Pioneer anomaly not gravitational

[Ich]

First, if I take a metric <br /> ds^2=dt^2-a^2(dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2 )<br />, with a=ut, and calculate the Einstein tensor, I get <br /> G^t_t=3t^{-2}<br /> , along with other diagonal elements that are nonzero. Doesn't this mean that the a=ut is observable?

(didn't check your math, but it seems wrong at first glance)
It is the minkowski metric if you use https://www.physicsforums.com/showthread.php?p=2283519#post2283519"

Also, there seems to be something missing from your argument, since it makes it sound as though expansion would be unobservable not just at the scale of the solar system but at the cosmological scale as well.

Expansion is not unobservable, but it has no physical effect. If all galaxies are tied to constant FRW coordinates, they are moving away from each other in the standard minkowski coordinates. That is surely observable, but it doesn't mean that physics is different. It just means that matter in the universe is arranged such that it is moving apart. No reason for e.g. a solar system to follow that trend, it's the same as if other galaxies were not moving apart. It simply doesn't matter what other things do.

 

[Old Smuggler]

First, if I take a metric ds^2=dt^2-a^2(dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2 ), with a=ut, and calculate the Einstein tensor, I get G^t_t=3t^{-2}, along with other diagonal elements that are nonzero. Doesn't this mean that the a=ut is observable?

FYI, Ich has a serious misunderstanding of differential geometry; he believes that by
linearizing the scale factor in the FRW models one always gets Minkowski space-time. That
this is wrong may be easily shown by calculating the Riemann tensor; it does not
vanish except for the very special case of an empty FRW model.

Unfortunately, a certain crackpot paper was published in American Journal of Physics
in August 2009, advocating an interpretation of the cosmic redshift based on this misunderstanding.

 

[bcrowell]

if I take a metric ds^2=dt^2-a^2(dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2 ), with a=ut, and calculate the Einstein tensor, I get G^t_t=3t^{-2}, along with other diagonal elements that are nonzero.

(didn't check your math, but it seems wrong at first glance)

I didn't do the math by hand. I used a computer algebra system, called Maxima. Of course that doesn't mean the calculation is right. It's conceivable that I set it up wrong, misunderstood the documentation for the software, etc. But in any case, here is my code and the output of the calculation:

Code:

load(ctensor);
ct_coords:[t,r,theta,phi];
a : u*t;
lg:matrix([1,0,0,0],
          [0,-a^2,0,0],
          [0,0,-a^2*r^2,0],
          [0,0,0,-a^2*r^2*sin(theta)^2]);
cmetric();
einstein(true);

Relevant part of output:

                                           3
(%t7)                            ein     = --
                                    1, 1    2
                                           t

                                           1
(%t8)                            ein     = --
                                    2, 2    2
                                           t

                                           1
(%t9)                            ein     = --
                                    3, 3    2
                                           t

                                           1
(%t10)                           ein     = --
                                    4, 4    2
                                           t

It is the minkowski metric if you use https://www.physicsforums.com/showthread.php?p=2283519#post2283519"

A change of variables can't change a nonzero Einstein tensor into a zero Einstein tensor.

Since your claim seems to be controversial, maybe you could show us a calculation that gives a vanishing curvature for this metric.

 

[Ich]

Hi and a happy new year,

sorry, I've been sloppy: What I've been referring to is the empty universe, with negatively curved space and a~t. Try your calculations (or the transformation) with this metric:
<br /> ds^2=dt^2-u^2t^2(dr^2+r^2 sinh(u r)^2 (d\theta^2+r^2\sin^2\theta d\phi^2) )<br />
(I hope it's correct)

 

[bcrowell]

Hi and a happy new year,

sorry, I've been sloppy: What I've been referring to is the empty universe, with negatively curved space and a~t. Try your calculations (or the transformation) with this metric:
<br /> ds^2=dt^2-u^2t^2(dr^2+r^2 sinh(u r)^2 (d\theta^2+r^2\sin^2\theta d\phi^2) )<br />
(I hope it's correct)

Happy new year to you, too!

The way you've written the metric, there'd be an over-all r^4 coefficient in the d\phi^2 term...?

