Thread: The law of gravitation and the equivalence principle

This equation attempts to show the relationship between gravitational mass and inertial mass.



It is a combination of the formula for relative acceleration (gravitational) and reduced mass (inertial):



If the equivalence principle holds true, then the ratio between gravitational mass and inertial mass is one to one. So it can be canceled out, revealing the familiar law of gravitation.



However, leaving the equation in it's expanded form can be very illustrative. For example, the relationship between gravitational mass and inertial mass in the universality of free fall can be demonstrated simply by showing the equation for the acceleration of the free falling body:



In the above equation it is easy to see why the free falling body will fall at the same rate regardless of it's own mass. Any change in the gravitational mass in the left fraction is balanced by the same change in inertial mass in the right fraction.

Another interesting aspect of this equation is that the proportional equivalence of gravitational mass and inertial mass is not a requirement.

Have you seen anything like this before?
What is your opinion?

I have been reading through a very old physics book ( Fundamentals Of Physics with SI Units ) I have had a quick look at inertia, it states in Newton`s first law that the mass of an object is a measure of its inertia. The mass of an object is of course a measure of the quantity of matter in the object. After reading through the post by " Osby " I have come to the conclusion ( rightly or wrongly ) that for any object in free fall, gravity and inertia are, 1- the same thing, 2- if not the same thing they both have the same effect. I will do some more reading.

Originally Posted by Dave Wilson

I have been reading through a very old physics book ( Fundamentals Of Physics with SI Units ) I have had a quick look at inertia, it states in Newton`s first law that the mass of an object is a measure of its inertia. The mass of an object is of course a measure of the quantity of matter in the object. After reading through the post by " Osby " I have come to the conclusion ( rightly or wrongly ) that for any object in free fall, gravity and inertia are, 1- the same thing, 2- if not the same thing they both have the same effect. I will do some more reading.

There is, in principle, the possibility that the term "mass" that occurs in theories of gravity (Newtonian gravity or general reltivity) is different from the mass term that occurs in dynamics INewtonian mechanics or reltivity, but F=dmv/dt). The first is called "gravitational mass" and the second is called "inertial mass". Experiments have been done to try to detect any difference. No difference has been found.

Thanks for the comments Dave and DrRocket.

Originally Posted by Dave Wilson

I have come to the conclusion ( rightly or wrongly ) that for any object in free fall, gravity and inertia are, 1- the same thing, 2- if not the same thing they both have the same effect. I will do some more reading.

My conclusion (right or wrong) is that gravitational mass and inertial mass are the same in proportionality (equivalence principle = true) and opposite in effect. Another conclusion I have reached is that in a gravitational interaction it is the total sum of gravitational mass that determines when, and it is the distribution of inertial mass that determines where.

Originally Posted by DrRocket

The first is called "gravitational mass" and the second is called "inertial mass". Experiments have been done to try to detect any difference. No difference has been found.

I think the best experiment to date is this one at University of Washington. They also have a nice web site. I have read that some gravitational theories predict a violation of the equivalence principle under some conditions. And it has even been argued that GR would allow for a violation of the EP for massive bodies.

Opinions? Can anyone find a problem with the OP?

Originally Posted by Osby

Opinions? Can anyone find a problem with the OP?

You did not justify your initial equation. The fact that it reduces to the usual Newtonian equation for gravitational force if inertial mass and gravitational mass are equal does not make it correct. Unless you can justify that equation it is not an expansion of the usual law. Calling it a modification is ad hoc without further justification. So, what is the basis ?

Normally you would start with Newton's law of universal gravitation for the force on one body, which we will take as small relative to a larger second body (or relative to some known inertial reference frame without any assumption on the relative size of the bodies).

where is the gravitational mass of the small body and is the gravitational mass of the large body.

One also has Newton's second law where is the inertial mass of the small body and is its acceleration, relative to the large body which we take as fixed.

Then and one sees that the acceleration is dependent on the ratio which is experimentally shown to be 1.

Originally Posted by DrRocket

You did not justify your initial equation. The fact that it reduces to the usual Newtonian equation for gravitational force if inertial mass and gravitational mass are equal does not make it correct. Unless you can justify that equation it is not an expansion of the usual law. Calling it a modification is ad hoc without further justification. So, what is the basis ?

I'm not sure what you're asking. Maybe calling it an expansion or modification is incorrect. It does represent a departure from the conventional. Note: I edited the first sentence of the OP.

Originally Posted by DrRocket

The fact that it reduces to the usual Newtonian equation for gravitational force if inertial mass and gravitational mass are equal does not make it correct.

