires acceleration of at least one of the two clocks.[3]

d) Reciprocal Lorentz contraction

Like reciprocal time dilation, reciprocal Lorentz contraction is also caused by the Einstein clock synchronization convention. If A and B are in relative motion, each observes the others lengths to be contracted in the direction of relative motion. When they are brought to relative rest again, the reciprocal length contraction disappears - this is unlike the case of relative aging, which is a lasting effect.

Like relative time dilation, Lorentz contraction can be viewed as simply a change in 'spacetime observation angle' (each views the others length at an angle in spacetime), which depends on relative speed. The formula for Lorentz contraction is essentially the same as the time dilation formula. Both these effects are contained in the Lorentz transformations[4] as special cases.

Twin "paradox" Variant

Using the above information, the classical 'twin paradox' can be twisted to a slightly more challenging one. Alice sets off from Earth on her long fast journey, with Bob staying at home. Some years after Alice have left Earth, she and Bob each opens a secret envelope, where they for the first time get instructions on how to complete the mission.

The instructions could be either (i) for Alice to return to Earth and for Bob to stay put; or (ii) for Alice to coast on and for Bob to leave Earth fast enough so that he can catch up and join Alice in space.

Without doing any math, firstly, who would have aged less in each of the two cases? Secondly, just before Alice and Bob opened their respective envelopes, who would you say have aged less up to that point?

@ralfcis: Before attempting to answer these questions, first make sure that you understand the discussion leading up to it. If not, keep on asking questions, but please stick to the baseline given. I do not want to waste time by analyzing some or other fancy relativistic scenario that you can dream up - such time can be more effectively spent by discussing the stated principles better.

-Jorrie

[1] Einstein's 2005 paper "On the Electrodynamics of Moving Bodies", Section I, $1:

"We have so far defined only an "A time" and a "B time." We have not defined a common "time" for A and B, for the latter cannot be defined at all unless we establish by definition that the "time" required by light to travel from A to B equals the "time" it requires to travel from B to A."

The two-way speed of light obviously does not suffer from this clock sync convention problem, because we need only one good clock and a definition of distance to measure the round trip (average) speed of light. The modern definition of distance is however dependent on the speed of light in vacuum, so there is some degree of conventionality in the modern value of the two-way speed as well. There is no doubt that is the same in all directions, though.

[2] The rigorously correct statement would require them to meet again in order to unequivocally establish who has aged less. It is however a sufficient requirement that the two twins must just be at rest relative to each other again in order to establish beyond reasonable doubt who has aged less. For example, they can each observe an event that is equidistant from the two of them and report their respective clock readings at the time of observing the event.

[3] It must be clearly stated that acceleration per se does not cause time dilation. Acceleration is required to change the inertial frame and the time spent in the new inertial frame determines the amount of elapsed time difference accrued - in other words, acceleration is the cause of the different spacetime paths.

Instead of acceleration, their could be a "time hand-off" by the away-twin to a third inertial observer, flying in the opposite direction, who then completes the home leg. The calculated elapsed time difference is still valid, because there is the same difference in spacetime paths.

[4] The Lorentz transformations are more general than just time dilation and length contraction. It gives the equations for Lorentz covariance, which in simpler terms means converting time and space intervals from one inertial frame to another in a consistent way. Space intervals and time intervals between two specific events together form the spacetime interval between the two events, which is the same for all inertial frames. When you have more space interval between events, you have less time interval and vice-versa, but in a squared (not linear) fashion.

PS. If there are any other relativity issues that you want raised to this 'Blog-level', please p/mail me with the request.

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How Zeons turned into something Useful

Posted June 30, 2015 6:00 PM by Jorrie

In the last Blog, How Aeons turned into Zeons, we derived a very simple formula for the expansion dynamics of the cosmos, normalized to a 'natural cosmological timescale' unit of 1 zeit = 17.3 billion years.

(2.1) H2 = 1 + 0.443 S3

where H is the fractional expansion rate and S is the 'stretch-factor', i.e. by how much distances have 'stretched' since light we observe has been emitted. It is basically the redshift factor z plus 1. By convention, S have a present value of 1.0; hence the present Hubble value H0 = (1.44)0.5 = 1.2 zeit-1. But what is the relationship between H and cosmological time?

