If gravity impacts length measurements, and length measurements impact gravity, how do we resolve net gravity? (paradox I can't resolve) Here's the concept.  We see a very dense 1.5 km radius asteroid, and my friend Charlie and I fly up in our spaceship to check it out.  I fly close on my bathroom scale, equipped with rocket thruster, and hover above the non-rotating asteroid.  I have 100 kg mass.
My scale reads 2.05864e15 Newtons, so gravitational acceleration g = 2.05864e13 m/s2.  I then ping the asteroid with a radio signal, and the round trip takes 2 microseconds, by which I calculate the distance to the asteroid as 300 meters, and to the astroid's center as 1800 meters.  I then use g = GM/r^2 to figure out that M = 1.00e30 kg.  I radio back to Charlie, who has been watching from a million miles away.
Charlie says, "You are wrong about the mass.  I've been watching via Zoom.  Due to the high gravity, your time is running slowly by a factor of 1.41.  The ping didn't take 2 microseconds, it took 2.82 microseconds.  I figure your value of R to use in GM/R^2 is 1923.4 meters.  That means the asteroid mass works out to 1.14e30 kg.
Who is correct?  What is the mass of the asteroid?  Things the two observers agree on:  the asteroid is 1.5 km in radius as measured from far away, the gravitational acceleration is 2.05864e13 m/s2 at a point where the local observer measures the distance to the surface as 1 light-microsecond and the more distant observer measures the distance as 1.42 light-microseconds.  What was the error of the incorrect observer?
 A: There's an unfortunate tendency in descriptions of relativity (special especially) to say that observers "disagree" about measured quantities, as though they'd actually get into an argument about it, along the lines of "$Δx=2\text{m}$!" "No, $Δx=3\text{m}!$"
If you make the reasonable assumption that participants in thought experiments understand the physical laws that they're applying, then this will never happen, because they'll understand that they're describing the same physical situation in different ways, and the only reason they've quoted different values for a quantity named $Δx$ is that they're using the name $Δx$ to refer to two different things. Hopefully, one of them will agree to change the name of their thing to $Δx'$, and then there will be no more "disagreement". Coordinates in relativity are like coordinates in Euclidean geometry, and there's nothing deeper in these "disagreements" than there would be in Euclidean geometry.
Given the extreme values involved here, you can't use a Newtonian approximation. Your proper acceleration is not $GM/r^2$ but $GM/(r^2\sqrt{1-r_s/r})$ (where $r_s=2GM/c^2$), and the ping time is not $2(r_2-r_1)/c$ but $2(r_2-r_1+r_s\ln(r_2-r_s)/(r_1-r_s))\sqrt{1-r_s/r_2}/c$. Solving these equations for $M$ and your radius is not entirely trivial but it's doable.
Charlie, knowing your proper acceleration, his own proper acceleration, and the ping time redshifted by a factor of $\sqrt{(1-r_s/r_3)/(1-r_s/r_2)}$, can solve for M and the two unknown radii. Using the ping time he measures as though it was your measured ping time will produce the wrong answer because it's just the wrong calculation.
I should mention one other issue here: the notion of "radius" is not entirely unambiguous. Normally in Schwarzschild coordinates you use a so-called "reduced circumference" which is $1/2π$ times the circumference of a circle centered on the massive body. If you want to use these formulas then you need to use a reduced-circumference radius for the asteroid ($r_1$). This is not "as measured from far away". In fact it makes no sense to talk about a radius measured from far away. You could measure the angular diameter of the asteroid from far away. Then you'd have to equate the measured angle to an angle calculated as a function of your position, the radius of the asteroid, and its mass (by calculating the geodesic path of a light beam that just grazes the asteroid). That would give you a system of one more equation in one more variable, which you could solve.