It may not be correct. That is the reason I am posting it here. I want to get the opinions of others. I am not trying to prove it correct. I am trying to find reasons that it could not be correct. If you think that it cannot be correct, can you tell me why?

I agree with the remainder of your post. But I would like to add the following comments.

Originally Posted by DrRocket

One also has Newton's second law where is the inertial mass of the small body and is its acceleration, relative to the large body which we take as fixed.

The acceleration is relative to the center of mass of and , which is the frame of reference. But you are correct that when one body is much more massive than the other then the more massive body is usually taken to be the frame of reference.

Originally Posted by DrRocket

Then and one sees that the acceleration is dependent on the ratio which is experimentally shown to be 1.

The key difference between the traditional equations and the ones presented in the OP is that the former requires the EP to be true and the latter does not. Please review this Wikipedia entry.

Originally Posted by Osby

[The key difference between the traditional equations and the ones presented in the OP is that the former requires the EP to be true and the latter does not. Please review this Wikipedia entry.

You are missing the point.

It is quite true that Newtonian gravity and Newtonian mechanics assume that gravitational mass and inertial mass are the same. It is also true that that assumption is one way of stating the equivalence principle.

But the discussion that I provided for you does not make that assumption.

What is missing in your formulation is any reason to believe that it is valid. That fact that it does not assume the equivalence principle does not make it viable.

So, what is your justification for the form of the equation that you propose ?

The burden of proof is on you as the proposer to show why it might be correct and not on someone else to show why it is not.

The Wiki article does not support your equations and in reality reflects what I showed you (with the proviso that what Wiki calles active and passive gravitational masses are the same as what I called gravitational mass).

So, it still falls on you to justify your equations. You are basically proposing a new law of gravity. You need to show two things. 1) That your equations are consistent with all experiments and 2) That your equations predict something correctly that is missed by accepted theory.

Originally Posted by DrRocket

What is missing in your formulation is any reason to believe that it is valid. That fact that it does not assume the equivalence principle does not make it viable.

So, what is your justification for the form of the equation that you propose ?

The equation is simply an alternative in case the EP is ever found to be not true. For example. The EP postulates that inertial mass and gravitational mass are proportionally equivalent regardless of material composition. Suppose that someday it is found, through experiment, that the ratio of gravitational mass to inertial mass for copper is just very slightly different than the ratio of gravitational mass to inertial mass for aluminum. In that case the law of gravitation would need to be modified, or a new one proposed. I am exploring the possibility that the equation in the OP could be the basis for that new equation.

Originally Posted by DrRocket

The burden of proof is on you as the proposer to show why it might be correct and not on someone else to show why it is not.

It is correct when the EP is true. I thought that was clearly demonstrated in the OP. But obviously, I cannot show that it would be correct if the EP were not true. That is an experimental problem. But right now I have no reason to think that it could not be true.

Originally Posted by DrRocket

The Wiki article does not support your equations and in reality reflects what I showed you (with the proviso that what Wiki calles active and passive gravitational masses are the same as what I called gravitational mass).

Yes, I know. That is the reason I linked to it.

Originally Posted by DrRocket

So, it still falls on you to justify your equations. You are basically proposing a new law of gravity. You need to show two things. 1) That your equations are consistent with all experiments and 2) That your equations predict something correctly that is missed by accepted theory.

1) I think that was demonstrated in the OP for the case of EP = true.
2) I do not know how to do this. As stated before, it seems to be an experimental problem. But I am open to suggestions.

Originally Posted by Osby

It is correct when the EP is true. I thought that was clearly demonstrated in the OP. But obviously, I cannot show that it would be correct if the EP were not true. That is an experimental problem. But right now I have no reason to think that it could not be true.

Your equation is trivial if the EP is true.

Beyond that all that you have done is throw out one of an infintie number of expressions that reduce to Newtonian gravity if the EP is true.

However, the EP is really one of the foundational postulate of general relativity, and general relativity shows that Newtonian gravity is not true, though a good approximation in many situations.

So what you have is an equation that has zero basis if the EP is not true, and that trivially reduces to something that is not true if the EP is true. What is the point of that ?

Originally Posted by DrRocket

Your equation is trivial if the EP is true.

That may be a matter of opinion.

Originally Posted by DrRocket

However, the EP is really one of the foundational postulate of general relativity, and general relativity shows that Newtonian gravity is not true, though a good approximation in many situations.

If Newtonian gravity is not true then why is it used to substantiate the EP? As I stated in a previous post, many gravitational theories predict an EP violation. It has even been argued that GR would allow for a violation with massive bodies. With all of the ongoing torsion balance experiments, and the proposed satellite experiments to test the EP, the possibility of an EP violation is a subject of mainstream science.

