Gravitation...

[cdexter_89]

it has been "proven" that in a vacuum, two objects of the same mass will fall at the same time...

but, it's also discussed that every matter has a gravitational force that affects other matters... that is, also related to it's mass...

i forgot the topic but that what it talks about...

so, does that mean that two falling objects in a vacuum does not necessarily fall at the same time?!?

[y_feldblum]

In Newtonian theory, the force exerted is proportional to product of the masses of the two bodies under consideration. But the acceleration of a body is inversely proportional to its mass. Thus, the two proportionalities cancel and the acceleration of a body under the influence of the gravity of another body will not depend on the first body's mass, though it will depend on the gravitating body's mass.

In general relativity theory, gravity is not a force at all. It might seem odd, but in relativity, bodies under the influence of gravitation simply move in a straight line, keeping exactly the same inertia, without accelerating. Thus, the speed at which it accelerates, being zero, does not depend on its mass.

[FaSheezy]

All gravity acts on all objects the same, regardless of their mass. An object would have to be incredibly massive to exert a gravitational force on another object. The equation is F = Gm1m2/r^2 G being the universal gravitational constant, m1 and m2 beings the masses of the two objects and r^2 being the distance between the center of mass of the two objects.

[The Guru Kid]

if M>m, the gravitational force on object M will be greater than on object m.

But since F=ma, a=f/m and the accelerations will stay the same.

in other words, the ratio of f/m will stay the same even though the actual numbers might change.

[JMeganSnow]

it has been "proven" that in a vacuum, two objects of the same mass will fall at the same time...

It's important to note that they will fall at the same rate, not time. They'll only fall at the same time if you drop them at the same time.

I confess to being more than a bit curious how this works under general relativity, y_feldblum. I thought "inertia" was the term for the tendency of an object at rest to stay at rest and an object in motion to stay in motion. So, wouldn't that mean that any object (barring some kind of reduction in mass) will always have the same inertia?

My knowledge of physics remains very shaky.

[jrs]

I thought "inertia" was the term for the tendency of an object at rest to stay at rest and an object in motion to stay in motion. So, wouldn't that mean that any object (barring some kind of reduction in mass) will always have the same inertia?

"inertia" is a synonym for "mass", and it should depend only on the identity of the object, not on external circumstances.

It is interesting that in Newtonian physics, mass is manifested in three different ways:

(1) active gravitational charge: how strong a gravitational field the object produces around itself;

(2) passive gravitational charge: how much the object is affected by the gravitational fields of other objects; and

(3) inertia: the resistance of the object to being accelerated by forces.

Newton's third law of motion (conservation of linear momentum) implies the equality of active gravitational charge and passive gravitational charge.

Einstein's equivalence principle implies the equality of passive gravitational charge and inertia.

[Alfred Centauri]

it has been "proven" that in a vacuum, two objects of the same mass will fall at the same time...

but, it's also discussed that every matter has a gravitational force that affects other matters... that is, also related to it's mass...

i forgot the topic but that what it talks about...

so, does that mean that two falling objects in a vacuum does not necessarily fall at the same time?!?

Actually, it has been observed (not proven) that, in a vacuum, two 'nearby' objects fall at the same _rate_ in a gravitational field _regardless_ of the mass of the individual objects. Within the context of Newtonian Gravity, this observation is 'explained' by setting the 'inertial mass' in F=ma equal to the 'gravitational mass' in F=GMm/r^2. It is the use of the symbol 'm' for both inertial mass and gravitational mass as well as the unqualified use of the word 'mass' that leads to questions like yours.

[Alfred Centauri]

In general relativity theory, gravity is not a force at all. It might seem odd, but in relativity, bodies under the influence of gravitation simply move in a straight line, keeping exactly the same inertia, without accelerating. Thus, the speed at which it accelerates, being zero, does not depend on its mass.

Allow me to point out that the term 'straight line' can be misleading when discussing GR. In Euclidian geometry, the shortest path between two points is along the 'straight line' connecting the two points. Within a curved geometry such as the surface of sphere, a straight line is somewhat ill-defined. In this case, we look to the property of 'shortest path' to define our generalized 'straight line' or geodesic. In the case of the surface of a sphere (such as the Earth), the geodesics are arcs of great circles.

GR is based on Special Relativity (SR) which joins space and time into a 4-dimensional structure called 'spacetime'. According to GR, the geometry of this spacetime is 'curved' and free (unaccelerated) objects trace out paths (world lines) that are geodesics, the generalized 'straight lines', of this curved geometry. These 4D paths in spacetime appear as accelerated motion when projected onto the 3D space we perceive.

[Alfred Centauri]

I confess to being more than a bit curious how this works under general relativity, y_feldblum.  I thought "inertia" was the term for the tendency of an object at rest to stay at rest and an object in motion to stay in motion.  So, wouldn't that mean that any object (barring some kind of reduction in mass) will always have the same inertia?