Anyhow, I'm perfectly willing to believe that expansion can be removed by a coordinate transformation in the case of an *empty* universe. In a universe with zero matter, radiation, or cosmological constant, the Einstein tensor vanishes everywhere, so it's reasonable to imagine that it's really just a Minkowski space expressed in strange coordinates.

But I don't understand how that would be relevant to your original statement:

Expansion per se (I mean unaccelerated, a~t) can be removed by a coordinate transformation. Therefore it can't have any physical effect on the solar system. What is relevant for solar system physics are gravitational sources that are unaccounted for in the standard calculations. There are two such sources of "cosmological origin", Dark Matter and Dark Energy. We should expect that Dark Matter dominates around the sun, producing indeed an anomalous blueshift - but several orders of magnitude below the observed effect. So cosmology is not the solution, at least not in GR or other metric theories consistent with observation.

Expansion of our actual universe (dominated by a mixture of matter and cosmological constant) is not removable by a coordinate transformation. The fact that it has no significant effect on our solar system ( http://arxiv.org/abs/astro-ph/9803097 ) is because cosmological expansion in general has no significant effect on small, gravitationally bound systems. A simple, closed-form example is Schwarzschild-de Sitter space, which describes an isolated black hole in a universe with no other matter, but a nonzero cosmological constant. The part of the metric close to the black hole is very close to a normal Schwarzschild metric, so, e.g., if you have a test particle in a circular orbit, the size of the orbit won't change significantly.

 

[Ich]

The way you've written the metric, there'd be an over-all r^4 coefficient in the d\\phi^2 term...?

Yeah, copy & paste error. Erase the last r², maybe it will work then.

But I don't understand how that would be relevant to your original statement

I made two claims:

Expansion per se (I mean unaccelerated, a~t) can be removed by a coordinate transformation. Therefore it can't have any physical effect on the solar system.

and

What is relevant for solar system physics are gravitational sources that are unaccounted for in the standard calculations.

If you're not happy with the empty universe, let's have al look at the equation of motion of a free particle in a general frw spacetime (if you're interested in toy models, https://www.physicsforums.com/showthread.php?p=2089458#post2089458" a funny derivation). In its low speed limit (enough for local physics), in comoving coordinates, it is:
a\ddot r + 2\dot a \dot r = 0
Pick an arbitrary origin and switch to cosmological proper distance x=a\,ras a space coordinate, with
\ddot x = \ddot a +2\dot a \dot r +a\ddot r
you get
\ddot x = x \ddot a / a
"Expansion per se" (\dot a) has vanished from the equation (claim 1).
Instead there's radial acceleration depending on ä, which looks formally like (Newtonian) gravitation from an uniformly distributed source. Using the second http://en.wikipedia.org/wiki/Friedmann_equations" , you find that it actually represents energy and pressure as the source of gravitation in FRW cosmologies (claim 2). (add: try to calculate orbit perturbations due to "expansion" with this Newtonian background as additional gravity source. It works.)
So, for solar system physics, the whole "expanding universe" influence reduces to a background "matter" distribution that you have to account for, and nothing more.

 

[bcrowell]

Ich, I just don't understand the point of what you're doing. We have a variety of arguments to show that in an FRW solution to standard GR, the solar system doesn't expand. Is the point here that you're describing a particular argument that you think is really sweet?

 

[Ich]

Ich, I just don't understand the point of what you're doing.

Just look at what triggered my first reply in this thread:

Even with no dark energy the expansion would still happen, just at a different rate, right? But either way I guess GR predicts that the local gravity in a bound system dominates over cosmic expansion

It is a widely held belief, even among professionals, that gravitationally bound systems withstand the tendency of "Expansion" to tear them apart.
My point is that there is no such tendency.

You couldn't follow my argument as it wasn't exactly well formulated - a~t is necessary, but not sufficient to remove expansion by a global coordinate transformation.
I tried to show the local argument that I had in mind when I wrote my first reply. But obviously, you already know that local physics is independent of \dot a and are now wondering what I'm arguing about.