Originally Posted by DrRocket

So what you have is an equation that has zero basis if the EP is not true, and that trivially reduces to something that is not true if the EP is true. What is the point of that ?

How do you know that? Are you basing that on Newton's laws being not true?

What is the point? I believe the equation could have some usefulness for illustration purposes. For example, when asked the question: Why does a heavy object fall at the same rate as a light object? Which equation would better illustrate the answer to that question? The traditional equation for acceleration, or the one shown in the OP?

Originally Posted by Osby

Originally Posted by DrRocket

Your equation is trivial if the EP is true.

That may be a matter of opinion.

Not really. Look at our equation. If gravitational mass and inertial mass are equal any damn fool can see that it reduces to Newtonian gravity. Anyhone who could not see that in under 1 second would flunk remedial freshman mathematics.

Originally Posted by DrRocket

However, the EP is really one of the foundational postulate of general relativity, and general relativity shows that Newtonian gravity is not true, though a good approximation in many situations.

Originally Posted by Osby

If Newtonian gravity is not true then why is it used to substantiate the EP? As I stated in a previous post, many gravitational theories predict an EP violation. It has even been argued that GR would allow for a violation with massive bodies. With all of the ongoing torsion balance experiments, and the proposed satellite experiments to test the EP, the possibility of an EP violation is a subject of mainstream science.

Of course Newton gravity is not true. If it were there would be no point in general relativity, which is vastly more complicated. It is also more accurate, as has been shown definitively by a large number of experiments. Newtonian gravity has been known to be only a good approximation since 1915. Where have you been ?

The EP is foundational to general relativity. Go read up.

Mainstream science constantly tests fundamental assumptions. The EP is no different. To day there is zero evidence of a violation.

Originally Posted by Osby

Originally Posted by DrRocket

So what you have is an equation that has zero basis if the EP is not true, and that trivially reduces to something that is not true if the EP is true. What is the point of that ?

How do you know that? Are you basing that on Newton's laws being not true?

Of course. That is fundamental physics.

Originally Posted by Osby

What is the point? I believe the equation could have some usefulness for illustration purposes. For example, when asked the question: Why does a heavy object fall at the same rate as a light object? Which equation would better illustrate the answer to that question? The traditional equation for acceleration, or the one shown in the OP?

Nobody cares what you believe. I know little kids that believe in the tooth fairy.

What is important is why anyone ought to believe the form that you propose for the equations of gravity in the even that the EP were not to be true. To be taken seriously you need either a theoretical reason for believing your equations or an experimental reason. Just putting together a bunch of symbols is not useful.

The fact that a heavy object falls at the same rate as light object (in a vacuum) has been known for centuries and is more than adequately described by Newtonian gravity and even more accurately by general relativity. Your equation explains nothing that is not explain more dearly by ordinary gravitational theories. Absolutely nothing.

Allow me to illustrate DrRocket's point.

I could easily say that the law should be: where is the smaller body.

Since EP seems to be true, this trivially reduces to Newtonian gravitation (which is known to be wrong).

If EP weren't true, why should this be the correct modification when I have no justification for it?

Originally Posted by DrRocket

Originally Posted by Osby

Originally Posted by DrRocket

Your equation is trivial if the EP is true.

That may be a matter of opinion.

Not really. Look at our equation. If gravitational mass and inertial mass are equal any damn fool can see that it reduces to Newtonian gravity. Anyhone who could not see that in under 1 second would flunk remedial freshman mathematics.

I misunderstood what you were referring to as trivial. Yes, I agree that the equation itself is trivial. But so are many other equations in physics. What is not trivial is what they tell us.

Originally Posted by DrRocket

Of course Newton gravity is not true. If it were there would be no point in general relativity, which is vastly more complicated. It is also more accurate, as has been shown definitively by a large number of experiments. Newtonian gravity has been known to be only a good approximation since 1915. Where have you been ?

I do not understand why you're coming down so hard on Newton's laws. Of course they are true (within limits). But I will admit that my knowledge of GR is limited.

You have been a very good and knowledgeable antagonist DrRocket. Which is what I was looking for. You have given me some things to think about. I would like to thank you for your time.

Originally Posted by MagiMaster

Allow me to illustrate DrRocket's point.

I could easily say that the law should be: where is the smaller body.

Since EP seems to be true, this trivially reduces to Newtonian gravitation (which is known to be wrong).

If EP weren't true, why should this be the correct modification when I have no justification for it?

Yes, I see your point. However, I also see a difference. I will think about this and reply later. It's getting late here. Thanks for the input MagiMaster.

Originally Posted by Osby

I do not understand why you're coming down so hard on Newton's laws. Of course they are true (within limits). But I will admit that my knowledge of GR is limited.

