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kairama15
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- TL;DR Summary
- Any object will move through spacetime along its geodesic. Since mass bends spacetime, an object initially at rest near the mass will move towards the mass along a geodesic. It would be useful to simplify the multidimensional nature of general relativity to help visualize how mass bends spacetime. We could use just 3 dimensions rather than all dimensions involved to help visualize how an object would move towards another heavy mass.
*Moving this thread from 'General Math Forum' to 'General Relativity Forum' in order to generate more discussion.*
Any object will move through spacetime along its geodesic. Since mass bends spacetime, an object initially at rest near the mass will move towards the mass along a geodesic. It would be useful to simplify the multidimensional nature of general relativity to help visualize how mass bends spacetime. We could use just 3 dimensions rather than all dimensions involved to help visualize how an object would move towards another heavy mass (as long as we only care about time, the distance from the heavy mass, and ignore the other angular components that aren't needed in a simple model).
There is a video on Youtube called Beauty of Geodesics
that visualizes how objects move along geodesics on a curved 3 dimensional surface. It may be very useful to visualize the 3 dimensional surface of the Schwarzschild metric like this to "see" how an object moves through curved spacetime generated by a mass.
A strategy would be to turn the line element of spacetime into a 3d graph. There are line elements of 3 dimensional surfaces like:
For the graph of a sphere: z=sqrt(r^2-x^2-y^2), there is a line element: ds^2=dr^2+r^2*dΘ+r^2*sinΘ*dΦ.
For the graph z=x+y, there is a line element: ds^2=dx^2+dy^2+dz^2. It's just the line element for a plane.
Is there a similar 3d graph that is associated with the Schwartzchild metric? The Schwartzchild metric is dτ^2= (1-2*G*M/(c^2*r))*dt^2 - (1-2*G*M/(c^2*r))*dr^2/c^2 (assuming dΘ and dΦ are not changing and are equal to 0). The 3d graph would be a function of r and t and its third coordinate would be τ.
...So instead of some 3d graph like z=x+y, we would get some 3d graph τ as a function of r and t. As long as we keep dΘ and dΦ equal to 0, the graph would be 3 dimensional.
So if there are 3d graphs associated with line elements for planes and spheres, is there a 3d graph associated with the Schwartzchild line element for space time τ? It would be quite beautiful to try to program a geodesic video of how a mass moves along a 3d surface of curved spacetime. I envision a stationary object next to the mass moving initially only along the t dimension, then slowly curving towards the mass as its geodesic along t and r changes.
We can't easily visualize how something moves through spacetime in 5 dimensions (spacetime,r,t,theta,phi), but we can visualize how something moves through spacetime in 3 dimensions (spacetime,r,t). Like an object initially at rest (r=ro) and only traveling along the time coordinate; and allowing the spacetime to curve it towards r=0 as r and t change on a 3 dimensional graph's geodesic.
Any object will move through spacetime along its geodesic. Since mass bends spacetime, an object initially at rest near the mass will move towards the mass along a geodesic. It would be useful to simplify the multidimensional nature of general relativity to help visualize how mass bends spacetime. We could use just 3 dimensions rather than all dimensions involved to help visualize how an object would move towards another heavy mass (as long as we only care about time, the distance from the heavy mass, and ignore the other angular components that aren't needed in a simple model).
There is a video on Youtube called Beauty of Geodesics
that visualizes how objects move along geodesics on a curved 3 dimensional surface. It may be very useful to visualize the 3 dimensional surface of the Schwarzschild metric like this to "see" how an object moves through curved spacetime generated by a mass.
A strategy would be to turn the line element of spacetime into a 3d graph. There are line elements of 3 dimensional surfaces like:
For the graph of a sphere: z=sqrt(r^2-x^2-y^2), there is a line element: ds^2=dr^2+r^2*dΘ+r^2*sinΘ*dΦ.
For the graph z=x+y, there is a line element: ds^2=dx^2+dy^2+dz^2. It's just the line element for a plane.
Is there a similar 3d graph that is associated with the Schwartzchild metric? The Schwartzchild metric is dτ^2= (1-2*G*M/(c^2*r))*dt^2 - (1-2*G*M/(c^2*r))*dr^2/c^2 (assuming dΘ and dΦ are not changing and are equal to 0). The 3d graph would be a function of r and t and its third coordinate would be τ.
...So instead of some 3d graph like z=x+y, we would get some 3d graph τ as a function of r and t. As long as we keep dΘ and dΦ equal to 0, the graph would be 3 dimensional.
So if there are 3d graphs associated with line elements for planes and spheres, is there a 3d graph associated with the Schwartzchild line element for space time τ? It would be quite beautiful to try to program a geodesic video of how a mass moves along a 3d surface of curved spacetime. I envision a stationary object next to the mass moving initially only along the t dimension, then slowly curving towards the mass as its geodesic along t and r changes.
We can't easily visualize how something moves through spacetime in 5 dimensions (spacetime,r,t,theta,phi), but we can visualize how something moves through spacetime in 3 dimensions (spacetime,r,t). Like an object initially at rest (r=ro) and only traveling along the time coordinate; and allowing the spacetime to curve it towards r=0 as r and t change on a 3 dimensional graph's geodesic.