Cosmological Time

Because of the instantaneous "interest on interest" nature of cosmic expansion, it follows an exponential pattern (eHt), provided that H remains constant over time. Presently H is still decreasing, so cosmological time is approximated by the following natural logarithmic equation:[1]

(2.2) t = ln[(H+1)/(H-1)]/3 zeit.

Let us put our present H0 = 1.2 zeit-1 in here (using Google or some other calculator) and check that we get the 0.8 zeit that we expect.

t_{now} = ln[2.2/0.2]/3 ~ 0.8 zeit.

Suppose we observe the redshift from a specific galaxy as z=1, which means that the light has left it when the wavelength stretch factor was S = z+1 = 2. Plugging S = 2 into eq. 2.1, we find that H = 2.13 zeit-1 when the light was emitted. The cosmic time was

tthen = \ln[(2.13+1)/(2.13-1)]/3 = 0.34 zeit after the start of the present expansion.[2]

The light from that galaxy obviously took 0.8 - 0.34 = 0.46 zeit to reach us - it is the so-called 'lookback time'.

Now that we have a feeling for the relationships, what is better than viewing it on a graph? Here you see the parameters mentioned so far (against time).

You can see our present time (0.8 zeit) clearly where the red and blue curves cross, i.e. where both S and a are unity. You can also see how H starts high and asymptotically approaches unity. The graphs were plotted using LightCone 7zeit, a variant of the standard LightCone 7 cosmological calculator.

I have 'sneaked in' the curve for the scale factor (a=1/S), but I did not give a direct relationship of it against time yet. You get it by solving eqs. 2.1 and 2.2 simultaneously, which is not a simple exercise at all. Fortunately, mathematics comes to the rescue and we have a nifty solution for a(t), i.e. the scale factor as a function of time:

(2.3) a(t) = sinh2/3(1.5t)/1.3

where sinh is the hyperbolic sine function. The 'hypersine' is obviously related to the natural logarithm ln(x) and the natural exponent (ex). The factor 1.3 is just scaling to make anow = 1 at tnow = 0.8 zeit, as is the convention. The importance of a(t) is that its curve shows how a decelerating expansion that gradually changed over to an accelerating expansion. The inflection point can be visually judged to lie between 0.4 and 0.5 zeit. It is actually at 0.44 zeit.

Proper Distance

Proper distance in cosmology is like measuring distance on a hypothetical frozen view of the cosmos, i.e. with expansion stopped. Obviously we cannot stop expansion, so how do we find proper distance? There are various methods, but once we have the present and long term Hubble values (H0 and H), we only need to measure the redshift of a source and then calculate its proper distance from where we are.

Because of the non-linear expansion, there is no precise analytical solution for the proper distance (that I know of). We have to numerically integrate in small steps; but fortunately the definite integral is rather neat:

(2.4) Dnow = ∫1S(dS/H) = 1S[dS/√(0.44S3+1)]

We have substituted H from eq. 2.1 in here, so that we have a distance in terms of the observable S = z+1. For calculation, you can use your favorite integrator, but a rather cool web based one is available at

http://www.numberempire.com/definiteintegralcalculator.php.

For our previous sample galaxy at S=2, I entered 1/sqrt(0.443x^3+1) into the Function box, with 1 in the "From" and 2 in the "To" boxes and then clicked "Compute" - it returned the answer as 0.64, which in our cas has the units lightzeit. So we receive the light when the proper distance to the galaxy is Dnow = 0.64 lzeit. Since we have used stretch S = 2, the proper distance when the photons left the galaxy was half of 0.64, i.e. 0.32 lzeit; so we say that Dthen = 0.32 lzeit for this galaxy. Recall that the light left that galaxy at t = 0.34 zeit, i.e. 0.46 zeit ago.

Here is a graph of Dnow and Dthen.

