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If I make the two statements:

Firstly, are these statements valid?

Secondly, if they are, does this not mean that special relativity and Newton's law of gravitation are the same thing? This would not make sence as special relativity should not account for gravity.

Or does 'weak gravitational force' as in the second statement not mean the same thing as 'in the absence of gravity'?

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I would say, that your two first statements are not wrong, i.e. you can defend them in a discussion if you explain what you mean by extension and special case.

Your conclusion, as you suspected anyway, is wrong however. Special relativity and Newtons law of gravitation are not the same thing. In fact they deal with different aspects of a physical theory.

Special relativity is better compared to Newtons laws of motion. Both try to answer how objects move through time and space in the presence or absence of external influences like forces.

Newton's law of gravitation is a way to get one of these forces that might act on bodies.

General relativity now combines both aspects into a single theory where a mass is moving 'forceless' in a geometry 'created' by all kinds of energy.

So if one says that general relativity is an extension of special relativity it possibly means that the equations of motion of a testbody look like the equations of special relativity in the right limit. If you state that Newtons law of gravitation is a special case of general relativity, you are stretching the fact, that in the right limit the 'creation' part of general relativity looks like Newton's gravitational potential.

So to put it very short and colloquial, you are looking at different sites of an equation in these two statements.

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  • $\begingroup$ I'm not sure what you are saying. Of course Newtonian Mechanics and General Relativity are very different. There is a great philosophy gap between them. What I am trying to do is to help the OP understand which part of physics these theories are concerned about. $\endgroup$
    – anonymous67
    Jul 5, 2014 at 15:59
  • $\begingroup$ If you tell me what part of my answer you are uncomfortable with I might be able to clarify. $\endgroup$
    – taupunkt
    Jul 5, 2014 at 16:04
  • $\begingroup$ It's just because of my bad English reading ability. Anyway I think we share the same idea but different language and expression. $\endgroup$
    – anonymous67
    Jul 5, 2014 at 16:08
  • $\begingroup$ I understand what you guys mean. In my question I didnt mean to imply that they are litteraly the same equations, I know they have completely different origins and interpretations etc. I rather meant they have the same consequences, but I understand your points, thank you. $\endgroup$
    – Jonny
    Jul 5, 2014 at 18:47
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I would say Newtonian gravity is the limit of general relativity as gravitation becomes weak.

See it this way: the limit when the temperature-amplitude is small of metal extension can be written as:

$$ L = L_0 (1+\alpha \delta t) $$

If $\delta t$ is quite large this equation is no longer applied.

Special Relativity doesn't care about gravity, like the classical mechanics doesn't care about the extension of metal.

EDIT: about the suggestion: where is the Newton's Law of Motion.

First, I think a theory of gravity always has two part:

  • How an object moves in a particular field of gravity

  • How an object creates its gravity field

In Newtonian mechanics, the first part is solved by the Newton's Law of Motion and the second part is solved by the Newton's Law of Gravity. (Although Newton never knew about gravity field, let's just pretend he did)

In special relativity, there may still be the Newton's Law of Motion (modified), and it deals with any kind of motion provided gravity is absent.

In general relativity, there is the Newton's Law of Motion (modified twice), and now it deals with every kind of motion, even in gravity. The most tremendous discovery is the part "How an object creates its gravity field". The Newton's Law of gravity is replaced by the Einstein Field Equations.

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I think it's better to think of these things in terms of the shape of space and time. Special relativity told us how coordinates transformed when space-time is flat. But space-time isn't necessarily flat, and general relativity told us how curved space works. General relativity then concluded that it's mass that creates this curvature - and thus gravity.

Newtonian gravity is the limit of when this curvature is very small, and you can treat space as almost flat. Then you get back Newton's gravity.

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Consider three theories:

$$L_A=1$$ $$L_B=1+h$$ $$L_C=1+h+h^2$$

Theory A is a special case of Theory C when $h$ is small, Theory B is a special case of C when $h$ is small, doesn't this mean A and B are the same?

This is not a perfect analogy, but an example as to why this sort of reasoning breaks down.

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