The feather, the hammer, and the moon?

[jajrussel]

Each cause spacial curvature to different degrees, and apparently the curvatures do not add together. The hammer and the feather follow the moons curved space at the same rate of acceleration.  It seems that the only effect individual curvature has is in how much energy must be used on the hammer or feather to change it's direction?

[Bufofrog]

48 minutes ago, jajrussel said:

Each cause spacial curvature to different degrees, and apparently the curvatures do not add together. The hammer and the feather follow the moons curved space at the same rate of acceleration. 

I am not sure what you are saying.  Are you surprised that a hammer and a feather fall at the same rate on the moon?  Do you think because the hammer has more mass it should fall faster?  If that is your question the reason is that the feather and the hammer are both << than the mass of the moon, so there contribution to the acceleration (or spacetime curvature) is nil, compared to the moon.

52 minutes ago, jajrussel said:

It seems that the only effect individual curvature has is in how much energy must be used on the hammer or feather to change it's direction?

That is not about gravity and the curvature of spacetime, that is just the force needed to change the direction of the mass.  Since F=ma the larger mass needs a higher force to achieve the same deflection as the feather.    

I hope I have not misunderstood what you were asking.

[swansont]

1 hour ago, jajrussel said:

Each cause spacial curvature to different degrees, and apparently the curvatures do not add together. The hammer and the feather follow the moons curved space at the same rate of acceleration.  It seems that the only effect individual curvature has is in how much energy must be used on the hammer or feather to change it's direction?

The feather and hammer attract each other, but this is quite small in comparison to their attraction to the moon. The curvatures do add, but it is negligible. But why bring GR into the discussion? This is more than adequately handled with Newtonian gravity. And if the issue is GR, there are probably more illuminating examples.

Direction change is a result of force, causing an acceleration. A description using energy is more complicated.

[jajrussel]

34 minutes ago, Bufofrog said:

am not sure what you are saying.  Are you surprised that a hammer and a feather fall at the same rate on the moon?  Do you think because the hammer has more mass it should fall faster?  If that is your question the reason is that the feather and the hammer are both << than the mass of the moon, so there contribution to the acceleration (or spacetime curvature) is nil, compared to the moon.

Well, yes. I'm sure that I'm not the only one who would have been surprised. I have accepted it but still, would like to know why the difference in acceleration is so negligible? And if there is a difference isn't it better to be told that there is a difference no matter how negligible than to be taught that there is no difference that the hammer and the feather fall at exactly the same rate. That seems like sensationalism rather than science. It makes a difference because one truth leads to another. When you simply say that both accelerate at exactly the same rate then what is the point of individual curvatures? 

Swansont is much faster than I, I need to stop and see what he has written.

Edited May 10, 2019 by jajrussel

[Eise]

1 hour ago, jajrussel said:

Each cause spacial curvature to different degrees, and apparently the curvatures do not add together. The hammer and the feather follow the moons curved space at the same rate of acceleration.  It seems that the only effect individual curvature has is in how much energy must be used on the hammer or feather to change it's direction?

I would explain it in the following way. 

Mass has 2 obvious properties: resistance against acceleration, and gravity. Both are proportional to the mass of the object (the feather and the hammer in this case):

Resistance against acceleration is described by Newton's law: F = m_hammer * a

Gravity is expressed by F = G M_moon * m_hammer / distance².

So now rewrite number one as. a = F / m_hammer.

So you know what a = (G M_moon * m_hammer / distance² ) / m_hammer =  G M_moon / distance² .

As you see, the mass of the hammer plays no role anymore. Of course with the feather it is the same. So both have the same acceleration.

For the curvature of the moon, it is also just the same for the feather and the hammer. Of course they both have their individual mass, and with that a very tiny gravity field. But it is negligible with the gravity of the moon.

 

[Strange]

The spacetime curvature caused by the feather and the hammer are insignificant compared to the curvature caused by the moon(*). So we can assume that the paths followed by the objects is purely determined by the mass of the moon. 

(*) This is equivalent to the Newtonian approximation of assuming that the masses are small and don't have any effect n the moon

[Bufofrog]

49 minutes ago, jajrussel said:

Well, yes. I'm sure that I'm not the only one who would have been surprised. I have accepted it but still, would like to know why the difference in acceleration is so negligible?

Another way to look at this is how much do you think the feather or hammer is causing the moon to accelerate towards them?  I think negligible is maybe even an over statement, don't you agree?

[Janus]

25 minutes ago, jajrussel said:

Well, yes. I'm sure that I'm not the only one who would have been surprised. I have accepted it but still, would like to know why the difference in acceleration is so negligible? And if there is a difference isn't it better to be told that there is a difference no matter how negligible than to be taught that there is no difference that the hammer and the feather fall at exactly the same rate. That seems like sensationalism rather than science. It makes a difference because one truth leads to another. When you simply say that both accelerate at exactly the same rate then what is the point of individual curvatures? 

Swansont is much faster than I, I need to stop and see what he has written.

It really matters what frame you are measuring the rate of "fall" by.  We tend to want to measure it as relative to the Large body ( in this case, the Moon).  So in that case, if you drop the feather and hammer separately, then the hammer will " fall" just the tiniest bit faster than the feather.  However, if switch to the frame for the barycenter of the Moon and object being dropped, then what is happening is that while the object dropped "falls" towards the barycenter, the Moon also "falls" toward the barycenter.  The object's acceleration is determined by the mass of the Moon, and the Moon's acceleration by the mass of the object.  For example a 1 kg object will accelerate at ~1.62 m/sec² towards the Moon, while the Moon accelerates at ~2.21 m/sec²  towards the 1kg object. 

In this view, both hammer and feather fall at the same rate, while the Moon would fall faster towards the hammer than the feather.   This means that the "closing acceleration" will be different for the hammer and feather when they are dropped singly. However, if you drop them together, side by side, then it will be the same for both.  With three object, the barycenter is determined by all three, and the acceleration of the Moon is determined by the combined mass of both feather and hammer.

When ever we teach a subject, it is generally better to build up from simple concepts and then add the complicating factors later.  When we first start learning to subtract, we are taught that you can't subtract a larger number from a smaller one, then later we are introduced to the concept of negative numbers. Later, we are taught that you can't take the square root of a negative number, and then along comes complex numbers.  

If you tried to teach all at once, it would  just be information overload.   Before you start dealing with GR and space-time curvature, you should already be thoroughly familiar with how this situation is dealt with under Newtonian rules.

[StringJunky]

3 hours ago, jajrussel said:

Well, yes. I'm sure that I'm not the only one who would have been surprised. I have accepted it but still, would like to know why the difference in acceleration is so negligible? And if there is a difference isn't it better to be told that there is a difference no matter how negligible than to be taught that there is no difference that the hammer and the feather fall at exactly the same rate. That seems like sensationalism rather than science. It makes a difference because one truth leads to another. When you simply say that both accelerate at exactly the same rate then what is the point of individual curvatures? 

Swansont is much faster than I, I need to stop and see what he has written.

Another way to look at it: The Moon, hammer and feather are moving in the same direction through space (co-moving) , all physically connected to the astronaut. When he lets go of two of them, the gravity of the Moon deflects the path of the two objects towards itself and the hammer and feather try to deflect the path of the moon towards themselves, with negligible effect.

Edited May 10, 2019 by StringJunky

[J.C.MacSwell]

They would actually fall at the same rate, if dropped together, and of course that was the experiment performed on the moon.