chrimony wrote: You say, "Without a more accurate definition of G, gravitational lensing can't even be tested, much less confirmed." It would seem very strange to me if all the scientists who have worked on GR missed such a basic problem. Would you agree that either you are making a gross error or they are?
Yes.
Sain84
Re: Relativity
CharlesChandler wrote:
chrimony wrote: You say, "Without a more accurate definition of G, gravitational lensing can't even be tested, much less confirmed." It would seem very strange to me if all the scientists who have worked on GR missed such a basic problem. Would you agree that either you are making a gross error or they are?
Yes.
That's incorrect. Gravitational studies of the Sun do not provide the mass of the Sun, they provide the product of G and Msun, GM. We know GM to much greater precision than we do G and hence M but that isn't important. That only means we can't determine M from GM to high precision but it does not affect our ability to test gravity because GM is the factor we need. And we know it very well.
Lack of precision of G leads to uncertainties in M but it does not affect our ability to test gravity. In the context of the Sun G only serves to calculate M, it is not used in an experimental context to derive predictions of gravitational effects.
Secondly errors in gravitational lensing are linear so if you only knew GM to for example 1 part in 10,000 then we would have an uncertainty in predicted light deflection of 1 part in 10,000. If out deflection is big say 1 arcsecond, then the error is 100 microarcseconds. If our deflection however is 10 miliarcseconds then the error is 1 microarcsecond. In this equation errors are linear. The numbers in this case are made up but even if GM was only known to 1 part in 10,000 your experimental uncertainty would be tiny.
CharlesChandler
Re: Relativity
Sain84 wrote: Gravitational studies of the Sun do not provide the mass of the Sun, they provide the product of G and Msun, GM. We know GM to much greater precision than we do G and hence M but that isn't important. That only means we can't determine M from GM to high precision but it does not affect our ability to test gravity because GM is the factor we need. And we know it very well.
And how do we know GM so well? By the amount of lensing, assuming GR?
Sain84 wrote: Secondly errors in gravitational lensing are linear so if you only knew GM to for example 1 part in 10,000 then we would have an uncertainty in predicted light deflection of 1 part in 10,000. If out deflection is big say 1 arcsecond, then the error is 100 microarcseconds. If our deflection however is 10 miliarcseconds then the error is 1 microarcsecond. In this equation errors are linear. The numbers in this case are made up but even if GM was only known to 1 part in 10,000 your experimental uncertainty would be tiny.
No, you don't get to say that the experimental variability, which is a relatively small number, axiomatically means that any effect that is measured is subject to a very small variability, and thus the variability can be neglected. You have to apply the variability to the total, and see if it is greater than the effect that was measured.
In other words, if you were measuring the boiling point of water, with a thermometer that was accurate to 1 degree out of 100, and if you unexpectedly found that your water sample boiled at 101 degrees C, meaning a discrepancy of 1 degree from the expected 100 degrees C, giving you reason to call it relativistic water, you wouldn't say that the discrepancy is 1 degree +/- 1/100 of degree, or 101.00 +/- 0.01 degree. It's 101 +/- 1 degree, which means that for all you know, the water actually boiled at 100 degrees C, but your thermometer just isn't accurate enough to detect variations like that.
Likewise, you don't observe a discrepancy of 10 milli-arc-seconds, and then realize that one of the numbers in the calcs is only accurate to 1 part in 10,000, and then conclude that the discrepancy is indeed 10 milli-arc-seconds, accurate +/- 1 micro-arc-second. That's ridiculous.
Sain84
Re: Relativity
CharlesChandler wrote: And how do we know GM so well? By the amount of lensing, assuming GR?
Almost any gravitational experiment can tell you GM. You can use Kepler's laws and observations of the planet's orbits or something like Shapiro delay. Even if you didn't know GM lensing can still be studied if you study how it changes across the sky. GM can therefore be determined from lensing as well. GM is easy to measure, any high school student can do it for the Earth. G wasn't determined for many years after newton because it is hard to measure, you need to know the mass of both bodies.
CharlesChandler wrote: No, you don't get to say that the experimental variability, which is a relatively small number, axiomatically means that any effect that is measured is subject to a very small variability, and thus the variability can be neglected. You have to apply the variability to the total, and see if it is greater than the effect that was measured.
