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Is Newton's equation for gravity wrong?


Samuel Daigle

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I have a question about Newton's equation for gravity. What would happen if you threw a ball with an initial velocity very close to the speed of light towards a black hole. We would assume that you would be very far from the black hole. Would the ball accelerate pass the speed of light before it reaches the event horizon? I did some calculation and found that the ball will indeed surpass the speed of light far before reaching the event horizon. Does this mean that F=GMm/r^2 is wrong because it breaks special relativity? I did found a solution on my own but I'm not even sure if it's right. I put the calculations and my solution to the problem on a word document, so if anyone is curious to see what I did, you can look at the word document.

Is Newton's equation for gravity wrong.docx

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6 minutes ago, Samuel Daigle said:

I have a question about Newton's equation for gravity. What would happen if you threw a ball with an initial velocity very close to the speed of light towards a black hole. We would assume that you would be very far from the black hole. Would the ball accelerate pass the speed of light before it reaches the event horizon? I did some calculation and found that the ball will indeed surpass the speed of light far before reaching the event horizon. Does this mean that F=GMm/r^2 is wrong because it breaks special relativity? I did found a solution on my own but I'm not even sure if it's right. I put the calculations and my solution to the problem on a word document, so if anyone is curious to see what I did, you can look at the word document.

Is Newton's equation for gravity wrong.docx

Newton's equations are not wrong, but inaccurate as relativistic velocities are approached. The ball, having mass, will never exceed the speed of light, but from various frames of references, time and space will vary.

 

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I see it like this, although it is simplified and not technically totally accurate:

There are 3 realms: Micro (QM), Normal (Earth) and Macro (Universe).

They all have their own unique set of rules called: Quantum mechanics, Newtonian and Relativity.

Obviously they can't be strictly separated like that. They do mix sometimes. But in the grand scheme they rule their domains.

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If Newton's equation are not wrong, is there a way to find the velocity of the ball  at any given distance from the black hole (from the point of view of an observer infinitely far away from the black hole)? What would I do differently in my calculation to find the velocity of the ball?

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1 hour ago, Samuel Daigle said:

If Newton's equation are not wrong, is there a way to find the velocity of the ball  at any given distance from the black hole (from the point of view of an observer infinitely far away from the black hole)? What would I do differently in my calculation to find the velocity of the ball?

You need a mathematician for that, and I aint one!  :P 

but one should be along in time.

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Sure it can be calculated; but gravitational time dilation would need to be considered.
Say you, a faraway observer, throws a light source towards the BH, that emits a pulse of light every second, you will note that as it approaches the BH's EH, the pulses are separated by increasing time intervals, and the intervals approach infinity at the EH. To a faraway observer, the speed of approach seems to slow down, and actually stop at the EH.

You should realize that if the speed of light could be exceeded by anything crossing the EH, then it wouldn't necessarily be 'trapped', as it would be able to travel 'out' of its own light cone ( and the EH is defined as the radius where escape velocity is equivalent to c ).

That means that BHs would be 'bright' ( not black ) with radiation from superluminal objects.
This clearly does not happen.

Just wondering, though, if cosmic rays approach the Earth at 99.99% of c , and are then accelerated by Earth's gravity, do your calculations result in their going superluminal also ?

Edited by MigL
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18 minutes ago, MigL said:

You should, first off, realize that if the speed of light could be exceeded by anything crossing the EH, then it wouldn't necessarily be 'trapped', as it would be able to travel 'out' of its own light cone ( and the EH is defined as the radius where escape velocity is equivalent to c ).

That means that BHs would be 'bright' ( not black ) with radiation from superluminal objects.
This clearly does not happen.

Just wondering, though, if cosmic rays approach the Earth at 99.99% of c , and are then accelerated by Earth's gravity, do your calculations result in their going superluminal also ?

 If the cosmic rays are high energy subatomic particle, my calculations should be the same because those particles have mass.  But if it's a high energy electromagnetic wave, then it would just follow the geodesic of space time. My problem only occur when the object in the gravitational field has mass

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Typically, cosmic rays are protons.
And they have energies ( and speeds ) approximately a million times higher than the LHC can achieve.

( sorry I edited/added to my post while you were composing your reply )

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9 hours ago, Samuel Daigle said:

I have a question about Newton's equation for gravity. What would happen if you threw a ball with an initial velocity very close to the speed of light towards a black hole. We would assume that you would be very far from the black hole. Would the ball accelerate pass the speed of light before it reaches the event horizon? I did some calculation and found that the ball will indeed surpass the speed of light far before reaching the event horizon. Does this mean that F=GMm/r^2 is wrong because it breaks special relativity? I did found a solution on my own but I'm not even sure if it's right. I put the calculations and my solution to the problem on a word document, so if anyone is curious to see what I did, you can look at the word document.

Newton's law for gravity is an approximation that works in most cases.

In the example you give, the ball will cross the event horizon at the speed of light (and this is true whether you drop it or throw it).

I haven't looked at your calculations yet, but you probably need to use the relativistic equation for acceleration (because velocities do not add linearly, as we approximate at low speeds, because of time dilation and length contraction). I don't think you have to use the full GR equations. 

EDIT: just skimmed through your word doc, and I am fairly sure that assuming velocities can be added is the source of your error.

I notice you start talking about Planck lengths and times. I don't know why that would be relevant. You define c in terms of Planck length and time, but the Planck length and time are already defined in terms of c!

Also, the Planck length is not "the smallest distance possible" (this is a common misconception).

But you then introduce time dilation and length contraction (which you omitted in your first calculation). And that is (presumably - still only skimming through it, I'm afraid!) why you now get the right answer. Nothing to do with Planck lengths; everything to do with relativity!

Edited by Strange
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48 minutes ago, Strange said:

Newton's law for gravity is an approximation that works in most cases.

In the example you give, the ball will cross the event horizon at the speed of light (and this is true whether you drop it or throw it).

I haven't looked at your calculations yet, but you probably need to use the relativistic equation for acceleration (because velocities do not add linearly, as we approximate at low speeds, because of time dilation and length contraction). I don't think you have to use the full GR equations. 

EDIT: just skimmed through your word doc, and I am fairly sure that assuming velocities can be added is the source of your error.

I notice you start talking about Planck lengths and times. I don't know why that would be relevant. You define c in terms of Planck length and time, but the Planck length and time are already defined in terms of c!

Also, the Planck length is not "the smallest distance possible" (this is a common misconception).

But you then introduce time dilation and length contraction (which you omitted in your first calculation). And that is (presumably - still only skimming through it, I'm afraid!) why you now get the right answer. Nothing to do with Planck lengths; everything to do with relativity!

Yeah, I would agree that the Planck length is not the most relevant thing for this subject, but it did helped me find other values later on. When I started to think about that problem, I started to think about the Planck length and wanted to see if it was related to special relativity, then I though it might have a link to gravity and so on...  I tried to solve this on my own, but I think I still have a little bit more to learn

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