I am unfortunately not familiar with the mathematics behind general relativity. However, on a heavy planet (say a sphere) gravity will bend space-time in a way that an object initially in rest, will experience over time, more and more of its time-dimension as a space-dimension towards the planet, hence causing it to accelerate towards the planet. (Video: https://www.youtube.com/watch?v=jlTVIMOix3I )
If I generalize this line of thought conceptually, this would mean that on the surface of the planet, there is more space (and less time) than an outside observer assuming a flat space-time would expect.
Is this correct? Is the surface of a heavy spherical planet bigger than $4 \pi r^2$? This does seem to match with the rubber sheet visualisations of curved space-time, which show a negative curved space around heavy objects. It also seems to match with the idea that time slows down falling in a black hole, while space tends to go to infinity.
But is this line of thought correct?
Because I'm also inclined to think that the all time-dimension are becoming space in the direction towards the planet. Therefore, there is more space around heavy objects, but not in a direction perpendicular to the direction of gravity. Therefore, the surface of a spherical planet would still be exactly $4 \pi r^2$, since all normal vectors on this surface are parallel to the direction of gravity.
So, if I would pull a rope through the planet and measure its length $2r$, and measure the surface of the planet $A$ walking around on the planet, $A>4\pi r^2$. Yes?