I didn't notice Old Smuggler's reply until now:

FYI, Ich has a serious misunderstanding of differential geometry; he believes that by linearizing the scale factor in the FRW models one always gets Minkowski space-time.

I admit that I may have given this impression in this thread, but I certainly feel myself being misrepresented by this account. In the https://www.physicsforums.com/showthread.php?p=2154352" Old Smuggler is referring to, I argued that you can express FRW spacetime locally, to first order, by Minkowski-like coordinates and that you'll find neighbouring galaxies to be moving away from each other in this frame. That's called Riemann normal coordinates, and not my own crackpot idea.
BTW, the alleged crackpot paper Old Smuggler mentions ishttp://arxiv.org/abs/0808.1081" , not exactly crackpot for my understanding.

 

[bcrowell]

It is a widely held belief, even among professionals, that gravitationally bound systems withstand the tendency of "Expansion" to tear them apart.
My point is that there is no such tendency.

Sorry, but I'm still completely lost as to what point of view you're advocating. We observe that the time for a laser beam to travel to the moon and back is essentially constant as judged by our clocks (subtracting out a recession related to ocean tide effects, which is well understood). We also observe data from CMB and supernova surveys that are consistent with a certain FRW model with a certain cosmological constant. Are you saying (1) that the lunar ranging data are wrong, (2) that the cosmological analysis is wrong, (3) that the cosmological models don't have to be interpreted in terms of expanding space, (4) that gravitational binding isn't the reason that the moon doesn't recede from us, ... or ...?

 

[Ich]

Sorry, but I'm still completely lost as to what point of view you're advocating.

Ok, I'll restate.
I say, answering JesseM's question, that the value of the Hubble parameter is completely irrelevant for solar system physics. By "completely irrelevant" I don't mean an effect heavily outweighed by "normal, baryonic," gravitation, I mean no effect at all.
You don't have to know whether the universe is expanding (H>0) or contracting (H<0) to calculate the perturbations for solar system orbits, at least not to leading order. It's enough if you know the local density of DM and DE, and Newtonian gravitation.

If you say "everybody knows that", ok, then I obviously misunderstood the posts in this thread. That's ok and I'll stop explaining. I keep explaining because I thought that you doubt this assertion.

As to your points: I don't challenge standard results. So to 1 and 2: no.

(3) that the cosmological models don't have to be interpreted in terms of expanding space

They may be interpreted in these terms, no problem with that. But I strongly disencourage using this picture when it comes to quasistatic, local physics. It means explaining loads of fictitious effects to get a null result, leaving the listener (and the narrator) in total confusion.

(4) that gravitational binding isn't the reason that the moon doesn't recede from us

Actually, yes, in a way. As long as the local DM density more than doubles the DE density, the additional effect to the knon "baryonic" gravitation tends to bind the system even stronger. This should be the case in the outskirts of a galaxy.
Rephrased: If there were no usual gravitation (Mass of Moon and Earth), and both were at rest wrt each other, the gravitation of Dark Matter around the Earth would pull the Moon towards the earth, winning against Dark Energy.
Ah, to avoid misunderstandings: I interpreted "receding" here as "in a larger orbit than without said effects". Of course, bound systems don't break up just because ther's a minor perturbation, they would simply settle in a new equilibrium. So my claim is that the Moon's orbit is a few microns (or whatever) smaller than without DM and DE.

 

[petm1]

Do you think they were just leaving the effective range of our solar systems' accelerated frame?

 

[lalbatros]

Ich,

This is not totally clear for me:

Expansion per se (I mean unaccelerated, a~t) can be removed by a coordinate transformation. Therefore it can't have any physical effect on the solar system.

Couldn't the Pioneer anomaly be wiped out by the same transformation?
In addition, I am not sure that removing the expansion would "remove physics", just as removing a rotation by going in a rotating frame doesn't remove the effect of rotation.
Could you clarify?

Michel

 

[bcrowell]

I say, answering JesseM's question, that the value of the Hubble parameter is completely irrelevant for solar system physics. By "completely irrelevant" I don't mean an effect heavily outweighed by "normal, baryonic," gravitation, I mean no effect at all.

