Why is the Principle of Equivalence considered a theorem?

[vinter]

I was reading the Principle of Equivalence which says that inertial mass is equal to the gravitational mass, though I am not very clear that why it should be written as a theorem. Here is what I read and the subsequent confusions that formed :-
First, to define the inertial mass, you fix a unit system. If you fix the SI system, you bring the object kept at the International Beauro of Weights and Measurements which is labeled 1 kg, apply a certain force on it and measure the acceleration; let it be a m/s^2. Now you bring another arbitrary object, label it m kg., apply the same force on it and measure its acceleration; let it be b m/s^2. Give a value to m such that m*b = 1 * a. This defines the inertial mass of the arbitrary object precisely.
Now, you assume that the gravitational force acting between two objects is directly proportional to the product of their inertial masses as defined above and inversely proportional to the square of the distance between them. Then you perform several experiments to find the constant of proprotionality. The constant you find this way will obviously be adjusted in a way that the masses you use in the gravitational force law formula are exactly equal to the inertial masses. And then, you suddenly become inquisitive and ask an important question - Are the masses coming in the gravitational force law equation equal to the inertial mass? To answer this, you again perform several experiments and find that they are, in fact, equal. You become excited and say that this is the beauty of the nature, state this thing as the "principle of equivalence" and subsequently conclude that God must be present (this last part was sarcasm). What the heck?

 

[Crosson]

If the inertial and gravitational masses were not equal, then the acceleration due to gravity would be proportional to the mass of the object! This is how we test the equivalence principle: check that all massive objects fall at the same rate in a vacuum.

The next time you think you have busted all of 20th century physics on a triviality, think harder.

 

[marlon]

There is another way of looking at this. When you jump outside a window and you fall down (you are in free fall) , you don't feel your own weight. But ofcourse you know two things : gravity is pulling you down and you have a certain mass. If you apply newto,'s second law you'd write 0 = -mg+m'a' where m' is the inertial mass and m'a' is the pseudoforce that you need to add because mg itself is NOT zero. the LHS must be zero because you don't feel your own weight as you fall down (it appears is if there are no forces acting on you)

This we have mg=m'a'. The equivalence principle states that :

1) if a'=g then m=m'
2) if m=m' then a=g'

In words :

Suppose you are in a closed room and you watch an apple fall down and you measure your weight. Now, consider to possiblities. Either the apple falls down because it is submitted to gravitation. You also can measure your weight mg because of gravitation. Secondly, if you are not on the Earth but the cabine you are in is moving upward with acceleration equal to g in magnitude, the apple will make the exact same movement relative to you AND you'd measure the exact same weight m'g.

marlon

 

[dextercioby]

Conceptually,it's not a theorem,but an axiom.Both in Newtonian physics of gravity and in General Relativity...

Daniel.

 

[vinter]

ok, I agree that I could not convey my point properly and that I don't know the spelling of Beureau. So here I make one more attempt :-
What I have understood about the principle of equivalence is that the mass that you use in the force law for gravity and the mass you use in the equation F = ma are same. Right? Now we know that the force law of gravitation is an empirical law, so it has been found experimentally. In doing that, we had a choice for the constant G and we chose it in such a way that if the masses taken in the force law for gravitation are EQUAL to the inertial masses, the force that we get is EQUAL to the gravitational force. So isn't the principle of equivalence just a matter of definition? Didn't we ensure that the gravitational mass is equal to the inertial mass as soon as we chose this particular value of G?

 

[Thomas2]

ok, I agree that I could not convey my point properly and that I don't know the spelling of Beureau. So here I make one more attempt :-
What I have understood about the principle of equivalence is that the mass that you use in the force law for gravity and the mass you use in the equation F = ma are same. Right? Now we know that the force law of gravitation is an empirical law, so it has been found experimentally. In doing that, we had a choice for the constant G and we chose it in such a way that if the masses taken in the force law for gravitation are EQUAL to the inertial masses, the force that we get is EQUAL to the gravitational force. So isn't the principle of equivalence just a matter of definition? Didn't we ensure that the gravitational mass is equal to the inertial mass as soon as we chose this particular value of G?

Yes, you are right. One should strictly speaking say that the inertial mass is proportional to the gravitational mass. They are only equal because the gravitational constant has been adjusted accordingly. However, this is the obvious thing to do, because the gravitational mass can only be defined through the gravitational force law which contains the gravitational constant as a free parameter. Since you have to fix the latter somehow, you might as well choose it such that the mass is identical to the inertial mass. Textbooks should be somewhat clearer in this respect.

 

[shyboy]

O yes, we can measure the gravitational mass in pounds and inertial mass in gramms.
Damn these French units!

 

[vinter]

Yes, you are right. One should strictly speaking say that the inertial mass is proportional to the gravitational mass. They are only equal because the gravitational constant has been adjusted accordingly. However, this is the obvious thing to do, because the gravitational mass can only be defined through the gravitational force law which contains the gravitational constant as a free parameter. Since you have to fix the latter somehow, you might as well choose it such that the mass is identical to the inertial mass. Textbooks should be somewhat clearer in this respect.