The red Dthen curve is also known as the cosmological lightcone. To the left of 0.8 zeit it represents our past lightcone and to the right of 0.8 zeit our future lightcone. Near t = 0.0 the universe was very dense and objects we now observe were actually very, very close to our space locality. Distances then grew much faster than the progress that photons could make in our direction and ancient photons were at first dragged away from us. At around t = 0.23 zeit, the fractional expansion rate dropped enough to allow photons to make progress and eventually reach our telescopes.

To summarize, we have now looked at a further three rather easy equations that allow us to calculate the most common values for the standard cosmological model.

Part 3 will deal with the Hubble radius and two cosmological horizons that we should take notice of.

=0=

End-notes

[1] The function eHt gives the factor by which cosmic distances increase in a time interval t, provided that H remains constant over this time interval. In the far future, H will be a constant, but in the early universe H changed significantly over periods longer than a million years. Hence the more complex general solution.

[2] To verify that the approximations are valid, I used the LightCone 7zeit calculator for a 'one-shot' calculation with Supper=2 and Sstep=0.

[3] The Dthen lightcone is where the LightCone 7 calculator got its name from.

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How Aeons turned into Zeons

Posted May 12, 2015 12:00 PM by Jorrie

Quite a lot has been written on this CR4 Blog about the standard Lambda-Cold-Dark Matter (LCDM) cosmological model and its equations. Arguably the most important equation of the model is the evolution of the expansion rate over cosmological time. In other words, how the Hubble constant H has changed over time. If one knows this function, most of the other LCDM equations can be derived from it, because it fixes the expansion dynamics.

The changing H is most simply expressed in this variant of the Friedman equation, an exact solution of Einstein's field equations for a spatially flat and perfectly homogeneous universe.

(1) H2−Λc2/3 = 8/3 π G ρ

Here H is the fractional expansion rate at time t, Λ is Einstein's cosmological constant, G is Newton's gravitational constant and ρ is the mass density equivalent of the changing energy density of matter and radiation at time t. This energy density includes dark matter, but no 'dark energy', because Λ appears as a constant spacetime curvature on the left side of the equation. The c2 converts square curvature (length-2) to a squared fractional expansion rate (time-2), to be on par with H2.[1]

As you can check, G ρ gives SI units of 1/s2, which is the natural unit of the squared fractional expansion rate. Since Λ is a constant spacetime curvature, it is convenient to replace Λc2/3 with a constant 'Hubble rate' H2 , which represents the square of the Hubble constant of the 'infinite future', when cosmic expansion will effectively have reduced matter energy density to zero.

(2) H2−H2 = 8/3 π G ρ

Since we can measure the present value of H, labeled H0 (H-naught) and also how it has changed over time, it allows us to use Einstein's GR to determine the value of H. If we assume that radiation energy is negligible compared to other forms (as is supported by observational evidence), then we can express eq. (2) as:

(3) H2H2 = (H02 H2 )S3

The present observed fractional expansion rate tells us that all large scale distances are now growing by 1/144 % per million years. S is the 'stretch factor' by which wavelengths of all radiation from galaxies have increased since they were emitted.[2] Since S3 tells us by how much the volume has increased, it also tells us by what factor density and hence spacetime curvature has decreased since the observed emission.

If the present H0 would have stayed constant, all large scale distance would have doubled in the next 14.4 Gy. This 'doubling time' is however slowly increasing, to eventually double all large scale distances every 17.3 Gy. Or stated differently, all distances will eventually grow at H = 1/173 % per million years.

The 17.3 Gy 'doubling time' is a sort of natural time scale set by Einstein's cosmological constant. An informal study by a group of Physics-Forums contributors suggested that the 17.3 Gy time-span could be a natural timescale for the universe.[3] For lack of an 'official name' for it, the group called it a 'zeon', for no other good reason than the fact that it rhymes with aeon.

One light-zeon is 17.3 Gly in conventional terms and H0 causes a doubling in distances every 14.4/17.3 = 0.832 zeon. This makes H = 1 per zeon and H0 = 17.3/14.4 =1.201 per zeon.[3] Our present time is 13.8/17.3 ~ 0.8 zeon.