I an not "coming down so hard on Newton's laws". They are the product of one of the most profound geniuses of all time, and they are very accurate in most applications.

But, just like Newtonian mechanics, Newtonian gravity was shown to be not completely correct by Einstein's special and general theory of relativity. We know beyond a doubt that Newton's laws are good appoximations valid in most situations but that they are not completely correct.

The point is that there is no point in proposing simple and unjustified modifications to Newton's law of gravitation for the purpose of addressing subtle issues, when at the level of the very best available precision you have to use general relativity, and at that level of precision Newton's equations are simply wrong.

Thanks for the illustration MagiMaster. I know now what DrRocket was asking for. However, I'm not sure I can provide an adequate answer. If I come up with something I will return to this thread. By the way, your equation is flawed because it does not include inertial mass for . But it served it's purpose.

Originally Posted by DrRocket

I an not "coming down so hard on Newton's laws". They are the product of one of the most profound geniuses of all time, and they are very accurate in most applications.

I can agree with that.

Originally Posted by DrRocket

The point is that there is no point in proposing simple and unjustified modifications to Newton's law of gravitation for the purpose of addressing subtle issues, when at the level of the very best available precision you have to use general relativity, and at that level of precision Newton's equations are simply wrong.

But I disagree with that. Correct me if I'm wrong, but high precision does not necessarily require GR. When measuring the value of G, or testing the equivalence of inertial mass and gravitational mass, it is Newton's equations that are used, not those of GR. And those tests and experiments require extreme precision. Subtle issues that affect Newton's laws would certainly affect those tests and experiments.

Originally Posted by Osby

But I disagree with that. Correct me if I'm wrong, but high precision does not necessarily require GR. When measuring the value of G, or testing the equivalence of inertial mass and gravitational mass, it is Newton's equations that are used, not those of GR. And those tests and experiments require extreme precision. Subtle issues that affect Newton's laws would certainly affect those tests and experiments.

OK as requested, you are wrong.

We already have experiments that disagree with Newtonian gravity, but not with general relativity.

If you are gong to chase unknown and only hypothesized effects that occur far to the right of the decimal point, you have to compare your experiment with the best available theoretical model. That is general relativity. GR and Newtonian gravity are close under most circumstances, but they do not produce identical results. So if you need to compare experiment to theory at the limits of precision, then you have to use the best available theory -- is GR.

There are other ways of formulating the equivalence principle than the equality of inertial and gravitational mass. General relativity is based on the equivalence principle in the form that acceleration is indistinguishable from gravity, and that implies that inertial and gravitational mass are equal. So to challenge the equivalence principle is to challenge GR. Clearly a challenge to GR requires comparison of the predictions of GR, not the Newtonian approximation, with experiment.

Originally Posted by DrRocket

We already have experiments that disagree with Newtonian gravity, but not with general relativity.

If you are gong to chase unknown and only hypothesized effects that occur far to the right of the decimal point, you have to compare your experiment with the best available theoretical model. That is general relativity. GR and Newtonian gravity are close under most circumstances, but they do not produce identical results. So if you need to compare experiment to theory at the limits of precision, then you have to use the best available theory -- is GR.

There are other ways of formulating the equivalence principle than the equality of inertial and gravitational mass. General relativity is based on the equivalence principle in the form that acceleration is indistinguishable from gravity, and that implies that inertial and gravitational mass are equal. So to challenge the equivalence principle is to challenge GR. Clearly a challenge to GR requires comparison of the predictions of GR, not the Newtonian approximation, with experiment.

That's interesting. Can you be more specific and give an example of how GR is used to measure the value of G? But regardless of whether it's used or not, I'm pretty sure that Newton's equations are used. So I find it interesting that GR uses a constant who's value is determined through the use of Newton's equations.

Originally Posted by Osby

Originally Posted by DrRocket

We already have experiments that disagree with Newtonian gravity, but not with general relativity.

If you are gong to chase unknown and only hypothesized effects that occur far to the right of the decimal point, you have to compare your experiment with the best available theoretical model. That is general relativity. GR and Newtonian gravity are close under most circumstances, but they do not produce identical results. So if you need to compare experiment to theory at the limits of precision, then you have to use the best available theory -- is GR.

There are other ways of formulating the equivalence principle than the equality of inertial and gravitational mass. General relativity is based on the equivalence principle in the form that acceleration is indistinguishable from gravity, and that implies that inertial and gravitational mass are equal. So to challenge the equivalence principle is to challenge GR. Clearly a challenge to GR requires comparison of the predictions of GR, not the Newtonian approximation, with experiment.

