Pioneer anomaly not gravitational

In summary, an analysis of the orbits of Neptune's satellites has shown that the Pioneer anomaly, which is a residual blue shift in the radio signals interpreted as an extra Sun-wards acceleration, cannot be explained by gravitational forces. This suggests that there may be a need to consider alternative models, such as Østvang's QMR theory, which is compatible with all classic solar-system tests. However, there is still not enough experimental data to draw any definitive conclusions, and there are many potential sources of error that need to be considered. Additionally, the possibility of a variable speed of light through space, as proposed by Einstein's view of a polarizable ether, cannot be ruled out. Further analysis and experimental data are needed to fully understand the Pioneer
  • #1
bcrowell
Staff Emeritus
Science Advisor
Insights Author
Gold Member
6,724
429
An analysis of the orbits of Neptune's satellites seems to show that the Pioneer anomaly can't be gravitational.

http://www.technologyreview.com/blog/arxiv/24541/
 
Physics news on Phys.org
  • #2
bcrowell said:
An analysis of the orbits of Neptune's satellites seems to show that the Pioneer anomaly can't be gravitational.

http://www.technologyreview.com/blog/arxiv/24541/
IMO, there were really never any reason to expect that the PA should involve any
real accelerations. That's just a specific interpretation of the PA, even if it is the popular one.
But the fact that the anomalous acceleration is very close to cH, would suggest that
the PA has something to do with mismodeling of null curves in an expanding Universe.
This does not work in GR, but alternative models exist. See, e.g., gr-qc/9910054.
 
  • #3
Old Smuggler said:
But the fact that the anomalous acceleration is very close to cH, would suggest that
the PA has something to do with mismodeling of null curves in an expanding Universe.
This does not work in GR, but alternative models exist. See, e.g., gr-qc/9910054.

Interesting. I took a quick, casual look at that paper, and at gr-qc/0112025, which describes his alternative theory of gravity. I don't claim to have examined it carefully enough to have even absorbed the general ideas of the theory. It is interesting that he claims to have a viable non-metric theory that is compatible with all the classic solar-system tests. However, he doesn't seem to have proposed any specific experimental tests other than the Pioneer Anomaly. It would be a lot more compelling if he could say, "Hey, here's this other empirical test where I predict something different from GR -- go measure this and see what happens."
 
  • #4
You may be interested in the discussion we have already had in physics forums, such as here: The Anomalous Acceleration of the Pioneer Spacecrafts, especially this analysis.

The unexplained residual is consistent with cH.

What actually was being detected was a residual blue shift in the radio signals, which is interpreted as an extra Sun-wards acceleration.

It could just as well be a time drift between atomic and ephemeris clocks.

Garth
 
Last edited:
  • #5
what is cH please?
 
  • #6
edpell said:
what is cH please?

The speed of light multiplied by the Hubble constant. IMO it's not very impressive that cH happens to be of the same order of magnitude as the anomalous acceleration observed in the Pioneer anomaly, but that seems to be one of the main empirical facts used to support Østvang's QMR theory.
 
  • #7
Garth said:
The unexplained residual is consistent with cH.

What actually was being detected was a residual blue shift in the radio signals, which is interpreted as an extra Sun-wards acceleration.

It could just as well be a time drift between atomic and ephemeris clocks.

Garth
The observations might be consistent with a variable speed of propagation of light through space that is polarized by gravitation of embedded masses. Such an interpretation of observations would be a game-changer and thus would be anathema to BB cosmology. If it's true, prepare to wait for a century or so before it can become "accepted".
 
  • #8
"The observations might be consistent with a variable speed of propagation of light through space that is polarized by gravitation of embedded masses. Such an interpretation of observations would be a game-changer and thus would be anathema to BB cosmology. "

The use of the conditional tense is appropriate indeed, as always when pure speculation without enough experimental data is involved.
 
