-2

Scenario, Let's say we have complete vacuum and nothing in the universe.
If we place 2 bodies at infinite separation and have HUGE masses ($m1, m2 \sim 10^{1000}$ kg or something):

They will attract.

When at infinity
They will attract but $F \to 0$ because $R \to \infty$ but it will still be present and since
$F = ma$, they will accelerate, which must increase over time as they get closer, both due to previous acceleration and due to $R$ getting smaller and smaller

After some time, $R$ might be $\sim 1000$ km. but at that time since, $F = \frac{G m_1 m_2}{R^2}$ and both $m1$ and $m2$ are of orders $10^{1000}$ kg, lets say $F$ comes as $10^{1900}$ N.

$F = ma$, but $m = 10^{1000}$ so $a = 10^{900}$
and since this is the acceleration at $1000km$ and the bodies were accelerating before it too, then they must have that acceleration too

So would not the bodies have velocities more than $3*10^8$ m/s at some time?

Qmechanic
  • 201,751
0xdead
  • 1
  • Possible duplicates: https://physics.stackexchange.com/q/24319/2451 and links therein. – Qmechanic Jul 29 '23 at 16:13
  • 1
    Even point particles with arbitrarily small masses falling toward each other exceed the speed of light as $r\to 0$, according to Newtonian mechanics. Heavy objects don’t fall any faster through vacuum than light objects do. – Ghoster Jul 29 '23 at 22:27

3 Answers3

4

Not really. You see, the fact that the speed of light is an universal limit is one of the main cornerstones of Relativity. Hence, when investigating these sorts of phenomena, you must consider relativistic effects and perform the computations using special relativity (in the absence of gravity) or general relativity (in the presence of gravity). Your analysis does not take these considerations into account, hence leading to this apparent paradox.

Your calculations were done with Newtonian mechanics, which is a theory that does allow arbitrarily large velocities. It pretty much matches the predictions of relativity for very small speeds and gravitational fields (such as those of day-to-day experience), but it leads to quite wrong results when you push it to such large masses, gravitational fields, and velocities. That is the reason this calculation leads to a wrong result.

Níckolas Alves
  • 17,567
0

The quick answer to your question is no, because as things accelerate it takes a greater and greater amount of force to cause further acceleration. There is no amount of force that can lead to faster than light travel - basically an infinite amount of force. BUT, keep in mind that right at the event horizon of a black hole, things do reach the speed of light. This is why light can't escape a black hole and why time stops (for an outside observer) for anything entering a black hole. You can do lot of research on that.

foolishmuse
  • 4,551
-1

No, and General relativity is why not. And object smaller then it's Schwarzschild radius will become a black hole. Here each mass will have a Schwarzschild radius of 1.5E957 Ly, or 1.5E973 meters. The resulting acceleration will never become large here.

blademan9999
  • 2,870