# What has general relativity got to do with special relativity?

What has general relativity got to do with special relativity? Arguably, special relativity is encompassed in the requirement that the dynamical laws of physics be Poincare invariant (as opposed to Galileo, or something else).

General relativity, on the other hand, can be characterised first off by the claim that free-fall motion is geodesic motion, and secondly that the connection determining the geodesics is curved in the presence of matter.

At least on these two readings, the theories are independent of each other in their basic assumptions. Indeed, gravity in Newton-Cartan theory arises from a curved connection while assuming a Newtonian absolute time.

Yes, in both SR and GR there is a Lorentzian metric. But I'm taking that to be a non-essential feature of GR, the essential one being that free-fall worldlines are geodesics.

Historically, it appears that some (Einstein I believe) thought of the two subjects as linked in their foundations, and somewhat interdependent. If this was believed at some point, what was the link supposed to be? Why are they both 'relativity' and not the theory of relativity (SR), and the theory of spacetime curvature (GR)?

Edit:

It seems the question has been misunderstood. Poincare invariance and geometrised gravity are independent (just recall the Newton-Cartan theory a.k.a. geometrised Newtonian gravity). My question is why historically geometrised gravity came up in the context of Lorentzian spacetime, not, for instance, a classical (Newtonian) spacetime.

• what about that in flat for GR space SR takes over? Jul 21, 2019 at 14:33
• Well GR is still centers around relative reference frames, right? Jul 21, 2019 at 14:38
• GR can be seen as a theory invariant for local Poincarè transformations. Actually when you do GR often you require only invariance under diffeomorphisms (local translations). But if you use the vielbein formalism then you can write a theory invariant under local Poincarè. Jul 21, 2019 at 14:42
• Possible duplicate of What is the connection between special and general relativity? Jul 21, 2019 at 15:46
• The question in your edit ("why historically did X appear before Y?") is not within our purview, as indicated in the guidelines already linked to ─ it belongs on History of Science and Mathematics. Jul 22, 2019 at 12:26

General relativity, on the other hand can be characterised first off by the claim that free-fall motion is geodesic motion, and secondly that the connection determining the geodesics is curved in the presence of matter.

At least on these two readings, the theories are independent of each other in their basic assumptions.

No $$-$$ you're missing a basic assumption on the side of GR, namely, that spacetime is locally Minkowski. Which is to say, that SR holds locally for sufficiently small regions of spacetime. Or, in other words, the fact that GR is built completely on top of SR.

Yes, in both SR and GR there is a Lorentzian metric. But I'm taking that to be a non-essential feature of GR, the essential one being that free-fall worldlines are geodesics.

Or in other words, your question is "assuming we ignore the core connection between the two theories, how are the two theories connected?". That's not a constructive or answerable question.

• Comments are not for extended discussion; this conversation has been moved to chat.
– Chris
Jul 22, 2019 at 22:21

Brief introduction

In Newtonian mechanics, the correlation between dynamical quantities and geometry was not that eminent. Sure, kinematics used geometry, but dynamics was good enough explained in terms of “forces”. The profound dependence of dynamical quantities with the underlying geometry was not that obvious. Just have a look into Galilean transformation for example $$\begin{cases} t=t'\\ x=x'+Vt'\\ y=y'\\ z=z' \end{cases} \tag{1}$$ It would be interesting if one wrote this in the form $$\left( \begin{array} &t\\ x\\ y\\ z\\ \end{array} \right)= \left[ \begin{matrix} 1&0&0&0\\ V&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix} \right] \left( \begin{array} &t'\\ x'\\ y'\\ z'\\ \end{array} \right) \tag{2}$$

But there is no much symmetry between space and time, and by those times there was no reason to believe that there was anything else there, especially because the absoluteness of time was completely established.

3 years after Einstein’s SR, with its radical changes in the concepts of space and time, Minkowski presented a reformulation of SR in terms of geometric concepts. In fact, most physical quantities seemed to be easily adapted to SR if they were written in the form of a 4-vector, including dynamical ones. An impressive symmetry between space and time transform, made people to believe that there was more in that than a simple coincidence.

Have a look into the homogeneous Lorentz group:

$$\left( \begin{array} &t\\ x\\ y\\ z\\ \end{array} \right)= \left[ \begin{matrix} -1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix} \right] \left( \begin{array} &t'\\ x'\\ y'\\ z'\\ \end{array} \right) \tag{3}$$

There is an astonishing symmetry between time and space here. More importantly, we have already accepted the idea that time was not that special and could change due to speed (position derivative) effects, but not with position itself, hence why not try to generalize this to include positional effects along with speed effects. The natural way to generalize it is by writing $$\left( \begin{array} &t\\ x\\ y\\ z\\ \end{array} \right)= \left[ \begin{matrix} F_1& F_2& F_3& F_4\\ F_5& F_6& F_7& F_8\\ F_9& F_{10}& F_{11}& F_{12}\\ F_{13}& F_{14}& F_{15}& F_{16}\\ \end{matrix} \right] \left( \begin{array} &t'\\ x'\\ y'\\ z'\\ \end{array} \right) \tag{4}$$

With $$F_i$$ being functions of position, time and dynamical quantities plus the condition that (4) should become (3) for small space-time volumes. This is the connection between GR and SR. It generalizes the Lorentz homogeneous group.

Now, it should be noted that Newtonian Gravity was not Lorentz invariant and thus was out of the scope of SR. A Scar in the theory that needed to be addressed and Einstein saw the opportunity to link the form of those $$F_i$$ by linking then to gravity.

As a final thought, one could consider SR to encompass the changes in physics due to speed (position derivative) and GR as including both changes due to position and its derivative speed.

PS: I used the notation $$c=1$$ and the metric signature $$(-,+,+,+)$$.

• Is equation 3 a misprint? Jul 22, 2019 at 23:57
• @JoshuaTilley. If you mean the absence of a $c$, that is due to unit system I used. See the PS at the end of the answer. Hope this was helpful. Jul 23, 2019 at 0:52