In other words, if you were measuring the boiling point of water, with a thermometer that was accurate to 1 degree out of 100, and if you unexpectedly found that your water sample boiled at 101 degrees C, meaning a discrepancy of 1 degree from the expected 100 degrees C, giving you reason to call it relativistic water, you wouldn't say that the discrepancy is 1 degree +/- 1/100 of degree, or 101.00 +/- 0.01 degree. It's 101 +/- 1 degree, which means that for all you know, the water actually boiled at 100 degrees C, but your thermometer just isn't accurate enough to detect variations like that.
Likewise, you don't observe a discrepancy of 10 milli-arc-seconds, and then realize that one of the numbers in the calcs is only accurate to 1 part in 10,000, and then conclude that the discrepancy is indeed 10 milli-arc-seconds, accurate +/- 1 micro-arc-second. That's ridiculous.
You're water analogy is correct but it goes against nothing I have done, I will demonstrate it. In a equation with a linear equation and only one source of error you just take the percentage error on the variable as the error on the result. Let's consider the "total error".
First we'll use volume of a cube for this example because everyone understands the formula. V=x*y*z Obvious. Now assume we know y and z perfectly and we only care about errors in x. Say x is measured at 100 cm with an error of +/- 1 cm. That is a fractional error of 0.01 or 1%. Under my linear propagation rule that means the fractional error in V is now 0.01 or 1%. If I set y=z=100 cm then I get the following. V=100*100*100 +/- 1% = 1000000 +/- 10000 cm^3
Now let's do it the long way and say that the error is +/- 1 cm so what range of volumes can I calculate. Taking upper and lower limits. Vupper=101*100*100=1010000 cm^3 Vlower=99*100*100=990000 cm^3 Taking the difference between them and the result we get +/- 10000 cm^3. Exactly what I said I would under my linear propagation rule.
So yes, if you have a predicted deflection of 10 milliarcseconds the uncertainty in the theoretical prediction is 1 micro-arc-second. Now what you are confused about is that this is only the theoretical uncertainty, it is not the measurement uncertainty. The measurement uncertainty is something else entirely but GM only affects the theoretical number. It's not ridiculous at all.
CharlesChandler
Re: Relativity
Sain84 wrote: G wasn't determined for many years after newton because it is hard to measure, you need to know the mass of both bodies.
Indeed. So how did Eddington determine in 1920 that gravitational lensing had been proved, before space-based measurements were available? I "think" that the answer is, "Imagination is more important than knowledge."
Sain84 wrote: In a equation with a linear equation and only one source of error you just take the percentage error on the variable as the error on the result.
That's a +/- 1% error in the total volume, given a +/- 1% error in the measurement of one of the sides, which is correct.
Sain84 wrote: So yes, if you have a predicted deflection of 10 milliarcseconds the uncertainty in the theoretical prediction is 1 micro-arc-second. Now what you are confused about is that this is only the theoretical uncertainty, it is not the measurement uncertainty. The measurement uncertainty is something else entirely but GM only affects the theoretical number. It's not ridiculous at all.
That is NOT correct. The "predicted deflection of 10 milli-arc-seconds" is the discrepancy — you don't multiply that by the deviation, to prove that your instrumentation is accurate to within 1 micro-arc-second!!! That would be a very neat trick indeed, if it was not sophistry. Nice try.
Sain84
Re: Relativity
CharlesChandler wrote: Indeed. So how did Eddington determine in 1920 that gravitational lensing had been proved, before space-based measurements were available? I "think" that the answer is, "Imagination is more important than knowledge."
You don't need space based measurements. GM can be calculated very well just with observations of the orbits of planets.
CharlesChandler wrote: That is NOT correct. The "predicted deflection of 10 milli-arc-seconds" is the discrepancy — you don't multiply that by the deviation, to prove that your instrumentation is accurate to within 1 micro-arc-second!!! That would be a very neat trick indeed, if it was not sophistry. Nice try.
This is where you are confused as I said, this has nothing to do with instrumental precision. This is how accurately you can predict the theoretical result. 10 mas is result, which I then multiply by the fractional error to get 1 uas, which is the precision with which you can make that prediction. The experimental error will be much larger but errors in GM have no bearing on that.
CharlesChandler
Re: Relativity
Sain84 wrote: GM can be calculated very well just with observations of the orbits of planets.
