Relativistic Mass Relative to a Stationary Observer

Question

I know what you’re thinking, “Not another question on Relativistic Mass.” I’ve spent the better part of a day going down the general and special relativity rabbit holes, and I can not find where this concept has been asked, so here goes.

If a very small, massive object was to be traveling close enough to light speed that it’s relativistic mass increased by several orders of magnitude, would it create gravitational effects relative to a stationary observer?

To be more specific, would an object with the resting mass of a paper clip be able to build up enough kinetic energy that it could bend spacetime, and pull something as massive as the Earth? The Earth being the stationary observer.

As a bonus question, would the near light speed object’s increase in relativistic mass cause gravitational lensing, observable from the stationary object?

A couple things I am aware of: I know the paperclip is only effected by its rest mass; I am purely interested in the frame of the stationary observer. The scientific community now uses “energy density” in place of “relativistic mass”, but I did not know how else to phrase the question. Similar questions have been asked, but every question I could find was asking about objects moving at the same relative speed, rather than one stationary and one moving.

Lastly, I am aware that, given the difference in velocity, the gravitational interaction between the two objects would only last fractions of a millisecond.

I am still a newer student of relativity, so there may be concepts I am unaware of or have misunderstood. I am still not 100% grasping how tensors fit in to this problem. Feel free to educate me as I am here to learn.

Answer 1

The question seems to be whether a small object could have an arbitrarily large gravitational effect if it were moving fast enough. I would say the answer is yes. However, that effect will not be the same as a stationary body with the same mass as the relativistic mass of the small object.

In general relativity, mass and energy affect the gravitational field via the stress energy tensor $T^{\mu \nu}$. This appears in the Einstein field equations $G_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T^{\mu \nu}$, relating the tensor to the curvature of space-time (expressed in $g_{\mu \nu}$ and $G_{\mu \nu}$).

You don't need to be too concerned with the (very complicated) mathematical details of this for now. However, it is helpful to note that $T^{0 0}$ represents relativistic mass density. This term is responsible for the Newtonian gravity in the classical limit. It can be increased arbitrarily as $v \to c$ for the small body.

That isn't the only effect increasing $v$ has on $T^{\mu \nu}$, though. Other terms representing mass and momentum fluxes will also grow. So the gravitational forces exerted by the small object won't be the same as for a very large body. For example, effects analogous to magnetism (which are usually negligible) will appear as $v$ increases. You will not end up with a black hole just by increasing $v$ for the small object.