We can easily normalize equation (3) to the new (zeon) scale by dividing through by H2 (which then obviously equals 1).

(4) H2−1 = (1.2012−1) S3 = 0.443 S3

or

(5) H2 = 1 + 0.443 S3 ----> !NB!

This remarkably simple equation forms the basis of a surprisingly large number of modern cosmological calculations, as will be discussed in a follow-on Blog entry.

Here is a graph of the normalized H over 'zeon-time', which is obviously the x-axis (courtesy PhysicsForums).

The blue dot represents our present time, 0.8 zeon and a Hubble constant of 1.2 zeon-1. The long term value of H approaches 1.

Any questions before we proceed?

Regards, Jorrie

[1] The traditional unit of the Hubble constant as used by Edwin Hubble is kilometers per second per Megaparsec. From an educational p.o.v. it was an unfortunate choice, because it seems to indicate a recession speed, while it is really a fractional rate of increase of distance. It is a distance divided by a distance, all divided by time. So its natural unit is 1/time, or simply time-1.

[2] 'Stretch factor' S = 1/a, where a is the scale factor, as used in the LightCone calculator. S is also simply related to cosmological redshift z by S=z+1.

[3] Science Advisor 'Marcus' led a group of PhysicsForums members in fleshing out of this "universal scale", based on the cosmological constant.

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HTRN's Cosmology Questions

Posted January 17, 2015 1:55 AM by Jorrie

in HTRN's thread about the rate of bubble rise and expansion in a fluid, he also saw analogies to cosmological expansion and he speculated:

"Maybe we are on the surface of a bubble, and expansion is a result of external as well as internal forces:
Reduction of outside "pressure"(what ever that is) and internal forces.
"

This is a common question and surprisingly difficult to answer in a compact fashion, so this will not be short and sweet. It was slightly off-topic in the original thread, hence this Blog entry.

The popular answer is simply that we have no indications that we are on the surface of any 'spacetime bubble', but rather that we are at the center of an expanding observable spacetime bubble. We think that dynamics of this expansion all come from 'forces' inside the observable universe, not from outside.

We know that there must have been some event (or series of events) in the past that caused the matter and radiation of our observable universe to expand (distances and wavelengths are increasing over time). Let's call it the 'initial cause'. The effect of the initial cause is that the universe would keep on expanding, but that the rate will slow down as time goes on. We have actually observed this slow-down in expansion rate over the first 7 or so billion years after the event, because we are 'looking back in time'.

We also know that something is presently causing the rate of this expansion to increase again; let's call that the present cause. Physicists are not sure, but it is possible that the initial and present causes boil doen to the same thing, Einstein's cosmological constant Lambda, just at two different energy levels. Lambda has a peculiar characteristic: it has part normal gravity and part 'anti-gravity', in that it has negative pressure coupled to positive energy density in a fine balance. Normal matter-energy has both its pressure and energy positive.

Positive pressure contributes significantly to the gravitational force inside a collection of matter, but negative pressure will lessen that gravitational force and may even overwhelm the energy density to give an overall expanding effect. Since Lambda is precisely equivalent to the energy density of empty space, space with no normal matter or radiation has no choice but to expand (or contract) exponentially. The "expand or contract" needs some clarification.

It is possible for any piece of space to be in static equilibrium, with just enough matter and vacuum energy to neither contract (under the energy density), nor expand (under the negative pressure). This is however an unstable condition, since any minute negative volume fluctuation will increase the energy density's gravity, but decrease the negative pressure, resulting in an exponential collapse of that space.* The exact opposite will happen for any minute positive volume fluctuation, resulting in exponential expansion.

The answer to the original question is then clear: space does not need anything 'outside it' in order to contract or expand. It has enough internal mechanisms. We are just not able to characterize those internal mechanisms from first principles yet, because it has a quantum-gravitational flavor. And that's still a tough nut to crack in a consistent manner.

There were more questions, but this is already a 'head-full', so I will rather reply to further questions as they arise.

-J

* This form of collapse could be the trigger for a cyclic universe, but more about that later.

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