Lorentzian geometry is an area of mathematics characterised by an inner product with a single minus sign. It is known as a difficult, slowly developing area of mathematics and this makes is suitable for amateurs.
Lorentzian geometry is a fascinating area of mathematics which has had very little development over the last century. Moreover, few professional mathematicians work in this area and the publication rate of articles is sufficiently low to allow them to be read by people with full time non-academic positions. This makes the Lorentzian geometry a good target for amateur mathematicians. You won't be over taken by research in the area and you'll be able to keep up with the professional community.
The area has one application, to the study of gravity and admittedly there are many many people working in this area. There is substantial money backing research in gravity (e.g. LIGO). But, this research is very focussed on gravity. There isn't much research on the periphery of this application and hence outside of this application is the area where amateur mathematicians should aim. As an example, existence and uniqueness results for differential equations in Lorentzian manifolds have only been proven in manifolds that are "globally hyperbolic". This assumption is ubiquitous. Thus the study of non-globally hyperbolic manifolds is an area where an amateur can make progress.
This is really the case because there is not much money for fundamental geometry in Lorentzian manifolds. The fads of mathematics have left the area behind, the big names in the field are all close to or have retired. In this environment it is difficult to convince granting bodies to fund research. Why should they when the mathematicians on the panels handing out the money see other priorities? The current "big problems" are far away for Lorentzian geometry. For professional mathematicians this situation is a warning sign, for the amateur it should be considered an invitation.
To be clear I'm talking about geometry not applications of geometry. The study of gravity is a huge area that Lorentzian geometry is applied to. There are vast numbers of physicists studying how various aspects of gravity with in the context of Lorentzian geometry. But, and this is a big but, virtually every result that is produced by these researchers assumes something called "Global Hyperbolicity" which essentially makes Lorentzian geometry trivial. They do it mainly because this assumption makes differential equations easy to work with. So, I'm talking about the kind of Lorentzian geometry that physicists working in gravity aren't interested it.
Lorentzian geometry is accessible. The lack of interest from the academic community mean that the very most recent advances have let the subject be. Amateurs will need to know differential geometry, analysis and some topology to about the equivalent of a 3rd year undergraduate. Getting contact with the bigger questions in the field can be achieved quickly. Indeed reading 3 books will get you to the point where the literature should be easy to read.
Even still the problems in Lorentzian geometry, when working without the assumption of global hyperbolicity, are hard. They are hard because Lorentzian geometry is very different from Riemannian geometry. The techniques and concepts and intuition that mathematicians develop in a standard course on differential geometry very rarely apply to Lorentzian manifolds. A different way of thinking is required. This makes the field fun and rewarding.
An example is important. I recently published a paper on when the distance defined by the Lorentzian inner product is finite. Despite the area being over 100 years old nobody knew when the Lorentzian distance was finite. The mathematics in the paper involves second year analysis, induction and some basic constructions in the field. Well within the ability of someone who has done some reading in the area.