I Any direct evidence of gravitational mass increase?

[Tiran]

None. Again, that would imply an absolute meaning to "speed". But if you apply a force to a fast moving object, $F=ma$ does not apply. The acceleration will not typically be parallel to the force, for example.

What do you mean it doesn't apply? How could a force applied to a fast moving object not effect the vector of that object?

 

[Nugatory]

What do you mean it doesn't apply? How could a force applied to a fast moving object not effect the vector of that object?

It does have an effect, but the equation $F=ma$ does not properly describe it: the acceleration is not necessarily in the same direction as the force, and the constant of proportionality between the force and acceleration is different for different directions and speeds.

 

[Tiran]

It does have an effect, but the equation $F=ma$ does not properly describe it: the acceleration is not necessarily in the same direction as the force, and the constant of proportionality between the force and acceleration is different for different directions and speeds.

Is this a claim that has anything to do with relativistic speeds, or are you just pointing out how vectors work?

 

[Ibix]

What do you mean it doesn't apply? How could a force applied to a fast moving object not effect the vector of that object?

It affects it, but not according to the Newtonian $F=ma$. The resultant acceleration is not, in general, even parallel to the force even with constant mass.

 

[Nugatory]

Is this a claim that has anything to do with relativistic speeds, or are you just pointing out how vectors work?

It is a claim about relativistic speeds and it is one of the bigger differences between relativistic and classical kinematics. In classical mechanics the direction of acceleration is always in the same direction as the force (if it weren't, we wouldn't be able to write $F=ma$, which only makes sense for scalars and vectors pointing in the same direction). In relativistic mechanics it is not.

As an aside, if we use the modern four-vector formulation (which was not known when relativity was first discovered, or no one would have bothered with the idea of mass increasing with speed) we can write the analogous vector equation $\vec{F}=m\vec{a}$ and it does work properly. But it's very different beast:
- The four-vectors are vectors in four-dimensional spacetime instead of three-dimensional space
- The acceleration four-vector $\vec{a}$ is defined as the time derivative of the velocity four-vector $\vec{v}$ as you'd expect, but $\vec{v}$ has a constant magnitude in all coordinate systems, and only its direction changes.
- The time derivative is with respect to a clock moving at the same speed and in the same direction as the object, at the moment that the force is applied. (That would be "proper time along the object's wordline" in the jargon).
- The $m$ that appears in $\vec{F}=m\vec{a}$ is the mass of the object as observed by an observer at rest relative to it, no adjustments for speed or time dilation needed to make everything come out right.

 

[PeroK]

Is this a claim that has anything to do with relativistic speeds, or are you just pointing out how vectors work?

It's the difference between classical three-vectors with three spatial components and relativistic four-vectors that also have a time component.

The equations of motion in relativity involve four-vectors and, indeed, we have:

$\textbf{f}$ $= m \textbf{a}$

For four-vectors.

 

[hkyriazi]

Something in your two posts gives me the impression that you are asking about gravitation in the context of Special Relativity, which does not deal with gravitation. If you are really asking about General Relativity, my apologies.

I tried to avoid GR by specifying that the rocket ship was non-accelerating. I'm trying to gain a fuller understanding of SR's relativistic mass increase.

 

[hkyriazi]

Well, in fact, there is a decreased gravitational effect. We could analyse the motion of a falling body using just SR and ignore GR:

[...]

Again, quite the opposite of what an advocate of relativistic mass might expect.

Thanks. A lot to grasp here. I'll print this out and chew on it a while, along with PAllen's response.

 

[hkyriazi]

There is two packed gas, say A and B, of the same amount. A is heated so average speed^2 of gas moecule is much higher than that of B. Say,
energy of A amount 1.1 kg c^2 > energy of B amount 1.0 kg c^2.
I reasonably assume that
energy of A as source of gravity 1.1 kg c^2 > energy of B as source of gravity 1.0 kg c^2
Honestly I do not know the experimental evidence achieved.

Good idea! This experiment gets at my question, and in a seemingly much more testable way.

 

[m4r35n357]

I tried to avoid GR by specifying that the rocket ship was non-accelerating.

Then you have this in reverse: GR is necessary for gravitation, it is not necessary for acceleration.

I'm trying to gain a fuller understanding of SR's relativistic mass increase.

There is no fuller understanding to be had, it is a dead end.