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Classical relativity, also known as Galilean relativity, is what you learn in high school; it has simple transformations between different reference frames, where every reference frame is on the same "time" but have different positions and speeds for different objects. And if you want to transform from one reference frame to another, you basically just add all the speeds and displacements, linearly. So if you're changing from frame A to frame B, then the speeds are all V_B = V_A + V_difference, and the positions are all x_B = x_A + x_displacement. It's nice and simple, mathematically.

Galilean relativity is really accurate in most circumstances, but not in all — specifically it becomes less and less accurate the faster you are moving. The experimental discovery of this led to the development of special relativity. Basically, physicists (including Einstein, Lorentz, and Poincaré, among others) realized that the transformation leaving time unchanged was wrong, and that the transformation laws weren't that simple. Lengths had to shorten (length contraction) and durations had to lengthen (time dilation) in order to preserve the speed of light across all reference frames. This makes special relativity a bit more complicated than Galilean relativity (it's no longer just adding speeds and displacements, you have to do multiplication and division of various extra factors now), but fortunately it's not too complicated — you can understand most everything in special relativity with just high school algebra, it's just not neat and linear anymore like Galilean relativity is. This new flavor of relativity was just called "relativity" by Einstein, and today is known as "special relativity" because it is a special case of general relativity.

Here's a really good short video illustrating the difference in reference-frame transformations between special relativity and Galilean relativity.

Not long after special relativity was developed, Einstein was able to further generalize special relativity, which lead to the development of "general relativity." General relativity is basically special relativity, but in curved spacetime (where special relativity only allows for flat spacetime). Because it can handle curved spacetime (and because it expresses a specific relationship between the curvature of spacetime and matter/mass), it can also model gravitational dynamics. Gravity then is not some extra force that needs to act on flat spacetime, it emerges naturally as a fictitious force (or "inertial force") due to the curvature of spacetime. General relativity is substantially more complicated, mathematically, than special relativity is — it involves non-linear partial differential equations, a special kind of abstract geometry called pseudo-Riemannian geometry, etc. But for modelling the most complicated gravitational situations (like black holes, neutron stars, and the cosmos as a whole) it is necessary. Special relativity and Newtonian gravity just don't give the correct answers together. General relativity became famous when it first predicted (or actually, retrodicted) the correct value for the precession of Mercury's orbit and for the deflection of light around the Sun from sources behind it.

Hope that helps!

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Special relativity takes place in flat space time (with Minkowski geometry), and doesn’t describe gravitational phenomena. (At least, not non-uniform gravitation; uniform gravitation is the same as an accelerating frame in flat space, via Einstein’s thought experiment about the man in the elevator in space)

General relativity takes place in possibly curved space time, and it describes how matter curves space time (matter causes gravity), and how the curvature of space time influences the motion of objects in it (gravity influences matter)

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Special or restricted relativity takes its name precisely because of a fundamental limitation: it applies only to agents that observe nature from frames of reference that move at constant speed, and in a straight line, relative to each other, or that they are at rest. These are called inertial frames of reference because in these the law of inertia, or Newton's first law, is valid: an object will remain at rest or moving in a straight line, at a constant speed, unless a force intervenes to change this condition. This, together with the experimental fact of the constancy of the speed of light, discovered by Michelson and Morley in 1887, leads to the two fundamental principles of special relativity:

The speed of light is constant in any inertial frame of reference.

The laws of nature must be the same in any inertial frame of reference, including the above fact, which is assumed to be a fundamental truth.

In other words, all inertial frames are equivalent when describing natural phenomena.

Conclusion:

Relative to an inertial frame of reference, all particles move in the same way.

Einstein used these two principles to build the entire edifice of special relativity, with its time dilation, increase of mass with speed, its famous formula of the equivalence between mass and energy, etc.

To better understand this, consider the following ideal example. We are in a vehicle on a straight and completely smooth road, to reduce any vibration. We move at a constant speed. If the vehicle is totally closed, without windows to the outside, we will not know if we are at rest or moving in a straight line. In other words, there is no distinction between a system at rest and one that is going at constant speed in special relativity. The movements of a system can only be detected in relation to those of others, hence the word relativity .. This fact was already known to physicists of the past, such as Galileo. In effect, without including the first postulate, it is called Galileo's principle of relativity and applies to very small speeds in relation to the speed of light. Haven't you noticed that when you are on a plane at 800 km/h, especially at night, it seems that you are stopped on the runway? We can move down the hall as if we were at home. Galileo's principle of relativity is the explanation.

Now, can Einstein's principle of relativity be extended to frames of reference that are accelerated relative to each other (non-inertial frames) and that do not travel at a constant speed, thus removing the constraint? When Einstein tried to incorporate Newtonian gravity into special relativity, problems arose that led him to answer this question, creating another revolution in science in its wake. Pondering the matter, he came up with what he described as the happiest thought of his life: if a person falls into a gravity field, he will not feel his own weight. Since he is falling freely, if he takes a coin out of his pocket and drops it, it will remain static with respect to him; that is, they fall at the same time. The observer interprets this as meaning that he is at rest, since he is not moving relative to the coin, and not falling into a gravity field: there is no field in its vicinity. This implies that without an object falling differently in a gravity field, the observer has no way of distinguishing whether it is at rest or falling into it. But it is well known that all objects fall the same way in a gravity field. This is the famous fact that Galileo describes in hisDiscorsi of 1638. And it is that in this field all objects fall with the same acceleration, regardless of their masses; in this case, observer and coin. Astronauts are a clear example of this. They are not floating in the space station, they are freely falling towards earth, along with the orbiting device and not feeling their own weight, they can move in all directions freely.

Going back to our observer, if he is at rest then he is in an inertial frame of reference. In other words, we are in an inertial reference frame in the presence of a gravity field. But we see that all the particles move in the same way.

Now, suppose, for example, a non-inertial frame of reference, far from any field, with an acceleration equal to that imparted by the earth's gravitational field to particles falling into it. Viewed from this frame, all particles will have the same acceleration (that of the frame). In other words, it seems that they move in a gravity field like that of the earth. There is no reason, as Einstein says, not to assume that the particles are located in a field of gravity, since they all move with the same acceleration and this is only possible if there is such a field. This applies to any acceleration.

Conclusion:

With respect to a non-inertial frame of reference, all the particles move in the same way, according to the property that they all move with the same acceleration. At the same time, from an inertial frame all particles move in the same way in the presence of a gravity field.

From here arises the fundamental principle of general relativity:

Equivalence principle :

A non-inertial frame of reference is equivalent to a gravity field.

Since non-inertial frames of reference describe rotational phenomena, for example, there are curvilinear trajectories involved. Obviously the trajectories are no longer straight lines, as in special relativity, but curves of minimum length called geodesics. And if a non-inertial frame of reference is equal to a gravity field, this naturally leads us to describe gravitational phenomena using spaces of different curvatures. In the end, as we have seen, special relativity makes its appearance in any gravity field in the vicinity of any observer immersed in it (frame at rest); thus general relativity contains it as a limiting case.

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