That's interesting. Can you be more specific and give an example of how GR is used to measure the value of G? But regardless of whether it's used or not, I'm pretty sure that Newton's equations are used. So I find it interesting that GR uses a constant who's value is determined through the use of Newton's equations.

Newtonian gravity is an approximation to general relativity, a good approximation under most circumstances, when the gravitational field is not too great.

G, the gravitational constant occurs in the formulation of general relativity. So to experimentally determine G using the Newtonian model is to determine G using an approximation to general relativity. If your level of precision is such that the less exact model of Newtonian gravity is acceptable then you can use the Newtonian model, but if you require higher precision then you need to use general relativity.

This is a totally different issue from the equivalence principle. General relativity also incorporates the equivalence principle, in a more integral way than does the Newtonian model.

Neither Newtonian gravity nor general relativity are compatible with your proposed modifications, unless inertial mass and gravitational mass are equal, In that case your equations trivially reduce to the Newtonian case. But if it were to be found experimentally that a very small effect were to result in a discrepancy with the Newtonian model it would not be immediately clear if the discrepancy was the result of a violation of the equivalence principle or because of the known small deviations between Newtonian physics and general relativity. It would all depend on the size and nature of the deviation. Thus far, the most sophisticated and precise experiments that are possible have found no such deviation, so one can reasonably assume that any future deviation that might be found would be very small and very subtle.

That is the reason that without some basis to support the form of the equation that your suggest there is no reason to consider it seriously. The reason that you do not see such an obvious and simple expression in the literature is that there is no reason to consider it. Physicists are not stupid and if there were any reason to consider such a thing, at least a hundred people would have already proposed it in the literature.

You will find in the literature several alternate theories of gravity. It is not that science is stuck on the Newtonian model or even on general relativity. You can find a fairly thorough look at alternate theories in one of the standard texts on general relativity Gravitation by Misner, Thorne and Wheeler. The bottom line is that general relativity has stood the test of a huge number of experiments. It is known to be extremely accurate over an extremely wide range of conditions, and is the accepted current best model of gravity.

General relativity is not the last word. It is also known that general relativity is incompatible with quantum mechanics. This is a problem when one is trying to describe the very early universe, a fraction of second following he big bang, and it is a problem when trying to describe the interior of a black hole, near the predicted singularity. Since quantum field theories are also known to be extremely accurate when gravity can be neglected, there is a great deal of research directed at the formulation of a theory that would include both quantum theory and gravity.

There is also a competing theory to general relativity that addresses gravitation that does not involve quantum theory. It is called Einstein-Cartan theory. The difference between Einstein-Cartan theory and general relativity is that Einstein-Cartan theory does not make the a priori assumption that spacetime is torsion-free. This results in a more complex mathematical structure, but it also results in a theory that can handle intrinsic spin and that does not predict at least some of the singularities that occur with general relativity. The predictions of Einstein-Cartan theory are,however, so close to general relativity under most circumstances that no experiment can distinguish between the two given the current limitations of measurement technology.

So, the bottom line is that Newtonian gravity works very well in most applications, and is used almost exclusively in engineering applications. It is certainly the model used to calculate orbits for satellites and planets for instance. It is the dominant model used in everyday astrophysics. But when extreme precision is needed, then the more accurate model of general relativity is used. Gravity is understood rather well, and deviation from Newton's approach is found in the geometric formulation of general relativity which is described using the rather sophisticated mathematics of pseudo-Riemannian geometry, and not with some simple algebraic "modification" of Newton's law. A lot of thought by a lot of very highly educated and intelligent people has gone into the study of gravity and they have not missed some simple alternative to Newton. This has been proved repeatedly by the extremely close agreement between general relativity and some extremely sophisticated and expensive experiments.

You are barking up the wrong tree.

If you want to understand this further you will have to read a serious physics book or two on the theory of gravity. Here are some titles:

This one is fairly easy reading

Relativity, Special, General and Cosmological by Wolfgang Rindler

These are not so easy but tell more of the story

General Relativity P.A.M. Dirac

General Relativity Robert Wald

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity by Steven 'Weinberg

This is probably the best, most encyclopedic and used the language of differential forms which has many advantages

Gravitation by Charles Misner, Kip Thorne, and John Archibald Wheeler

I have taken note of a couple of the above mentioned books. I will have a look for them on Amazon.

Originally Posted by DrRocket

Newtonian gravity is an approximation to general relativity, a good approximation under most circumstances, when the gravitational field is not too great.

So do you mean that since the gravitational fields involved in the measurement of G are so weak that the use of Newton's equations are more than adequate? If that is the case then I understand. And does that mean that my original statement that high precision does not necessarily require GR is true? Since the fields involved are so weak a very high degree of precision can be obtained simply by using Newton's laws?