  • #9
lalbatros said:
"The observations might be consistent with a variable speed of propagation of light through space that is polarized by gravitation of embedded masses. Such an interpretation of observations would be a game-changer and thus would be anathema to BB cosmology. "

The use of the conditional tense is appropriate indeed, as always when pure speculation without enough experimental data is involved.
We have plenty of experimental data, and not just from the Pioneers, but from other probes that are at least partially spin-stabilized. If the apparent "acceleration" is not real, and is not caused by gravitational forces on the probes themselves, we must be prepared to consider that the radio signals could be propagating at something other than c. Einstein's view of space as a polarizable ether was not well-received by his contemporaries nor by modern-day physicists, but it was a view that he held from 1920 onward.

http://www.bartleby.com/173/22.html
Einstein said:
In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its result hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light).

Einstein "On the Ether" 1924 said:
The ether of the general theory of relativity differs from that of classical mechanics or the special theory of relativity respectively, in so far as it is not 'absolute', but is determined in it locally variable properties by ponderable matter.
 
  • #10
turbo-1,

How far have you analyzed your experimental data?
Could you make them available as well as your analysis?

I am very far from being a specialist on this topic and I have never seen the full data. I even don't know how I could process them!
That's why I am so sceptical, considering the many effect and sources of error to be taken into account and listed by Turyshev (see for example: http://arxiv.org/abs/physics/0502123). And this list is maybe far from being complete. For example, what would be the impact of disturbances caused by (very) small charge densities dispersed in space? Has this been studied? I would be interrested to know the impact on the experimental data.

For me, the Pioneer anomaly is a very interresting topic, but I doubt it is a "clean enough" experiment to bring us to big conclusions, specially considering the small value of the anomaly.
 
  • #11
The Pioneer anomaly is being studied by the Planetary Society. A few years ago, years' worth of magnetic data-tapes were recovered, and it's unclear whether they have completed their analysis, from information on their web site. However, a recent publication by Turyshev says:

Initial efforts to explain the Pioneer anomaly focused on the possibility of on-board systematic forces.While these cannot be conclusively excluded (Anderson et al. 2002; Turyshev et al. 2005), the evidence to date did not support these mechanisms: it was found that the magnitude of the anomaly exceeds the acceleration that these mechanisms would likely produce, and the temporal evolution of the anomaly differs from that which one would expect, for instance, if the anomaly were due to thermal radiation of a decaying nuclear power source.
http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.0399v1.pdf

While analysis to that level is pretty impressive, there is another more compelling reason to discount systematics. The anomaly is not confined to the Pioneer craft. Other probes also show evidence of anomalous Sunward acceleration, and when probes have different masses, cross-sectional areas and thermal signatures (due to the various ways RTGs are mounted) it is difficult to come up with scenarios in which the anomaly can be produced for each of them. As in the quote above, if thermal radiation contributed to the anomaly, one would expect that the effect would diminish as the RTGs decayed.
 
  • #12
turbo-1 said:
While analysis to that level is pretty impressive, there is another more compelling reason to discount systematics. The anomaly is not confined to the Pioneer craft. Other probes also show evidence of anomalous Sunward acceleration, and when probes have different masses, cross-sectional areas and thermal signatures (due to the various ways RTGs are mounted) it is difficult to come up with scenarios in which the anomaly can be produced for each of them.
Do all the different probes show the same magnitude of anomalous acceleration as a function of distance from the Sun, or is it different for different probes?
 
  • #13
The Pioneer Anomaly acceleration is constant from the orbit of Saturn outwards and equal on both the Pioneer probes.

Inside the orbit of Saturn other effects would have masked the effect, as indeed they have on other deep space probes, although signs of the effect have been detected on the Voyager probes as well. The Voyager spacecraft 's three-axis stabilization maskes the effect.

The Flyby anomalies are different, though they may be connected.
 
  • #14
Garth said:
The Pioneer Anomaly acceleration is constant from the orbit of Saturn outwards and equal on both the Pioneer probes.

