# What happens when an object being accelerated by gravity begins to approach light speed? [closed]

Imagine you have a pair of 2-dimensional circular portals, with one placed perfectly above the other (practically the same as portals in the Portal games). A spherical object is held between the portals and then dropped, enabling it to fall endlessly. The room this occurs in is a perfect vacuum and the sphere has 0 sideward velocity relative to the portals.

Normally an object accelerates until terminal velocity, but magically there isn't a single gaseous particle in the room.

Does the acceleration of the object begin to slow as it reaches a notable fraction of the speed of light, as more and more energy is required to accelerate it further? Or does the object accelerate at the same rate (as gravity provides force, not energy) before reaching a hard cutoff at 99.999...% the speed of light?

I know relativity has a lot to do with observers, so I guess the observer would be someone looking into the room through a glass window. Perhaps they have a laser shining across the path of the falling object, and in that way measure the speed of the object (by measuring the amount of time the laser is blocked for divided by the sphere's diameter).

• Such portals are science fiction, and we normally close questions about them as non-mainstream. But a question asking about indefinite constant acceleration in special relativity is perfectly fine. Commented Apr 7, 2023 at 16:53

This is going to be a crappy (i.e., non-mathematical) partial answer, but I hope it will help a little:

Whenever you talk about things moving at relativistic speeds, it's important to be explicit about frames of reference. Observers who are moving differently from each other will disagree on things like, how fast an object is moving, how long an object is, how much time elapses between events. Sometimes they cannot even agree on the order in which different events happen.

Let's say, for sake of argument, the the "object" is a person wearing a space suit (because of the vacuum.) The falling test subject will feel no acceleration at all—they are freely falling. They seem to be falling through an endless "tunnel of hoops" where the hoops are the edges of the portal.

As the subject falls, they count how many hoops they fall through per second. That number will increase without bound. At first, the hoops-per-second will increase because the hoops are flying by at ever increasing speed. But, that speed will asymptotically approach (and never exceed) $$C$$. As the speed gets close to $$C$$, the hoops-per-second will continue to increase though because the distance between the hoops will seem to get closer, and closer, asymptotically approaching zero.

Meanwhile, there's an experimenter in the "lab" coordinate system, who's looking in from the side. From the experimenter's point of view, the speed of the falling test subject will never exceed $$C$$, and the number of times-per-second that the subject falls through the portal will never exceed $$C/l$$ where $$l$$ is the vertical distance between the entrance and the exit of the portal.

The "lab" observer sees the number of "hoops-per-second" reach a limit. But the falling test subject sees it increase forever. How is that possible? The discrepancy is resolved if the "lab" observer is able to see the falling test subject's wrist watch. As the speed of falling approaches $$C$$, the "lab" observer will see the wrist watch slowing down. The falling test subject, sees nothing unusual about their own watch, but the experimenter sees the time between ticks of the second hand grow longer and longer, again without bound. If the lab observer counts number of hoops per tick of the subject's watch, they'll get the same number that the falling test subject experiences.

I'm sure that there's more that could be said about this (including mathematical equations), but I'm not the person who can say it with authority.

Your object will never reach the speed of light, but instead it will approach it asymptotically since every time it passes through the portal, it will need more and energy to gain (less and less) speed. It would require infinite amount of time (energy) to go to v=c, so it will never reach it. No hard cutoffs. About the acceleration rate, the thing is more subtle. You don't have only a spatial velocity but also a temporal one, the rate of ticking of your own clock. It is this velocity which stars to absorb your acceleration, making one tick of the clock more and more distant from the next one. This is indeed why you take infinite time to get to spatial c, is due to the absorption of acceleration into your temporal velocity.

Note: here acceleration decreases your temporal velocity, so that it tends to zero as your spatial one approaches c, so if your temporal velocity goes to zero, your clock basically stops and you will never reach c

• Re, "...your clock basically stops..." If you are the falling object, and you look at your own wrist watch, you will not see it stop. You will not even see it slow down. The observer looking in through the window will see your watch slow down, but that's because you and the external observer will not agree on what time it is or, on certain other measurements. You cannot give meaningful answers to questions about relativistic motion without a clear understanding of the frame of reference in which each different measurement is taken. Commented Apr 7, 2023 at 17:43
• Yes i know that, he specifically asked about the window observer. If he asked about the object itself i would have used a Rindler coordinate frame, but it wasn't required by his question, so it seemed overkill to me Commented Apr 7, 2023 at 17:46
• Sure, but when you're talking to a newbie who does not share your understanding of the subject, then their thoughts may not be based on the same unspoken assumptions that underlie your argument. Commented Apr 7, 2023 at 18:01
• That's true, i was assuming some a priori basic knowledge of special relativity hahahaha Commented Apr 7, 2023 at 18:09