# How is special relativity explained by general relativity?

To be more specific about this, I am under the below assumptions and then will explain my question further. Please let me know if any of the assumptions are incorrect.

(1) Special relativity describes the different in the experience of time between two observers in inertial reference frames.

(2) Given that we know the speed of light is constant in any inertial reference frames, Lorentz transformations tell us how the travel through space/experience of time is changed between each observers frame.

(3) General relativity describes how the force that we know as gravity is not actually gravity and more the bending of space-time around objects with mass. Where the straightest path will be a curved on around the object.

(4) This bending not only gives the perception of gravity towards an object but will equally bend the time as well so that the time the person moving near the mass will experience will be much less comparative to a very distant observer.

(5) Special relativity is a specific case of general relativity where the effects of gravity is negligible

Thus my question; how does general relativity explain time dilation between two non-inertial reference frames where gravity isn't negligible. I.e. the twins paradox in terms of general relativity. Is it the same and is it the acceleration "bending" spacetime in the same way that mass is?

• Concerning point 5 it would be better to say special relativity deals with flat spacetime. Dec 3, 2023 at 1:20
• No, acceleration does not bend space time, but there are many analogies between effects due to gravity vs acceleration. One example is the existence of "horizons". Dec 3, 2023 at 1:53
• Special relativity is to general relativity exactly as linear algebra is to calculus. It is a special case which locally well approximates any general case.. Dec 3, 2023 at 5:50
• To understand how time dilation works in GR you need first to understand the root cause of time dilation. Have a read through What is time dilation really?. Dec 3, 2023 at 8:44
• Take a smooth curved spacetime: special relativity is the restriction of GR on a small enough neighbourhood of an event (i.e. on the "tangent space"). To appreciate this, you may check the "tetrad formalism". Dec 3, 2023 at 12:36

Let me expand on the comment by WillO.

In order to be able to formulate newtonian mechanics we grant Pythagoras' relation:

$$r^2 = x^2 + y^2 + z^2 \tag{1}$$

Interestingly, back when Isaac Newton composed the Principia he went through stages of formulating sets of axioms, and at one stage he had six of them, among them an assertion that velocities and accelerations add and subtract according to the rules of Pythagoras' theorem.

I'm guessing back then that concept was cumbersome to communicate because there was no standard convention of vector representation. Mathematicians were drawing diagrams, with lengths of line representing various magnitudes.

My best guess: in the end Newton decided that his audience would totally grant that space is euclidean, so he didn't make that an explicit axiom of his theory.

With the benefit of hindsight we see that it does matter, because the transition from pre-relativistic physics to relativistic physics is a transition of metric: to the Minkowski metric.

The Minkowski metric subsumes the Euclidean metric.

As we know: the Minkowski metric relates spatial dimensions and the time dimension.

(It's not that Newtonian mechanics doesn't make an assertion that relates spatial coordinates and time coordinate; it does. Newton's first law: an object in inertial motion will in equal intervals of time cover equal intervals of distance.)

General statement about theory of motion:
In order to formulate a theory of motion at all one must formulate the appropriate metric.

Another example of a metric would be a metric for distances between two locations on Earth. For distances that are small relative to the size of the Earth the Euclidean metric is sufficient. For larger distances you have take into account that the journey is actually along the surface of a sphere.

Note especially: the metric for distances-along-the-surface-of-a-sphere does not explain the Euclidean metric. It's just that at distances that are small relative to the size of the Earth the difference between using either of the metrics becomes negligably small.

The transition from special relativity to general relativity was another change of metric, a most radical one.

The metric of special relativity is not subject to change. The concept of the Minkowski metric is that it is the same everywhere and always.

In terms of GR: that which is described by the metric is thought of as an entity that is subject to change. In terms of GR there is a reciprocal relation: spacetime acts upon matter, and is being acted upon by matter.
As John Archibald Wheeler formulated it:
Gravitational mass is telling spacetime how to curve
Curved spacetime is telling matter how to move

(There are various versions of that in circulation; the exact wording is not essential. It is the underlying idea that counts.)

(I use the expression 'GR metric' here, but there is of course no overall GR metric. For every case one must find its bespoke metric. Example: the Schwarzschild metric.)

The metric of GR does not explain the Minkowski metric.

It is the case that GR is a deeper theory than SR, but not in the sense that it explains all of SR.

Both in terms of GR and SR the Minkowski relation must be granted in order to formulate the theory at all.

At scales that are small relative to the extend of curvature of spacetime the difference between using the Minkowski metric or the GR metric becomes negligably small.