If you know the mass of one object, and the orbital characteristics, you can calculate the mass of the other object. But what if you don't know the mass of either object? Then you only know the relative masses. They could both be heavy, or light, but either way, you'd get the same orbits. You can measure the mass of an object on Earth by its inertial force, and then come up with an estimate of the Earth's mass, given the gravitational force. And then you can estimate the mass of the Sun, given the mass of the Earth, and its orbital characteristics. But none of that is any more accurate than the benchmark measurement, which was of the object that you weighed here on Earth. And that's going to give you the precision to estimate milli-arc-second deflections due to the supposed gravitational lensing?
Sain84
Re: Relativity
CharlesChandler wrote: If you know the mass of one object, and the orbital characteristics, you can calculate the mass of the other object. But what if you don't know the mass of either object? Then you only know the relative masses. They could both be heavy, or light, but either way, you'd get the same orbits. You can measure the mass of an object on Earth by its inertial force, and then come up with an estimate of the Earth's mass, given the gravitational force. And then you can estimate the mass of the Sun, given the mass of the Earth, and its orbital characteristics. But none of that is any more accurate than the benchmark measurement, which was of the object that you weighed here on Earth. And that's going to give you the precision to estimate milli-arc-second deflections due to the supposed gravitational lensing?
In Kepler's laws the mass of the planet falls out when one mass is much larger than the other. The third of Kepler's laws states that the square of the period over the semi-major axis cubed is constant for all planets. The value of the constant includes GMsun. When a planet becomes too large the assumption breaks down and you get two body dynamics. This can still be solved with Newtonian mechanics but you get a ratio of masses times the gravitational constant. In the case of the Earth we can break this problem by measuring GM here for known masses. You can do it with satellites elsewhere now too. If you want to include other bodies it can only be solved numerically. These add complexity but at the same time precision.
As I said there are many ways to calculate GM not just these methods but as I said the precision measurements already exist, you don't need much.
CharlesChandler
Re: Relativity
Sain84 wrote: As I said there are many ways to calculate GM not just these methods but as I said the precision measurements already exist, you don't need much.
Were they thinking about using these or something? The last time I checked, gravity could only be estimated to within a factor of 1.2 × 10−4. Or is that what you mean by super-duper-high-precision measurements?
Sain84
Re: Relativity
CharlesChandler wrote:
Sain84 wrote: As I said there are many ways to calculate GM not just these methods but as I said the precision measurements already exist, you don't need much.
Were they thinking about using these or something? The last time I checked, gravity could only be estimated to within a factor of 1.2 × 10−4. Or is that what you mean by super-duper-high-precision measurements?
That's the precision in G not in GM. You don't need G separately for this, that's what we've been talking about. GMsun is 1.32712440018 x 1020 +/- 8 x 109. Really quite astonishing precision. We know GM far better than we know M just because of the difficulty of measuring G. So you see these errors are much, much smaller than experimental errors when measuring lensing.
CharlesChandler
Re: Relativity
Where can I find GMsun defined?
Sain84
Re: Relativity
CharlesChandler wrote: Where can I find GMsun defined?
It's derived from the current JPL ephemeris. It's a numerical model which simulates the gravitational interactions of the planets along with other bodies used to predict their locations, orientations and other parameters at any time. It is used in spacecraft navigation. Kepler's laws will give you a good number for GMsun but to get really good numbers you need to account for all the major gravitational bodies in the system, that's where this numerical model is used. The constants like GMsun are fitted as part of the simulation. Data is taken from a huge number of sources described in the document.
When I read papers like that, full of fancy heuristics, it always makes me think of the state of the geocentric theory just before Copernicus proposed the heliocentric theory. I remember watching a documentary that described a planetarium that was constructed to predict the celestial motions based on Ptolemy's work. It was a pretty fancy apparatus, instantiating all of the tweaks that had been applied to the geocentric theory, to absorb as many of the inaccuracies as possible. Copernicus looked at that, and at the fact that no matter what they did, they just couldn't get the last little bit of inaccuracy out of it, and concluded that they were missing something. Mother Nature doesn't do random or arbitrary. Things happen for reasons, and layer upon layer of heuristics doesn't get you closer to the truth. Sometimes you have to take a step back, and wonder what you're missing. So when I see a paper where they achieved an "exact solution" with a bunch of fancy heuristics, I wonder why it took such fancy heuristics. Mother Nature does physics, not statistics. Anyway...
When the predictions of a best-fit numeric model confirm the predictions of a totally different model (i.e., GR), I call that coincidence. You'd have to show me where GR predicted the motions of the celestial bodies to nine places past the decimal point, of sufficient accuracy to be worth something in explaining gravitational lensing. But it doesn't.