Originally Posted by DrRocket

G, the gravitational constant occurs in the formulation of general relativity. So to experimentally determine G using the Newtonian model is to determine G using an approximation to general relativity. If your level of precision is such that the less exact model of Newtonian gravity is acceptable then you can use the Newtonian model, but if you require higher precision then you need to use general relativity.

Is it possible to use the Einstein field equations to measure the value of G?

Originally Posted by DrRocket

Neither Newtonian gravity nor general relativity are compatible with your proposed modifications, unless inertial mass and gravitational mass are equal, In that case your equations trivially reduce to the Newtonian case.

Ok, this is the kind of response I was looking for in my OP. How and why would the equation be incompatible with Newtonian gravity if the EP were not true (other than the obvious)?

Originally Posted by DrRocket

But if it were to be found experimentally that a very small effect were to result in a discrepancy with the Newtonian model it would not be immediately clear if the discrepancy was the result of a violation of the equivalence principle or because of the known small deviations between Newtonian physics and general relativity.

That's a good point. One I have not considered. However, my initial reaction is that this would not apply to EP violations involving "composition dependent gravity", since those violations would affect both the Newtonian model and GR equally. But I'm not sure because I don't know enough about GR.

Originally Posted by DrRocket

That is the reason that without some basis to support the form of the equation that your suggest there is no reason to consider it seriously. The reason that you do not see such an obvious and simple expression in the literature is that there is no reason to consider it. Physicists are not stupid and if there were any reason to consider such a thing, at least a hundred people would have already proposed it in the literature.

FYI - I know that physicists are not stupid, and I know that I am not smart. But I cannot help but think for myself. Why must I study something and not have my own ideas? If I discover that I am wrong, or that I have misconceptions, then I have learned something in the process. If I were a young person and I wanted to make a career out of this then things would be different. (I certainly wouldn't be on an internet forum). However, I am in this both for enjoyment and to learn. But anyway, thanks for answering my original question "have you seen anything like this before?".

Thanks for the book references. I think I remember reading a few pages from Gravitation on Google Books. I'm reading "Concepts Of Mass In Contemporary Physics And Philosophy" right now. I've been avoiding GR because I feel it may be too difficult for me. But I'll check out some of the books in your list, especially the first one.

Originally Posted by Osby

Originally Posted by DrRocket

Newtonian gravity is an approximation to general relativity, a good approximation under most circumstances, when the gravitational field is not too great.

So do you mean that since the gravitational fields involved in the measurement of G are so weak that the use of Newton's equations are more than adequate?

No.

I mean that G is a fundamental physical constant that is required to describe gravitation.

You do not measure G. You measure some gravitational event from which you infer a value of G, using an overall model of gravitation.

We have two models of gravitation, one is Newtonian gravity. The other is general relativity. They try to describe the same physical phenomena.

So, now you measure some phenomena. From that you do some calculation to figure out what the value of G is. If you do not need to be too accurate in determining G, then the Newtonian model will work just fine. But if you need a more accurate value for G you need to use the most accurate gravitational model available, and that is general relatvity. If you calculate different values using Newtonian gravity and general relativity, then general relativity wins. They should be close and differ only somewhere pretty far to the right of the decimal point. If both calculations produce a surprise then we need a new theory entirely.

Basically you start with the measurement, which is fixed, and dump it into a calculating machine to get G. To get the best value of G you need the best calculating machine. If you don't need that much precision then the simple machine is good enough.

Here is the Newtonian equation used to determine the value of G using a torsion balance. It is taken from the Instruction manual of a Pasco torsion balance model AP-825.



where,

= torsion constant for the band
= angle through which the band is twisted
= distance between centers of two masses
= length of lever arm of pendulum bob crosspiece

My question: Can the Einstein field equations be used in a similar manner to determine the value of G? It does not have to be a torsion balance. It can be any method. The point is that the Einstein field equations are used instead of the Newtonian equations.

Originally Posted by Osby

Here is the Newtonian equation used to determine the value of G using a torsion balance. It is taken from the Instruction manual of a Pasco torsion balance model AP-825.



where,

= torsion constant for the band
= angle through which the band is twisted
= distance between centers of two masses
= length of lever arm of pendulum bob crosspiece

My question: Can the Einstein field equations be used in a similar manner to determine the value of G? It does not have to be a torsion balance. It can be any method. The point is that the Einstein field equations are used instead of the Newtonian equations.

One more time.

The Einstein inclusions imply the Newtoian equations in the limit of small gravitational fields. Any time you use the Newtonian equations successfully you have also used the Enstein field equations. It is only when you need greater accuracy that you need to solve the equations more precisely.