Inside the orbit of Saturn other effects would have masked the effect, as indeed they have on other deep space probes, although signs of the effect have been detected on the Voyager probes as well. The Voyager spacecraft 's three-axis stabilization maskes the effect.

The Flyby anomalies are different, though they may be connected.
If there's no clear evidence that the magnitude of the anomalies for different probes are too similar to be explained by quirks of engineering, would you then disagree with turbo-1's statement that "when probes have different masses, cross-sectional areas and thermal signatures (due to the various ways RTGs are mounted) it is difficult to come up with scenarios in which the anomaly can be produced for each of them"?
 
  • #15
JesseM said:
If there's no clear evidence that the magnitude of the anomalies for different probes are too similar to be explained by quirks of engineering, would you then disagree with turbo-1's statement that "when probes have different masses, cross-sectional areas and thermal signatures (due to the various ways RTGs are mounted) it is difficult to come up with scenarios in which the anomaly can be produced for each of them"?
Part of my skepticism of design-dependent systematics arises from the fact that even though the anomaly can be masked in some probes that are not fully spin-stabilized, the anomaly still results in some blue-shifting. If off-gassing, leaking thruster fuel, or thermal radiative pressures can somehow be responsible for the anomalies, why is the thrust always directed away from the Sun resulting in braking? Why no acceleration out of the system resulting in red-shifting?

We don't have enough probes out there to make the lack of red-shifting a "smoking gun". Dedicated spin-stabilized probes with centrally-located RTGs might help clear this up.
 
  • #16
Sounds like a worthwhile experiment send out 10 probes in different directions. All the probe needs to do is ping once per ten seconds and point towards Earth (that is point the antenna part towards Earth). Say 20Kg per probe 200Kg total. But at so respectable velocity (we don't have all century). Anyone have a feel for the cost?
 
Last edited:
  • #17
Will Pluto Express gives us usable data on this question? On their web page it says it is three axis AND spin stabilized. I guess it depends how often they fire the reaction wheel dumping thrusters? Anyone know how they will use the probe?
 
  • #18
edpell said:
Sound like a worthwhile experiment send out 10 probes in different directions. All the probe needs to do is ping once per ten seconds and point towards Earth (that is point the antenna part towards Earth). Say 20Kg per probe 200Kg total. But at so respectable velocity (we don't have all century). Anyone have a feel for the cost?

If you look at http://en.wikipedia.org/wiki/New_horizons#Spacecraft_subsystems you can get a feel for sizes of the different stuff that has to go aboard this type of probe. New Horizons is 465 kg, and a lot of the equipment looks like it would also be needed for a dedicated pioneer-anomaly probe. E.g., it's got a really huge radio dish.

There's some more info here about possible dedicated missions or other ways of exploring the Pioneer anomaly: http://en.wikipedia.org/wiki/Pioneer_anomaly#Further_research_avenues

If I were peer-reviewing a proposal for a new dedicated probe, the question in my mind would be this: if it's not a gravitational effect, then what are the chances that it's even interesting physics at all?
 
  • #19
While analysis to that level is pretty impressive, there is another more compelling reason to discount systematics. The anomaly is not confined to the Pioneer craft. Other probes also show evidence of anomalous Sunward acceleration,

We have a total of five spacecraft outside Saturn's orbit, two of them Pioneers, and that's really not enough data to go on.

. If off-gassing, leaking thruster fuel, or thermal radiative pressures can somehow be responsible for the anomalies, why is the thrust always directed away from the Sun resulting in braking?

One common theme is that each one of the five probes (two Pioneers, two Voyagers and New Horizons) necessarily consists of a big-*** parabolic dish facing the Earth (i.e., with good precision, the Sun), and some hardware bolted on the back side of the dish.
 