Do you think the calculations in this paper http://arxiv.org/abs/astro-ph/9803097v1 are incorrect? They get an effect that's nonzero, but much too small to measure in the case of, e.g., the solar system.

By the way, I emailed Dag Oestvang to ask a couple of questions about the current empirical status of his quasi-metric relativity (QMR). He says that it predicts an effect on Neptune's satellites from cosmological expansion that is bigger than the GR prediction, but still too small to measure, and therefore consistent with Iorio's analysis. He also says that there are hidden model-dependent assumptions in the analysis of the solar system data, so although QMR predicts expansion of the solar system, it's not inconsistent with existing radiometric data.

 

[yuiop]

I have a question. At what distance apart are two bodies no longer considered to be gravitationally bound? The gravitational equations suggest the influence of gravity of spatially infinite and if gravity waves travel at the speed of light, then any celestial body we can see is gravitationally interacting with us. Maybe there is a difference between being "gravitationally bound" and "gravitationally interacting". Obviously escape velocity is a factor to be considered. If a body recedes from us at greater than the required escape velocity, is it considered to no longer be gravitationally bound even though it is presumably still gravitationally interacting with us? If DE and DM exist and the evidence suggests they do, then presumably they must have some effect within the Solar system even if that effect is too small to detect. No?

 

[bcrowell]

I have a question. At what distance apart are two bodies no longer considered to be gravitationally bound?

Take a look at this paper http://arxiv.org/abs/astro-ph/9803097v1 . It's only an approximation that bound systems don't expand. The more tightly bound they are, the better the approximation is.

 

[Ich]

Couldn't the Pioneer anomaly be wiped out by the same transformation?

The transformation does nothing except removing coordinate artifacts. Solar system physics, including the pioneer trajectory, isn't done in cosmological coordinates (cc) anyway, so there can't be confusion from a misinterpretation of those.

In addition, I am not sure that removing the expansion would "remove physics"

(the following section deals with the empty universe, as my original statement was intended to)
No, it shows that there is no such physics in the first place, which can't be seen easily in cc.
In cc, you have galaxies "at rest" and "space expanding" between them. It is not clear, neither from the mental picture nor from the equation of motion, how this expansion changes the laws of local physics. It is widely believed that things are totally different from what we know, and that calculations are tedious and subject to interpretation.
In Minkowski coordinates, there is standard spacetime and the galaxies are moving away from each other. It is therefore immediately clear that physics is the same, no matter whether some galaxies are moving outward or not.
That's not different physics, it's just the appropriate description for the purpose of analyzing local dynamics. cc doesn't predict something different, it's just a total mess.

 

[Ich]

Do you think the calculations in this paper http://arxiv.org/abs/astro-ph/9803097v1 are incorrect?

No, why should I? You surely noticed that their eq. 2.8 is exactly what I posted here, and therefore the results should be the same.
In detail:
They do exactly what I'm promoting in this thread (and in the other thread Old Smuggler referred to):
They go to Riemann Normal Coordinates, a LIF, as they call it, and do their calculations there.
They lament the often obscure use and misuse of different coordinate systems that makes results difficult to understand and compare (see e.g. eq. 4.1, a result that's not in line with the other results, except that "r" may mean something completely different here).

However, I'm going a step further, by connecting \ddot a / a with the matter density via the Friedmann equations. That's also not "new physics" or original research, but it it very instructive.
Because, if you do so, you see that you can reproduce their result for two body systems with ordinary Newtonian gravity, if you add the averaged mass density of the universe as a perturbation. Their result of growing orbits immediately explains itself as being due to a constantly decreasing amount of matter within the orbit of the planet in their model universe.
You see further that the assumption of an homogeneous, isotropic universe doesn't make sense in this context. For example, it is reasonable to assume that the dark matter density in the solar system is much higher that the universe average, and that it remains essentially constant, as it is bound to our galaxy cluster. Therefore, in our universe, with WIMP-like CDM, the perturbation of the orbit is negative and constant. In a de Sitter universe, it would be positive and constant.
So

It's only an approximation that bound systems don't expand.

is not correct IMHO. The result that systems expand itself stems from an approximation, namely that the matter density in the region of interest equals the averaged matter density of the universe.

