# Is the expansion of the universe relative to our clocks and rulers?

1. Aug 13, 2013

### JackMartinelli

Is the expansion of the universe relative to our clocks and rulers?

If so, it seems correct to include relativistic effects. And yet I am told that this is wrong.

Regards,

Jack Martinelli

2. Aug 13, 2013

### marcus

What relativistic effects were you thinking of including?

Were they SR effects (the 1905 theory with fixed non-expanding geometry, that applies locally at small scale where expansion is negligible)?

or were they GR effects (the 1915 theory which allows geometry to be dynamic, distances and angles to change etc., no speed limit on speed of distance growth etc.)?

3. Aug 13, 2013

### Chronos

GR does not respect our clocks or rulers.

4. Aug 13, 2013

### timmdeeg

Yes, in the expanding universe distances between comoving objects (galaxies) increase relative to (thought) rulers. However, as Marcus already mentionend, Special Relativity can't be applied in curved space-time (considering cosmological distances).

5. Aug 13, 2013

### WannabeNewton

It is relative to the measuring apparatuses carried along a preferred time-like congruence: that given by the worldlines of observers comoving with the Hubble flow. Relativistic effects are intrinsic to GR and SR can be applied locally.

6. Aug 13, 2013

### JackMartinelli

I have been considering that expansion velocities of volumes can treated relativistically.

Of course you run into a problem with the phrase "fixed non-expanding geometry". What does it mean in an objective sense? Same with small scale.

GR says "mass tells space how to bend".... I have no idea how it communicates with spacetime. And because it seems to lead to a dead end as far as completely describing matter and fields ... I'm not sure its a good epistimological theory.

7. Aug 13, 2013

### JackMartinelli

Yes ... somewhere I read a quote by Poincare that said something to the effect "... clocks and rulers are put in by hand....". And we know that a ruler is a line like object. A theory is verified by clocks and rulers... seems to me that a theory that begins with the objects we verify it with are more likely to be correct.

8. Aug 13, 2013

### JackMartinelli

If the curvature is the result of relative expansion ala SR. Its too late.

Say I have r' = r * sqrt( 1- v^2/c^2 ) ...

where v = Hr'

r is not large or small ... its just some length that increases uniformly with respect to some other length ... say some material length. Doesn't that imply that the unit of length in the expanding frame actually shrinks? That is, "causes" curvature of the frame.

9. Aug 13, 2013

### Mordred

One thing to make clear on the first statement. Expansion velocities. Referred to as recessive velocity can at far distances exceed the speed of light. However that velocity is observer dependant according to Hubbles law. "The greater the distance, the greater the recessive velocity". In other words those 2 and 3c recessive velocities are measured from say Earth. However if you were to teleport to those galaxies their recessive velocity would be the same as our local nearby galaxies.

So using GR with those high above c velocities would be incorrect. They are not in actuality moving with inertia at 2 or 3c. In fact the space between them and us is simply increasing.
Recessive velocity in this regard misleading as the term velocity is dependant upon inertia. Would be more accurate to simply think of it as recessive distance. Instead of recessive velocity. But were historically stuck with the term.

10. Aug 13, 2013

### Mordred

These two articles show how the mathematics of the FLRW metric is applied to geometry and expansion.

11. Aug 13, 2013

### Mordred

The other article

12. Aug 13, 2013

### WannabeNewton

There is a quantity related to volume and expansion that you can look at and it is inherently relativistic (everything in GR is so there's no need to worry). This quantity is conveniently called the expansion and is usually denoted by $\theta$. The definition itself is a tad bit technical: let $\xi^a$ be a time-like vector field such that the integral curves of $\xi^a$ never intersect one another and such that they fill all of space-time (or some proper open subset of space-time). We call $\xi^a$ a time-like congruence. Physically, we imagine $\xi^a$ as representing the worldlines of fluid particles that never collide with one another.

