= 27d567bc2d48d9f40e9ccc20ea370b15)
Alejandro1 comments on Stupid Questions February 2015 - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (198)
The current definition of the gravitational constant maximizes the simplicity of Newton's law F = Gmm'/r^2. Adding a 4π to its definition would maximize the simplicity of the Poisson equation that Metus wrote. Adding instead 8π, on the other hand, would maximize the simplicity of Einstein's field equations. No matter what you do, some equation will look a bit more complicated.
Absolutely, and Planck's constant maximizes the simplicity of finding the energy of a photon from its wavelength, and π maximally simplifies finding the circumference of of a circle from its diameter. But in all those cases, it feels to me like we're simplifying the wrong equation.
ETA: To be explicit, it feels like there should be a 4π in Newton's law. The formula is calculating the gravitational flux on the surface of a 3-dimensional sphere, and 3-dimensional spheres have a surface area 4π times their radii.
Saying that this is what the formula intrinsically does, amounts to saying that field lines are more fundamental/"real" than action-at-distance forces on point particles. But in the context of purely Newtonian gravity, both formulations are in fact completely equivalent. (And if you appeal to relativity to justify considering fields more fundamental, then why not better go for simplifying Einstein's equation and including 8π in G?)
Yep :-). I don't know enough the physics to back that up, but that's what my gut tells me. A more educated version of me might be able to say something "the vocabulary of forces is 'shallow'; the vocabulary of fields is deeper; the vocabulary of group symmetries is deeper still." I certainly do not have the depth of understanding to make that sort of statement with any authority. If you know enough physics to correct me or clarify, please please do.
If somebody who groks relativity told me that this is the right thing to do, I would believe them (ETA mentioned on Wikipedia). I'd be curious where the factor of 2 comes from in the Newtonian approximation.
I can take a stab at explaining this. Both the Poisson equation and the Einstein equation have the general form
In the Newtonian case, F is the gravitational potential. In the Einstein case, it is the spacetime metric. This is a quantity with a simple, natural, purely "mathematical" definition that you cannot play with and change redifining constants; it measures the distance between events on a four-dimensional curved spacetime. "Matter source" in the Poisson equation stands for mass density, and in the Einstein equation it stands for a more complicated entitity that reduces to exactly mass density in the limit when Newtonian physics holds. So the ratio of the constants in each equation is determined by the ratio of how "spacetime metric" and "gravitational potential" are related in the Newtonian limit of GR.
In Newtonian physics, the gravitational potential is that whose first derivatives give the acceleration of a test particle:
This is considered a phyiscs law combining both Newton's law of gravity and Newton's second law of motion. In GR, the spacetime metric also has the (purely mathematical) property that (in the limit where velocities are much smaller than the speed of light, and departures from flat space are small) its gradient is proportional, with a factor 2, to the acceleration of geodesic (minimum length) trajectories in spacetime:
So if we make the physical assumption that test particles in a gravitational field follow geodesics, then we can recover Newtonian gravity from GR. (The whole reason why this is possible is the equivalence principle, the observation that all forms of matter respond to gravity in the same way.) Since small perturbations to a flat metric have to be identified with twice the Newtonian potential, this is where the extra 2 in the Einstein equation comes from.
Wow, you have a good point. I always use the concept of surface area (and considering spheres of equal total force) to remember why the r on the bottom is squared. Putting the surface area into the formula is like replacing a factor that raises questions with the answer to those questions.
More importantly, π is the area of the unit circle. If you're talking about angles you want τ (tau), if you're talking about area you want π. And you always want pie, ha ha.