Physics majors,
I understand that gravity deforms the spacetime through which it acts. In other words its presence changes the domain of its own propagation. What sort of mathematics do you guys use to represent that sort of thing where the solution to the problem changes the domain of the problem itself?
I'm trying to gain a more intuitive understanding of a solid mechanics problem in engineering, where stresses acting through a solid body deform the shape of the solid, further changing how the stresses act through the body. Over here we use some wonky PDE discretization techniques known as the finite element method, but I'm curious about how you guys go about solving similar problems to this.
t. mechanical engineering major
>>7940124
>What sort of mathematics do you guys use to represent that sort of thing where the solution to the problem changes the domain of the problem itself?
pde
>>7940124
>I understand that gravity deforms the spacetime through which it acts. In other words its presence changes the domain of its own propagation. What sort of mathematics do you guys use to represent that sort of thing where the solution to the problem changes the domain of the problem itself?
The wording of your question is weird and not really true. Gravity IS the deformation of spacetime.
To represent this we use what we always use. The principle of least action, which uses the calculus of variation to get Euler-Lagrange equations. The only difference is that now we're using the action S for spacetime itself instead of for an object moving through spacetime.
>>7940124
In general relativity spacetime is described by differential geometry - specifically (pseudo-)Riemannian geometry. Differential geometry describes geometric spaces called differential manifolds, such as smooth curves and surfaces, and their higher dimensional generalisations. The specific details of distances and angles in this space are given by a metric tensor on the space, from which properties like the distance between points or the curvature can be derived - this is Riemannian geometry.
The mathematics of Riemannian geometry is usually (in physics, at least) described by tensor calculus, which you might be familiar with. A tensor is like a higher dimensional version of a vector, so a rank 0 tensor is a scalar, rank 1 is a vector, rank 2 can be represented by matrices etc. The metric tensor isn't actually a tensor, but a tensor field i.e a matrix at every point in spacetime. The Riemann curvature tensor is a rank 4 tensor field.
In GR, you either impose conditions on the metric using the Einstein field equations, or using the action principle that >>7940135 mentioned, which is also defined in terms of tensors. You can view the Einstein equations as a system of 10 PDEs, but the geometric interpretation in terms of tensors means you can apply all these geometric and topological techniques. The situation in electromagnetism was similar - Maxwell's equations were originally 20 differential equations in 14
(scalar) variables or something like that, but it reduces to 4 equations in 4 (vector) variables, which also emphasises the geometrical interpretation of E&B as vector fields on space. In GR we reduce 10 PDEs to 1 tensor equation, which also has a nice geometrical interpretation as matter causing spacetime to deviate from flatness. You could also apply tensor calculus to EM to reduce Maxwell's equations to 2 tensor equations, combining the E&B fields into one tensor field called the field strength tensor. This also has a geometrical interpretation.
>>7940532
tl;dr Riemannian geometry and tensors.
>>7940124
it's just pdes with tensors. im sure you -could- try to solve the efe using fem but it'd probably be a total nightmare. personally i don't know the actual computational algorithms used -- that's covered by numerical relativity, which is the only place you'll get actual solutions (or care about them desu) -- i don't do numerical relativity, and you need supercomputers anyway. the more pen-and-paper physicists work out certain properties and analyze the efe under certain conditions.
>>7940124
>what sort of mathematics
If only there was some sort of widely accepted mathematical theory to describe gravity...
>>7940732
There are many competing mathematical theories of gravity Newton, Einstein, quantum, Barnett,...
>>7940124
>What sort of mathematics do you guys use to represent that sort of thing
Geometric Analysis
>>7940744
None of these are "competing" mathematical theories. Especially not Newton's, which is undoubtedly incorrect, although useful as an approximation. Einstein's theory has been proven many times and has yet to be disproven or broken in any experimental setting. That doesn't mean it's complete, since it's still incompatible with quantum mechanics, but it's undoubtedly the best model we have and is perfectly adequate for nearly all conditions. Any other theories are far too theoretical and lacking of any experimental evidence to be considered truly "competing" theories to GR
>>7940757
Barnett gravity is highly competitive as an alternative to more standard theories the Einstein gravity. There is a great deal of interest in its development in the community. You should do some more reading; your undergradness is showing.
