Is special relativity a special case of general relativity, qualitatively?

Since Einstein name his theory Special Relativity and General Relativity, there should be some expected relationship between them, particularly "Special Relativity" being a special case of the more general "General Relativity". However, I can't seem to relate them in anyway; Special relativity concerns the fact that the speed of light is universal and General relativity is mainly about the curvature of spacetime.

BTW, there is already another similar question on Stackexchange: Reducing General Relativity to Special Relativity in limiting case , but I am in for a more qualitative explanation.

• What about the answers there does not satisfy you? SR is really just GR with the Minkowski metric as vacuum solution. Sep 7, 2014 at 16:18
• @ACuriousMind Can you explain in terms of special relativity and general relativity as I have defined above? I know it's not the only definition and it would be ultimately the same, but me and my friends are physics enthusiasts; we don't know much about the Minkowski metric and those similar advanced concepts. Sep 7, 2014 at 16:25
• Just a note, Einstein didn't name it "Special Relativity". Originally, he just called it relativitätsprinzip. After GR was published, everyone started referring to his first theory as special relativity
– Jim
Sep 7, 2014 at 16:30
• I don't think that SR is a special case of GR. To quote somebody who shall not be named "There is nothing special about SR as it holds for all forces and there is nothing general about GR as it is about only one force i.e., gravity." Sep 8, 2014 at 0:24

Yes, special relativity is a special case of general relativity. General relativity reduces to special relativity, in the special case of a flat spacetime. I.e., general relativity reduces to special relativity, in the special case of gravity being negligible, for example in space far from any objects, or when considering a small enough piece of space in freefall that gravity is unimportant to the problem.

Like special relativity, general relativity also assumes that the speed of light is universal. However, when spacetime is curved, the universality of the speed of light can only be applied locally, within regions of spacetime that are small enough that the effects of gravity aren't important within the region.

• Nit pick: I think your answer would read better if it was worded "General relativity is special relativity, in the special case of a flat spacetime" (or maybe even "General relativity reduces to special relativity in the special case ...") rather than "Special relativity is general relativity, in the special case ..." Sep 7, 2014 at 18:09
• @DavidHammen I agree, your suggested wording is clearer. I changed my answer to use that wording. Thanks for the suggestion! Sep 7, 2014 at 18:28
• It should also be pointed out that this is one way of stating the equivalence principle: that spacetime is locally Minkowski, i.e., SR is always a valid local approximation to GR.
– user4552
Sep 7, 2014 at 18:29

Here's a qualitative argument for why special relativity is a special case of general relativity.

When you first learn special relativity it tends to be introduced using the two postulates that Einstein started with. This is a perfectly good basis for SR, but it causes no end of intuitive problems for students. Just search this site for examples. It also conceals the link between SR and GR and IMHO makes it harder to learn GR.

My preferred going in point is that both SR and GR are geometrical theories, that is they describe the geometry of spacetime. SR is one particular geometry while GR allows for many different geometries.

The basis of GR is Einstein's equation:

$$G_{\mu\nu} = 8\pi T_{\mu\nu}$$

The quantity $T_{\mu\nu}$ decribes the matter/energy distribution, and the quantity $G_{\mu\nu}$ describes the geometry of spacetime. Basically we feed in a matter distribution and solve the equation to calculate a quantity called the metric.

For example if you take $T_{\mu\nu}$ to be a spherically symmetric object like a star then the metric you'll end up with is the Schwarzschild metric that describes black holes. If you take $T_{\mu\nu}$ to be a uniform distribution of matter then you'll get the FLRW metric that describes the universe as a whole. And if you take $T_{\mu\nu}$ to be zero, i.e. no mass present, then you'll end up with the Minkowski metric that describes special relativity (actually there are several metrics that correspond to no matter, but the Minkowski metric is the simplest and best known).

So this is why SR is a special case of GR, because it's one of the solutions to the equations of GR.