The expansion $\theta$ is then defined as $\theta = \nabla_{a}\xi^{a}$. $\theta$ can be interpreted physically as follows: pick any fluid particle in the congruence and consider an observer comoving with this particle. The observer is carrying with him a Fermi-Walker transported measuring apparatus (a clock and three mutually orthogonal, non-rotating meter sticks). At a given instant of proper time $\tau$ on the observer's clock, the observer defines a very small sphere around him consisting of infinitesimally nearby fluid particles in the congruence. $\theta$ then measures the rate (given by the proper time $\tau$ as read on the observer's clock) at which the volume of the sphere decreases or increases i.e. the rate at which the infinitesimally nearby particles expand away or contract towards the fluid particle that the observer is comoving with. In fact, one can directly show that $\nabla_a \xi^a = \frac{1}{V}\xi^a \nabla_a V = \frac{1}{V}\frac{dV}{d\tau}$ where $V$ is an infinitesimal volume carried along the worldline of the chosen fluid particle.

For an FLRW universe, we can calculate $\theta$ in comoving coordinates: $\nabla_a \xi^a = \nabla_{\mu}\delta^{\mu}_{\tau} = \Gamma ^{\mu}_{\mu \tau} = \partial_{\tau}\ln \sqrt{|\det(g_{\mu\nu})|}$. We find that $\theta = 3\frac{\dot{a}}{a}$ where $a(\tau)$ is the scale factor. Here the time-like congruence $\xi^a$ represents the worldlines of the galaxies mentioned by timmdeeg way above.

Last edited: Aug 13, 2013
13. Aug 13, 2013

### timmdeeg

kindly let me add and where $\frac{\dot{a}}{a}$ is the hubble parameter, which describes the expansion rate of the universe.

14. Aug 13, 2013

### marcus

I see. Thanks for responding!

Would you like me to tell you how fast a given volume is expanding? Say a cube which is 1000 lightyears on a side? I mean according to the standard cosmic model (which is solidly relativistic, being based on gr). I expect you might want to get familiar with the standard model that cosmologists currently use, since in order to think critically and question effectively you need familiarity with the "baseline" model in use.

So if you would like to know, just to get a feel for the standard model numbers, please say. It's very easy to calculate (really just one short line of arithmetic) and would be a pleasure.

15. Aug 13, 2013

### timmdeeg

Curvature is the result of gravity. The dynamics of expansion depend on the ratio of attractive and repelling energy density.

A ruler length doesn't shrink physically but due to expansion compared to increasing distances.

16. Aug 13, 2013

### JackMartinelli

Interesting but not what I was looking for ... How do you know a clock measures time & why does a ruler measure length. What makes a clock a clock. And how do you define an abstract (thought) ruler and clock that can be included in the math. A measurement gives us the magnitudes we verify a theory with... but I don't see anything in your discussion where this happens.

17. Aug 13, 2013

### JackMartinelli

GR is not complete. Read ch. 88 of Gravitation by MTW.

I'm sure the standard model is as solid as Bohr's model of the atom.... Bohr's model gave all the right answers but was wrong.

I can look up recession velocity in wikipedia. How about telling me what a unit of length is and a unit of time.

18. Aug 13, 2013

### Mordred

Not sure what you are getting at with your last two replies. Both time and length are affected by GR observer dependancies. In cosmology we use cosmological time. With a reference point being the CMB. The second article I posted show how distances are categorized on cosmological scales. The post by Wannabenewton shows the GR affects on both.

19. Sep 17, 2013

### Romulo Binuya

Since 1967, the unit of time *second* has been defined to be:
"the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom." and the unit of length *meter* is based on that *second* and the speed of light *c*.

I assumed that since they could build atomic clocks they are aware of time dilation due to gravity and relative motion and therefore incorporated that effects in the definition of *second*. I don't think incorporating the relativistic effects in the definition is wrong. That standard unit of time and length I'm guessing is being used in all domain of physics.

20. Sep 18, 2013

### Chronos

Assuming time is a constant, using atomic clocks as a measurement device is perfectly sensible. Unfortunately, we have no unambiguous way to objectively define time.