More precisely, inertia is defined as that property of an object that resists _changes_ in motion. In Newtonian mechanics, a _change_ in the motion of an object is due to a force acting on that object. The _magnitude_ of that change in motion is proportional to the applied force and is inversly proportional to the inertial mass of the object. This inertial mass is a fundamental property of the object and is constant.

In relativistic mechanics, the inertia of an object is _not_ constant but is instead a function of velocity. At zero velocity, the intertia is given by the 'rest mass' of the object. As the velocity of the object increases, the inertia increases without bound as the velocity approaches the speed of light.

[jrs]

In relativistic mechanics, the inertia of an object is _not_ constant but is instead a function of velocity. At zero velocity, the inertia is given by the 'rest mass' of the object. As the velocity of the object increases, the inertia increases without bound as the velocity approaches the speed of light.

You are trying to preserve the Newtonian equation

a = F/m

where

a is the acceleration vector;

F is the force vector; and

m is the "mass".

This is a mistake. To do so you would have to make mass not only a function of speed, but it would also have to depend on the angle between the force vector and the velocity vector. This is unreasonable.

The correct approach is to first calculate the linear momentum:

dp = F*dt

where

d is the differential operator;

p is the linear momentum vector;

F is the force vector; and

t is time.

In words, the linear momentum vector is the time integral of the force vector.

Then you calculate the velocity via:

v = p/sqrt(m*m+p*p/(c*c))

where

v is the velocity vector;

sqrt is the square-root function;

m is the mass; and

c is the speed of light in a vacuum.

With these equations, mass is an invariant.

[Alfred Centauri]

You are trying to preserve the Newtonian equation

a = F/m

I disagree as I'm well aware that the product rule must be used. Starting with:

p = mv

where m = (gamma) m_0 and m_0 is the mass measured in the rest frame.

We have...

F = dp/dt = m dv/dt + v dm/dt

= m a + v dm/dv dv/dt

= (gamma) m_0 a + v m_0 d(gamma)/dv dv/dt

Thus, the force vector and acceleration vector are not parallel as in Newtonian mechanics. In the special case where the force and velocity vectors are parallel, the above simplifies too:

F = (gamma)^3 m_0 a

(I've used scalar quantities since we're in 1-D)

Now, taking as given that inertia is the resistance to change in motion for an applied force (F / a), we have that the inertia is:

F / a = (gamma)^3 m_0

Which clearly increases without bound as the velocity tends to c. Of course, m_0 is a constant.

Regards,

Alfred

[jrs]

The equation is F = Gm1m2/r^2, G being the universal gravitational constant, m1 and m2 beings the masses of the two objects and r being the distance between the center of mass of the two objects.

This Newtonian equation is only an approximation. I think that the approximation could be improved for fast moving objects in the solar system as follows:

F = (G*M/(r*r))*{sqrt[m*m+p*p/(c*c)]+p*v/(c*c)}

where

F is the magnitude of the gravitational force exerted on the object by the Sun;

G is the universal gravitational constant;

M is the mass of the Sun;

r is the distance between the Sun and the object;

sqrt is the square-root function;

m is the mass of the object;

p is the linear momentum of the object;

v is the velocity of the object; and

c is the speed of light in a vacuum.

Notice that this does not merely replace the mass by the "moving mass", but also has another term which has the effect of a further doubling of the force on objects moving near the speed of light.

[jrs]

F = (G*M/(r*r))*{sqrt[m*m+p*p/(c*c)]+p*v/(c*c)}

I should have qualified this by saying that it only holds when the object is moving perpendicular to the direction to the Sun. If it is moving towards the Sun or away from the Sun, then the sign of the second term is reversed. So you get:

F = (G*M/(r*r))*{sqrt[m*m+p*p/(c*c)]-p*v/(c*c)}

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I recently found a really good book on Gravitation --

"General Theory of Relativity" by P.A.M.Dirac, Princeton Landmarks in Physics, written in 1975, copyright 1996 by Princeton University Press.

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Let the normalized gravitational potential of the Sun be:

U = G*M/c*c*r

If we assume that U*U is negligible, then the primary effects of the Sun's gravity are:

(1) time intervals become longer by a factor of 1+U, i.e. clocks near the Sun slow down relative to clocks far away;

(2) objects near the Sun become shorter (and thinner and narrower) by a factor of 1-U.

Combining these together we find that the speed of light decreases

to c-2*U*c.

[markb]

I think gravity is the energy of the graviton. Newton's calculations stop when we start calculating things were gravity comes instantaneously. They are accurate as is f*d=w. It is accurate only to like 13 decimal places in slow speeds and not really accurate when going at a significan fraction of light. Einsteins theory of General relativity explained gravity. But i still wonder... What is the graviton if there is one?