You say in a comment above that you're not familiar with the Minkowski metric. All you need to know is that it contains everything you need to know about SR. Time dilation, length contraction, failure of simultaneity, the constant speed of light and lots more can be derived from it. My own personal view is that starting with the Minkowski metric is the best way to understand SR. Just look at all the questions I've answered by invoking the Minkowski metric.

• +1, even though I disagree with you that starting with the Minkowski metric is the best way to understand SR. Some parts of SR are tractable to a 16 yr old algebra-based physics student. Almost all of SR is tractable to an 18 or 19 yr old who has taken introductory calculus-based physics. You don't need to know differential geometry to understand special relativity. Sep 7, 2014 at 18:06
• I don't think this is as good an answer as Red Act's. There is the issue mentioned parenthetically, which is that you can have vacuum spacetimes that are not Minkowski. In addition, one can write down metrics that don't look like the Minkowski metric, but are in fact the Minkowski metric written in different coordinates. Curvature is really the issue here. the Minkowski metric is the simplest and best known I don't think this works. I could write the Minkowski metric in coords that make it look complicated. The criterion is flatness, not simplicity.
– user4552
Sep 7, 2014 at 18:27
• Being a 15-year-old student taking calculus-based physics and having studied special relativity, I can confirm @DavidHammen 's comment! Metrics can serve as an interesting foray for a future quest to learn general relativity, but it's much easier to start elsewhere to learn special relativity. Most of the sources I've read (even college lecture notes I've come across) don't start with the Minkowski metric, but with a discussion of reference frames and Lorentz transformations. Sep 7, 2014 at 18:30
• @Rennie said "Minkowski metric is the best way to UNDERSTAND special relativity", he didn't say nothing about learn for first time special relativity. I agree to there is a better understanding of special relativity when you think about it in terms of Minkowski metric. Sep 7, 2014 at 20:06
• I think it's a bit more complicated than that, isn't it? Even after you take all matter out of the Einstein equations, you are still left with a highly non-trivial gauge problem, that does not exist in the way we calculate in SR. The price one has to pay for that is that problems with acceleration are not consistently treatable in the usual form of SR. Sep 7, 2014 at 21:31

Special relativity should be a special case of general relativity, qualitatively. General relativity should deal with any problems special relativity has with acceleration. However the following is a problem with special relativity related to acceleration that general relativity may not deal with. Consider, in special relativity, an observer in an inertial frame of reference and two entities located in the first of a couple of moving inertial frames of reference with respect to the observer. Suppose the two moving frames are moving in the same direction with different speeds. Suppose the two entities are separated in distance in the direction of the motion of the two frames of reference.

Suppose the two entities move similarly from the first moving frame to the second moving frame. With respect to the observer the distance between the two entities changes due to this change in the inertial frame of reference which contains them. The size of the change in distance between the entities is an effect of special relativity that depends on how far apart they were before they moved from one frame to the other and depends on the difference in speeds of the two frames of reference. If there is no limit on how far apart they were before the move to the second frame, then a problem occurs. There is a problem if this distance is great enough to have the change in the distance between them large enough to have them change distance separation faster than the speed of light with respect to the observer. The problem results from

1. the amount of change in distance related to their changing the frame that contains them and
2. the amount of time taken to make the change.

The general relativity over this situation in special relativity should deal with such problems related to acceleration if special relativity is a special case of general relativity.

Imagine you have a surface, and locally you approximate the surface by the tangent plane. The surface may be curved, but locally you may think it is flat (if you are a small animal walking on the surface).

Special relativity is the reduction of General Relativity to the "tangent plane". In General Relativity the spacetime is a curved 4D manifold, and the tangent space is the Minkowski space of Special Relativity.

So Special Relativity is general Relativity made local, exactly as Euclidean geometry is the local approximation to the global geometry of a (say) spherical surface.

As long as you are $$$$small enough'' (namely, you do not feel tidal forces), you always live in the tangent space and locally the laws of Physics are the ones of Special Relativity.