A closed laboratory free falls and approaches the event horizon of a Black Hole. Are measurements of physical constants affected?

Question

A closed laboratory is in perfect free fall and hence is an inertial frame. This lab is falling toward the event horizon of a Black Hole. The lab is sufficiently small and the event horizon is sufficiently large such that tidal effects can be ignored. The physicist is initially unaware of the lab's proximity to the event horizon.

As the laboratory approaches the event horizon, the doomed physicist conducts experiments to determine the speed of light and other physical constants based on time (that have units that contain some function of the second), such as G.

1. Are the physicist's measurements of c, G, etc. in this inertial frame affected by Gravitational Time Dilation?

2. If not, is there any experiment the physicist could perform that would reveal that the lab is in free fall in an extremely powerful and increasing gravitational field?

-----Responses-----

Safesphere: My question was "Would measurements be altered (ignoring tidal effects) as the lab approaches the event horizon?" The correct (your) answer was: Yes, measurements would be effected by tidal effects that are unavoidable in the radial direction no matter how small a lab of finite size and how straight the local edge of the event horizon. My question turned out to be ill-posed, thank you for the correction.

"then you would measure the speed of light as near zero at the ceiling while normal at the floor."

Excellent point, my understanding that is what a physicist/measuring equipment on the floor would find with the ceiling at the event horizon.

To expand on this point, let's clarify and look at the instantaneous moment that a lab of height H(in that moment) was approaching the event horizon of a non-rotating black hole of mass M and Schwarzchild radius R such that the floor just touches the event horizon and the rest of the lab remains outside the horizon. G and c have seconds$^2$ and seconds respectively in the denominator of their units so should go in the same direction.

The experiment performed is 3 sets of two "small" spheres of mass m held firm with their centers 1 meter away from each other at the floor, center, and ceiling. These spheres are released at the moment the floor touches the event horizon. (Acknowledging Safesphere's comment that this instant only lasts ~0 seconds) our physicist's measuring equipment at the center of the room is somehow able to observe the infinitesimally small distance traveled in this instant and calculate the corresponding force from each of the 3 sets.

Given time dilation, I think the calculated forces would be:

Floor --> 0 (though light from this experiment would never reach the observer)

Center --> $Gm^2/1^2 = Gm^2$

Ceiling --> $\dfrac{1 - \dfrac{2GM}{(R + H)c^2}}{1 - \dfrac{2GM}{(R + H/2)c^2}}Gm^2$. So G would be observed to be a large value at the ceiling.

So values of c in a different experiment observed from the center of the room would be calculated to be floor = 0 (hence could not be observed), center = c, and

Ceiling --> $\sqrt{\dfrac{1 - \dfrac{2GM}{(R + H)c^2}}{1 - \dfrac{2GM}{(R + H/2)c^2}}}c$. So c would be observed to be large value.

From Dale's answer, I gather that if the lab were small enough and far enough from the event horizon such that radial tidal effects were not measurable, time dilation of observed experiments would be exactly canceled out by the slowness of the physicist's clock such that any physicists inside the closed lab would not be able to determine their predicament.

Answer 1

Locally there is no measurement of c or G or anything else that will detect the horizon. Locally the horizon is just an ordinary null surface with nothing to distinguish it from any other null surface.

Answer 2

The fundamental constants are independant of the time dilation effect.

In principle , for a point object, crossing the horizon for the infalling object is not a significative event. It is done without problem, just as crossing any part of the space-time. Nontheless, the lab is not a point and so is affected by the gravity of the black hole in such way that by the phenomenon of spaghettification, it is possible to know that we are infalling toward a black hole, at some point. Just like it is possible to deduce the presence of the moon due to the changing sea level, it is possible to know we are falling in a black hole by the deformation of the lab.