Originally Posted by DrRocket

One more time.

The Einstein inclusions imply the Newtoian equations in the limit of small gravitational fields. Any time you use the Newtonian equations successfully you have also used the Enstein field equations. It is only when you need greater accuracy that you need to solve the equations more precisely.

Well, I need greater accuracy. How do I do it?

Originally Posted by Osby

Originally Posted by DrRocket

One more time.

The Einstein inclusions imply the Newtoian equations in the limit of small gravitational fields. Any time you use the Newtonian equations successfully you have also used the Enstein field equations. It is only when you need greater accuracy that you need to solve the equations more precisely.

Well, I need greater accuracy. How do I do it?

Then you go get a specialist in numerical solutions of the Eiinstein field equations, very carefully measure and define your problem and run the computer for a while.

There are very few k nown exact solutions of the field equations. They usually have to be solved numerically.

For any situation that you will find on Earth, where gravity is smalll by the standards of general relativity, the solution will agree with the Newtonian solutions to very many decimal places.

I'm confused.

Originally Posted by DrRocket

Originally Posted by Osby

But I disagree with that. Correct me if I'm wrong, but high precision does not necessarily require GR. When measuring the value of G, or testing the equivalence of inertial mass and gravitational mass, it is Newton's equations that are used, not those of GR. And those tests and experiments require extreme precision. Subtle issues that affect Newton's laws would certainly affect those tests and experiments.

OK as requested, you are wrong.

Originally Posted by DrRocket

Originally Posted by Osby

Originally Posted by DrRocket

Newtonian gravity is an approximation to general relativity, a good approximation under most circumstances, when the gravitational field is not too great.

So do you mean that since the gravitational fields involved in the measurement of G are so weak that the use of Newton's equations are more than adequate? If that is the case then I understand. And does that mean that my original statement that high precision does not necessarily require GR is true? Since the fields involved are so weak a very high degree of precision can be obtained simply by using Newton's laws?

No.

Originally Posted by DrRocket

For any situation that you will find on Earth, where gravity is smalll by the standards of general relativity, the solution will agree with the Newtonian solutions to very many decimal places.

Originally Posted by Osby

I'm confused.

Yes, you are.

It is a matter of how high a precision you need.

For most purposes Newtonian gravity works just fine. It will measure G quite well for instance.

But you started this thread with a modification of Newtonian gravity that would violate the equivalence principle of both the Newtonian model and general relativity, and that is sufficiently subtle that a host of very sensitive and very sophisticated experiments have been unable to detect even the slightest violation.

So, if you are looking for something that subtle, you need to look at the very best theoretical model.

You have also mixed together the issues of the determination of the constant G with the possible violation of the equivalence of inertial and gravitational mass. Those are two different issues.

Given the precision of available instrumentation, a measurement of G based on the Newtonian model will be as accurate as the instruments will permit. In principle, if instruments were very much better and one wanted to measure G to an extreme level of accuracy you would reach a point at which you would have to use a general relativistic model. But we are a long way from that point right now.

As far as a violation of the equivalence principle goes, there is no experiment with current technology that would detect a violation using either the Newtonian or general relativistic model, and there is zero reason to believe that the equivalence is not exact anywayt. But if one had instrumentation from the future and if one did detect a variation based on the Newtonian model, the deviation would be so slight that one could not conclude that a real violation had occurred without doing the necessary calculation to determine if general relativity had also been violated, simply because we know that Newton's model is not quite right.

Hence, a modification of Newtonian gravitation for the purpose of evaluating the lack of equality of inertial and gravitational mass is not useful.

Note that this same line of reasoning would apply if there were a more accurate theory than general relativity which also involved the equivalence principle. We would have to use that model for experiments requiring the ultimate in accuracy. General relativity is singled out here because it is the best model that we have at the moment.

Originally Posted by DrRocket

Originally Posted by Osby

I'm confused.

Yes, you are.

Finally! something I'm right about.

Originally Posted by DrRocket

You have also mixed together the issues of the determination of the constant G with the possible violation of the equivalence of inertial and gravitational mass. Those are two different issues.

Mixing the two together is normal because they are interrelated. Have you ever looked at an actual experiment? Here's one:
http://iopscience.iop.org/1742-6596/...9_1_012019.pdf
Read the last sentence of the abstract then read section 1.2.

Originally Posted by DrRocket

As far as a violation of the equivalence principle goes, there is no experiment with current technology that would detect a violation using either the Newtonian or general relativistic model, and there is zero reason to believe that the equivalence is not exact anywayt. But if one had instrumentation from the future and if one did detect a variation based on the Newtonian model, the deviation would be so slight that one could not conclude that a real violation had occurred without doing the necessary calculation to determine if general relativity had also been violated, simply because we know that Newton's model is not quite right.