Last edited:
  • #20
bcrowell said:
If I were peer-reviewing a proposal for a new dedicated probe, the question in my mind would be this: if it's not a gravitational effect, then what are the chances that it's even interesting physics at all?
Well, that would depend on your definition of "interesting physics". Some people apparently
think that new physics that is not "mainstream new physics", i.e., not coming from some simple extension of mainstream theory, is uninteresting. Since most gravitational
explanantions of the PA may be classified as mainstream new physics, evidence that the PA
is not gravitational may diminish the general interest in the PA and thus the chances for
a dedicated mission to test it.

More radical explanations of the PA may still be viable, e.g., based on the notion that the
cosmic expansion has observable consequences in the solar system. This confronts the
"well-known" mainstream position that the cosmic expansion does not apply to
gravitationally bound systems. But it is very difficult to unlearn something that is taken for
granted. Thus some radical explanations may seem even more farfetced than what is
really the case, also reducing the motivation for a dedicated test mission.
 
  • #21
Old Smuggler said:
This confronts the
"well-known" mainstream position that the cosmic expansion does not apply to
gravitationally bound systems.
That's not exactly the mainstream position. The mainstream position is that within star systems and galaxies and clusters, gravity is just strong enough to dominate "dark energy", while the gravitational attraction at larger distances is too weak to counteract dark energy.
 
  • #22
Al68 said:
That's not exactly the mainstream position. The mainstream position is that within star systems and galaxies and clusters, gravity is just strong enough to dominate "dark energy", while the gravitational attraction at larger distances is too weak to counteract dark energy.
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 (I wonder if any numerical simulations have been done to check exactly how small the effects of expansion should be within the solar system according to GR)
 
  • #23
Al68 said:
That's not exactly the mainstream position. The mainstream position is that within star systems and galaxies and clusters, gravity is just strong enough to dominate "dark energy", while the gravitational attraction at larger distances is too weak to counteract dark energy.
I know what the mainstream position is. Since I was talking about observable consequences
of the cosmic expansion in the solar system, where the mainstream position is that there
cannot be any, I didn't see any reason to include all the buts and ifs appearing if one is
talking about galaxy clusters etc.
 
  • #24
JesseM said:
But either way I guess GR predicts that the local gravity in a bound system dominates over cosmic expansion (I wonder if any numerical simulations have been done to check exactly how small the effects of expansion should be within the solar system according to GR)
Several studies have been done on the effects of the cosmic expansion within the solar
system according to GR, and also the effect of a cosmological constant of appropriate
magnitude. They all show that such an effect is incredibly small at best, and certainly
not measurable. I don't have a reference at hand right now, but you may try to search
for author(s) Vollick, Faraoni or Cooperstock at the arXiv.
 
  • #25
JesseM said:
Even with no dark energy the expansion would still happen, just at a different rate, right?
Sure, but without dark energy, the rate of expansion would be dominated even by the weak gravity over large distances, ie the rate would be slowing.
But either way I guess GR predicts that the local gravity in a bound system dominates over cosmic expansion (I wonder if any numerical simulations have been done to check exactly how small the effects of expansion should be within the solar system according to GR)
I wonder about that, too, but I would think that it would be negligible, since compared to intergalactic space, the gravitational forces within the solar system are huge, as well as within galaxies and clusters.

I would think if dark energy was strong enough to have any measurable effect within the solar system, it would have expanded the distance between galaxy clusters to way beyond observable distances long ago. Especially if dark energy's effect increases with distance as is suspected.
 
  • #26
The PA is not cosmological expansion detected within the solar system, the anomalous acceleration is in the opposite direction!

Garth
 
  • #27
Garth said:
The PA is not cosmological expansion detected within the solar system, the anomalous acceleration is in the opposite direction!
Garth
It seems that you have the cosmic redshift in mind. But no sane person would identify
the apparent anomalous blueshift inferred from the PA with the cosmic redshift.

However, the cosmic expansion may have another, more subtle effect, namely an
anomalous delay of radio signals used in the telemetry. Since the PA essentially appears
from the calculation of trajectories of the probes, such an anomalous delay could be
interpreted as if the probe's coordinate velocity were slowing down anomalously, thus
indicating an anomalous acceleration towards the observer.

