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Weirdly, I didn't find an answer or even close to one with a search through the forum. I probably missed it.

Portals! Speed thing go in, speedy thing go out. Lots of sci-fi fun.

I've got a hair-better than layman's understanding of physics, but I know when I'm wildly out of my depth. Here's the scenario.

In low earth orbit (call it the point where there's 90% of earth's gravity) a Sufficiently Advanced SpaceshipTM (SAS) opens up a pair of portals, one on top of the other. The SAS is able to keep all debris, dust, air particles, and everything outside of the portals, which are connected to each other. A perfect vaccuum. The SAS then drops a perfectly round pebble down, and lets it accelerate. The start of the portal and the end of the portal are fairly close to each other, and there is no noticeable different in the gravitational pull between the start and the end.

Let us assume gravity of 8.8 m/s2 (it should TECHNICALLY be 8.82, but let's try to work with nicer numbers). Newtonian physics gets us the following velocities over time:

528 m/s after 1 minute 31,680 m/2 after 1 hour 760,320 m/s after 1 day 5,322,240 m/s after 1 week 277,516,800 after 1 year (365 days)

Wait a minute.

The speed of light in a vaccuum is 299,792,458 m/s, and I'm about to hit that with my portal loop. Bring on the relativity!

This is where I crash and burn super hard. Not only do we have an accelerating object, but it's an accelerating object inside a modest gravity well. I know it starts to get super wonky at this point, although a glance at the energy equations suggests that it continues to gain energy at a steady rate. However, the energy is split between kinetic energy, and... momentum? Mass? Both? I do know we'll never breach the speed of light barrier, but I'm dead curious how the general relativity impacts the scenario. I just don't know nearly enough to get my curiosity satisfied.

It's been a while, and I'd love some help knowing roughly how fast the pebble is going at various points, how much energy it has, and, of course, at what point the SAS should release the pebble for various degrees of Somebody is going to have a Very Bad Day.

To be clear: This is different from Is there a formula that gives the position of an object depending on the time, but which doesn't allow the object to surpass the speed of light? because we're dealing with a gravity well. I think that makes it different enough. Earth's gravity well isn't particularly strong (as I understand it), but over the potential timelines involved (100 years?) I suspect it makes a difference.

Thank you!

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  • $\begingroup$ This question is similar to: Is there a formula that gives the position of an object depending on the time, but which doesn't allow the object to surpass the speed of light?. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. $\endgroup$
    – John Rennie
    Commented Oct 8 at 14:12
  • $\begingroup$ Re, "...because we're dealing with a gravity well." I suspect that you'll need more than just your gut feeling that "that makes it different enough." Can you say more about why you think that the formulas in the answers on the other question are not valid when $a$ stands for acceleration due to gravity? $\endgroup$
    – Solomon Slow
    Commented Oct 8 at 14:40
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    $\begingroup$ Extra credit: How much energy must the portal machine supply for each jump if it obeys conservation of energy? $\endgroup$
    – Solomon Slow
    Commented Oct 8 at 14:44
  • $\begingroup$ @SolomonSlow Okay, sorry for the 'gut feeling' part. That should properly translate to 'my limited education included these elements, and I am piecing together these elements into a misunderstood picture of all the involved elements'. My limited understanding of general relativity is it handles accelerations when near c, which has a secondary element of warping space-time, which includes a second effect. I could be completely off. $\endgroup$
    – Selkie
    Commented Oct 8 at 15:09
  • $\begingroup$ @SolomonSlow Extra credit! Alright, in the scenario the SAS has made a portal via a wormhole, creating a small distortion. The initial cost of the portal is quite high, but once it's created and stable, there is either no upkeep cost, or a persistent upkeep cost independent of the amount of mass passing through. If conservation of energy must be properly applied, and the system is otherwise closed, the problem becomes to my layman's eye to be trivial. E=MGH. Since the energy accumulated by the rock is the only energy entering the system, it's the same energy to maintain the portal. $\endgroup$
    – Selkie
    Commented Oct 8 at 15:09

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