Instrumentation from the future is not required. Did you know that the equivalence of active gravitational mass has only been laboratory tested to a precision of . It was the Kreuzer experiment of 1968. Modern technology has far surpassed that level of precision. Look it up.

There's no point in continuing to argue the equations of the OP. You've made up your mind that it's pointless garbage. I will consider your argument.

Originally Posted by Osby

Instrumentation from the future is not required. Did you know that the equivalence of active gravitational mass has only been laboratory tested to a precision of . It was the Kreuzer experiment of 1968. Modern technology has far surpassed that level of precision. Look it up.
.

You are missing the point entirely.

The question that you have raised is not the precision with which the equivalence of inertial mass, active gravitational mass and passive gravitational mass (this is the first time that you have raised the issue of active vs passive gravitational mass by the way) have been measured. The issue is the precision that would be required to differentiate between differences among those mass types and a discrepancy between Newtonian gravity and general relativity, if one were to be found. A gross discrepancy would invalidate both theories. But a gross discrepancy is extremely unlikely given the accuracy with which delecate phenomena have been predicted to date. A very small discrepancy would require that you make your comparison against the best available theory -- general relativity.

Also you keep switching back and forth between the issue of measuring G and the issue of differentiating between the hypothetical mass types. Those are totally different issues.

G measurements can be made with suitable accuracy using the Newtonian model now and for a long time to come. In principle there might come a time when the precision of measurement was so high that you would need to use general relativity, or maybe some better theory that would have been developed by that time. The measurement of G is just the determination of a number, and one can live with some uncertainty in that number (i.e. you don't need infinite precision).

With regard to the equivalence of the mass types the situation is a bit different. The conceptof mass is important in modern physical theoreis. In fact relativity has told us that mass and energy are the same thing. (] A discrepancy in this area would be a major blow to general relativity. It would not be just some simple adjustment of a simple equation -- GR does not work that way. It is not just important that inertial mass and gravitational mass be approximately equal, itis important that they be the same thing. GR has been validated by many tests. So, I think I can say with quite a bit of confidence that if there is any difference it will show up as a tiny and very subtle discrepancy requiring utmost precision and the best available gravitational model to quantify it and to assure that the discrepancy is real and not an experimental error.

Originally Posted by DrRocket

You are missing the point entirely.

The question that you have raised is not the precision with which the equivalence of inertial mass, active gravitational mass and passive gravitational mass (this is the first time that you have raised the issue of active vs passive gravitational mass by the way) have been measured. The issue is the precision that would be required to differentiate between differences among those mass types and a discrepancy between Newtonian gravity and general relativity, if one were to be found.

Yes, you have already raised this issue and I acknowledged.

Originally Posted by Osby

Originally Posted by DrRocket

But if it were to be found experimentally that a very small effect were to result in a discrepancy with the Newtonian model it would not be immediately clear if the discrepancy was the result of a violation of the equivalence principle or because of the known small deviations between Newtonian physics and general relativity.

That's a good point. One I have not considered. However, my initial reaction is that this would not apply to EP violations involving "composition dependent gravity", since those violations would affect both the Newtonian model and GR equally. But I'm not sure because I don't know enough about GR.

So if you really want to help me out, you can confirm or deny that "composition dependent gravity" would affect GR equally to Newtonian gravity or not.

Edit: Never mind. I see that this is somewhat of a misinterpretation on my part. As I've said before, my knowledge of GR is limited. So it's going to be difficult for us to have a discussion in that context. I have studied a lot of torsion balance tests and experiments and I have never encountered the application of GR. But if a violation of the equivalence of gravitational mass and inertial mass is ever detected, I am not denying that it could be because of deviations between Newtonian gravity and GR. It will be yet another opportunity to verify the validity of GR - or destroy it.

Originally Posted by DrRocket

Also you keep switching back and forth between the issue of measuring G and the issue of differentiating between the hypothetical mass types. Those are totally different issues.

It was not about differentiating between the mass types. It was about the measured value of G and testing the equivalence of gravitational mass and inertial mass. Evidently you did not read the links I provided. And the remainder of your post further confirms that you did not read the links. Any difference in the ratio of gravitational mass to inertial mass from one composition of matter to another would show up as different measured values for G between equal inertial masses of those two compositions of matter.

You might be interested in this on-going experiment that will, among other things test the strong equivalence principle a bit further.

http://arxiv.org/PS_cache/arxiv/pdf/...710.0890v2.pdf