Exactly, that's what I was trying to say. Thanks for making it clear.
So as you said, the value taken for G is what makes gravitational mass exactly equal the inertial mass. But the fact that we can make such a thing happen by a mere adjustment of the proportionality constant requires that the two masses be at least proportional to each other and this requirement must be fulfilled by nature itself. My understanding so far is that the principle of equivalence states that this, indeed, is true. Is it correct?

 

[Thomas2]

Exactly, that's what I was trying to say. Thanks for making it clear.
So as you said, the value taken for G is what makes gravitational mass exactly equal the inertial mass. But the fact that we can make such a thing happen by a mere adjustment of the proportionality constant requires that the two masses be at least proportional to each other and this requirement must be fulfilled by nature itself. My understanding so far is that the principle of equivalence states that this, indeed, is true. Is it correct?

Yes, it is correct (see http://www.mathpages.com/home/kmath582/kmath582.htm for an account of Newton's own work on this). In principle it would therefore be possible that gravity acts only on a certain percentage of any mass. This would not make any difference whatsoever for experiments (unless one could separate these components somehow).

 

[AntonVrba]

This is a very interesting point made here.

Further if you consider that G is THE physical constant that has the higherst uncertaintiy (i.e least accurate) of 1.5 x 10^-4 . in comparison to Planck's constant 10^-7 and electric charge 10^-8.

The irony is that Planck's constant units are J-s or and the Units of Joule are KILOGRAM metres^2/time^2 but then we do not know what a kilogram is other than a lump of metal stored in a safe.

Maybe just maybe gravity is not that what we think it is -otherwise G would have been better defined by now.

So

The next time you think you have busted all of 20th century physics on a triviality, think harder.

Crosson I would suggest keep an open mind - closed minds stop developments

 

[marlon]

Crosson I would suggest keep an open mind - closed minds stop developments

Accepting bull**** is not the same as being open minded. Correcting false claims is not the same as being narrow minded. When are people going to get that ?

marlon

 

[shyboy]

well, once the Earth rotation was considered as a ***. I think that it is important to be able to defend the controversial thesis. Then, even if you obecting the quantum mechanics, like Einstein did, the science will still benefit.

Unfortunately, we need to have somebody who will pursue the "dead ends", because in some part the progress depends on luck. Of course, you should remmeber that the price for the failure is your wasted life.

 

[Integral]

well, once the Earth rotation was considered as a ***. I think that it is important to be able to defend the controversial thesis. Then, even if you obecting the quantum mechanics, like Einstein did, the science will still benefit.

Unfortunately, we need to have somebody who will pursue the "dead ends", because in some part the progress depends on luck. Of course, you should remmeber that the price for the failure is your wasted life.

Such objections must be based upon thorough knowledge of the current state of the theories. If you do not know or understand what is known by those educated from a field you simply cannot make a meaningful contribution. You are arguing from ignorance and blind faith, not knowledge and experimental evidence.

 

[whozum]

I don't understand the debate here. It seems everyone is just saying 'Hey there's this thing called G and it relates the force of gravity to acceleration of freefall'.

 

[AntonVrba]

this thread is going off-topic but never mind - the thread is about equivalence - let it be equivalence of theories.

Such objections must be based upon thorough knowledge of the current state of the theories. If you do not know or understand what is known by those educated from a field you simply cannot make a meaningful contribution. You are arguing from ignorance and blind faith, not knowledge and experimental evidence.

That is the most snobbish and establishment like statement I have read! What happens if there are some fundamental flaws in the current theories?

I disagree with you, by having a free mind you can walk along a new path and not be hindered by "so-and-so said such-and-such". As long as any new theories proposed confirm the observed then they are valid to be investigated. Remember the fool invented it because he did not know it was impossible.

I honestly think some theories have big flaws. Let's take the vitual particles (W bosuns) and the uncertainty principles of Heisenberg and the deduction in quantum mechanics that allows a particle to temporarily "borrow" energy provided that it is relinquished within the time determined by Heisenberg's equations. This is just a nice way of "the educated in the field" describing the observed without really knowing what is happening by using an otherwise good theory to get out of a hole.

Instead of "borrowing" energy, possibly the energy is present in a yet undetectable form (the virtual particles) let's say the compliment of EM radiation, or gravity-electric or gravity-magnetic or whatever it may be.

By postulating a new energy form or radiation it would then be possible to have alterante explanations to the observed - and as such could also be valid and should be debated and not outright rejected just because the proposal was made from outside the establishment.

A simple question - do we know what a neutrino really is?

 

[vinter]

Hey, relax everybody! I posted this thread just because I had this confusion. Obviously in today's time, considering the large amount of work done by so many greats in the past few centuries, if a novice like me, by some thought process, reaches a point which contradicts some very established theorem, or even a point where the full scenario seems absurd, most probably his thought process was wrong, not the actual theorem. And what I wanted to know here was just the flaw in my thought process. I never had any aim of proving the current theory wrong, though it may have sounded so.
Just as an example, consider some very well set paradox. If someone comes up with a situation where it seems that the principle of conservation of energy is flawed, of course, there is something wrong in the situation he has come up with and not in the principle itself. And in such a case, one would prefer to sit and try to find what's wrong in it instead of saying something like "obviously you are wrong. do you think you are so smart that you can prove the principle of energy conservation wrong?"