 

[Garth]

It's only an approximation that bound systems don't expand.

is not correct IMHO. The result that systems expand itself stems from an approximation, namely that the matter density in the region of interest equals the averaged matter density of the universe.

Actually Ich there is an issue here.

Orbital dynamics in GR are calculated in the Schwarzschild solution of the one body case.
That solution is embedded in flat non-expanding Minkowskian space, i.e. g_{\mu\nu} \rightarrow \eta_{\mu\nu} as r \rightarrow \infty

There is a question about what the result would be if the Schwarzschild solution was embedded in an expanding cosmological metric.

However, the problem with such an approach is that any consequential motion or acceleration would be expected to be in the opposite sense to the PA, i.e. outwards and not sunwards.

Garth

 

[Ich]

There is a question about what the result would be if the Schwarzschild solution was embedded in an expanding cosmological metric.

As long as you assume spherical symmetry, if you cut out a spherical cavity, you will have flat spacetime inside (see e.g. the paper cited by bcrowell and references therein).
So that means that the reason for any deviations from flatness must lie within that cavity.
If you "refill" it with Dark Matter, it will have negligible pressure and velocity within some fairly big radius, so you won't make much error if you treat it as homogeneous, istropic and non-moving (as measured in the LIF) dust. You can handle that in Newtonian approximation.
It's even easier with Dark Energy, as it is has no rest frame. There is an exact solution for the whole problem, http://arxiv.org/abs/gr-qc/0602002" , which reduces to an effective negative mass density in Newtonian approximation.

Of course, you would not want to embed the Schwarzschild metric in an expanding metric. I don't see how this should be done.
You'd have to embed it in appropriate static or quasistatic coordinates. These deviate only slightly from Minkowski coordinates at galactic scales, and can be managed as a perturbation of the original Schwarzschild metric.

However, the problem with such an approach is that any consequential motion or acceleration would be expected to be in the opposite sense to the PA, i.e. outwards and not sunwards.

As I said earlier, you'd actually expect sunward acceleration if you include dark matter. But it's ~5 OOM too small to explain the anomaly, IIRC.

 

[heldervelez]

https://www.physicsforums.com/showpost.php?p=2522588&postcount=23" (not inertial) can translate in fictious forces not accounted in the analysis.
I think that we don't have the required equations.

 

[bcrowell]

Would that be:
The kinematic origin of the cosmological redshift
Emory F. Bunn and David W. Hogg
American Journal of Physics -- August 2009 -- Volume 77, Issue 8, pp. 688-694
Issue Date: August 2009

I don't think the Bunn and Hogg paper is crackpot material. They argue for an unusual interpretation of the cosmological expansion, but there's nothing actually incorrect about it. It's just a matter of taste which interpretation you prefer. For more on this topic, see http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.2 (subsection 8.2.5).

 

[FreeThinking]

I don't think the Bunn and Hogg paper is crackpot material...

Sorry 'bout that. I found a link to the article in an earlier post and deleted my post, but obviously not before you replied. I do appreciate your efforts anyway. Having had a script to AJP long ago, I would be very surprised if anything published in it was really "crackpot", which is why I wanted to know exactly which article it was.

Thanks.

 

[HarryWertM]

Have the simplest aspects of GR been considered in regard to the Pioneer Anomaly? EG: clocks [& radio oscillators] run faster in lower gravity fields; photons are blue shifted when approaching gravity fields [like the Sun]; lengths change; etc.?

 

[Jonathan Scott]

Have the simplest aspects of GR been considered in regard to the Pioneer Anomaly? EG: clocks [& radio oscillators] run faster in lower gravity fields; photons are blue shifted when approaching gravity fields [like the Sun]; lengths change; etc.?

The analysis has gone WAY beyond the simplest aspects of GR, which is always taken into account in the coordinate system used to describe this sort of trajectory accurately within the solar system. The paper referenced in the original post rules out a whole class of gravity theories, of which Newtonian and GR are just special cases.