This explanation does not work in GR, but it has been shown to work in another theory.
The bottom line is thus that the interpretation of the PA is crucially theory-dependent,
and that the PA really might be the cosmological expansion detected within the solar system.
 
  • #28
Old Smuggler said:
It seems that you have the cosmic redshift in mind. But no sane person would identify
the apparent anomalous blueshift inferred from the PA with the cosmic redshift.

However, the cosmic expansion may have another, more subtle effect, namely an
anomalous delay of radio signals used in the telemetry. Since the PA essentially appears
from the calculation of trajectories of the probes, such an anomalous delay could be
interpreted as if the probe's coordinate velocity were slowing down anomalously, thus
indicating an anomalous acceleration towards the observer.

This explanation does not work in GR, but it has been shown to work in another theory.
The bottom line is thus that the interpretation of the PA is crucially theory-dependent,
and that the PA really might be the cosmological expansion detected within the solar system.
You have to be very careful in your language to avoid confusion.

To say "cosmological expansion detected within the solar system" is to say the same thing as "cosmic redshift".

I have already said What actually was being detected was a residual blue shift in the radio signals, which is interpreted as an extra Sun-wards acceleration. It could just as well be a time drift between atomic and ephemeris clocks.

Which, as you have pointed out, does not work in GR.

What I have argued on these Forums for several years now, such as here and here: Does the PA show that DM/DE is dynamically important in the outer solar system? is that because the PA anomalous blue shift of radio signals, once non-gravitational effects have been allowed for, is equal to cH to within observational errors, then the effect may well be cosmological in nature and an indication that GR may have to be modified in some way.

Garth
 
  • #29
JesseM said:
Even with no dark energy the expansion would still happen, just at a different rate, right?
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 http://arxiv.org/abs/gr-qc/0602002" [Broken].
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.
 
Last edited by a moderator:
  • #30
Ich said:
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.

I'm sure you're right, but there are a couple of things I don't understand about this argument.

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

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.
 
  • #31
First, if I take a metric [itex]
ds^2=dt^2-a^2(dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2 )
[/itex], with a=ut, and calculate the Einstein tensor, I get [itex]
G^t_t=3t^{-2}
[/itex] , 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.
 
Last edited by a moderator:
  • #32
bcrowell said:
First, if I take a metric [itex]ds^2=dt^2-a^2(dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2 )[/itex], with a=ut, and calculate the Einstein tensor, I get [itex]G^t_t=3t^{-2}[/itex], 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.
 
Last edited:
  • #33
bcrowell said:
if I take a metric [itex]ds^2=dt^2-a^2(dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2 )[/itex], with a=ut, and calculate the Einstein tensor, I get [itex]G^t_t=3t^{-2}[/itex], along with other diagonal elements that are nonzero.

Ich said:
(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:
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:
Code:
                                           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

Ich said:

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.
 
Last edited by a moderator:
  • #34
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:
[tex]
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) )
[/tex]
(I hope it's correct)
 
  • #35
Ich said:
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:
[tex]
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) )
[/tex]
(I hope it's correct)

Happy new year to you, too!

The way you've written the metric, there'd be an over-all [itex]r^4[/itex] coefficient in the [itex]d\phi^2[/itex] 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:

Ich said:
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.
 

Similar threads

  • Special and General Relativity
2
Replies
39
Views
9K
  • Special and General Relativity
Replies
13
Views
3K
  • Special and General Relativity
Replies
9
Views
2K
  • Cosmology
Replies
14
Views
1K
  • Special and General Relativity
Replies
11
Views
3K
  • Special and General Relativity
Replies
29
Views
1K
  • Special and General Relativity
Replies
7
Views
2K
  • Special and General Relativity
Replies
7
Views
3K
Replies
42
Views
3K
  • Special and General Relativity
3
Replies
101
Views
7K
Back
Top