Does an object with kinetic energy have increased mass? If so, how is this reconciled in relativity?

Question

Let's imagine that you are standing in a field. You have a ball that weighs 10 grams. You throw this ball at 40 kph. If I understand things correctly, this ball now has kinetic energy relative to your still reference frame (and I suppose you to it?).

Since E=mc^2, and this ball has increased "energy", wouldn't the mass of this ball be more than the original 10 grams?

If the answer is yes, then I imagine you would measure this increased mass in your reference frame, but in its own reference frame it would be the original 10 grams.

The big implication with my follow up question about relative reconciliation is the difference in the gravitional field exhibited within the different reference frames. How would this difference be reconciled?

Answer 1

The short answer is "yes," and if you want the long answer, the answer is that the question you are looking at is a the core of general relativity. The difference between special relativity and general relativity is that general relativity deals with accelerating frames, while special relativity only covers the simpler cases where the frames do not accelerate with respect to the inertial frames. In your simple case, the "rest frame" for the ball would be accelerating (if you think about it in special relativity terms. In general relativity it's a shade different)

As for gravity, it is not a force in general relativity. It is the observed effect of the distortion of space and time. Veritasium has a reasonably accessible cover of that material. In general relativity, these apparent inconsistencies are resolved by asserting that the reference frame attached to the ball is in a non-accelerating(!!) reference frame, and contorting the shape of space and time to make it work.

A full handling of general relativity, of course, calls for calculus over tensors.

And, as an aside, since you mention the energy of the system, it is worth noting that the energy of the system is conserved within a frame, but it is not conserved across frame conversions. So it's okay that, from your perspective, the ball has some energy because it's moving, and from the ball's perspective you and all of the Earth have a gargantuan amount of energy. Those viewpoints are describing different frames of reference, so it's okay that the total energy is different in each frame.

Answer 2

The short version of this is "the source of the gravitational field in general relativity is not the mass, it is the stress energy tensor", which is a two-component matrix over the four spacetime coordinates whose values are, relative to some coordinate system:

$T_{tt} = $ Energy density

$T_{ti} = $ The ith component of 3-momentum density

$T_{ii} = $ The pressure felt by an infinitesimal square in the ith direction

$T_{ij} = $ The sheer on a an infinitesimal square oriented in the ith and jth direction

So, if you shift to a reference frame where a stationary point mass is now moving, you won't just change the $T_{tt}$ component, which refers to the energy density, you will change these other components. The stress-energy tensor knows the difference between a stationary mass and a moving one.

This is similar to the situation in Electromagnetism in special relativity, where you rewrite the Maxwell's equations that contain a source as:

$$\nabla_{a}F^{ab} = j^{b}\;\;\;\;\;\;\nabla_{a}\left(F^{ab} - F^{ba}\right) = 0$$

where j is the 4-dimensional charge density, which knows the difference between a stationary charge and a moving charge, and $F_{ab}$ is built from the components of the E and b fields.