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The gravitational constant $G$ has units of $N m^2 kg^{-1} = m^3 kg^{-1} s^{-2}$. Since seconds appear as units, time dilation would seem to affect the value of $G$ for different observers. That is, the "second" of one observer may differ from that of another observer, and that would seem to affect $G$. Similarly, there might be a discrepancy in the meaning of "meter" between two observers due to length contraction. Does this mean that the value of $G$ can be different for different observers, or do the discrepancies somehow cancel out?

(Added to sharpen question Dec. 28)

Given a box with time dilation (like the traveling ship in the Twin Paradox), it seems you can show empirically & theoretically that G must have a different value in the box. For simplicity, set aside relativity, and just take the box to be an apparatus in the laboratory, with a knob to set time dilation to a value k between 0 and 1. When 1 sec passes outside the box, 1 SEC passes inside, with 1 sec = k SEC.

Now, put a toy car that moves at a constant speed of v m/sec into the box, set time dilation to k. Start the box and the car at the same time, and let the box run for 1 sec. When we open the box and measure with a ruler, the car has traveled a distance of kv meters. So the velocity of the car while in the box, measured entirely in the frame outside the box is kv m/sec. In other words, anything moving at a constant speed v outside the box moves at the slower speed kv inside the box. That's partly what it means that "time passes more slowly" in the box.

If we assume the box is large enough to hold the earth, and we do similar experiments to measure the acceleration g, we find it to be 9.8 m/SEC^2 in the box, which translates to 9.8 m/(1/k sec)^2 = k^2 9.8 m/sec^2.

According to this reasoning, G really is different inside the box, when measured using the ordinary (non-dilated) meter and second units of our world. Is this calculation wrong?

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If you use Newtonian dynamics to calculate the behaviour of objects moving at relativistic speeds then you're going to get the wrong answers. One approach is to try and fix up the calculation e.g. by using concepts like relativistic mass. However this doesn't work well and causes endless conceptual problems (after 100 years students are still being perplexed by the concept of relativistic mass).

The sane approach is simply to use relativistic dynamics instead. Then your calculations all work fine and you don't have tinker with the values of constants in the way you suggest.

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  • $\begingroup$ I sharpened the question $\endgroup$ – user90510 Dec 27 '16 at 21:43
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No, the starting point for relativity is the equivalence principle, an assumption along the lines of (because this is not a precise definition, and there are various ones) that the laws and physical constants of physics are seen to be the same in any frame of reference; (the exact opposite of your question, so you are in a highly relevant place).

Time Dilation is an effect observed in order for the equivalence principle to be upheld.

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