- #31
- 10,447
- 1,610
re: needing more than the stress-energy tensor
In GR, the stress-energy tensor is the density of energy and momentum. It is somewhat similar to the charge density in E&M. In E&M, one can interpret the "source" of electromagnetism as a scalar, charge, which has a density given by a rank-1 tensor, the four-current.
In GR, one can interpret the "source" of gravity as energy and momentum (a four-vector) which has a density given by a rank-2 tensor, the stress-energy tensor.
Thus the four-current, multipled by a vector representing a volume, gives the charge density (a scalar) contained in that volume in E&M. The stress-energy tensor, a rank 2 tensor, mutliplied by a vector representing a volume, gives the energy-momentum (a four-vector) contained in that volume.
In order to solve either E&M, one needs more than the distribution of charge. One also needs additional boundary conditions. E&M is linear, so that one can add any homogeneous solution of Maxwell's equations to an inhomogenoeous solution, and the result will still be a solution of Maxwell's equations. In E&M, it's usually more convenient to deal with the electromagnetic potential rather than the fields. In terms of the potential, one talks for instance about the solution to Poisson's equations (in a region where there is charge) or Laplace's equation (in a region that is charge free), but one needs boundary conditions, for instance Direchlett boundary conditions (see http://en.wikipedia.org/wiki/Boundary_condition) to specify a unique solution.
Gravity is not linear, so it's not as simple as it is for E&M, but one still needs both the stress-energy tensor AND the boundary conditions. Just knowing the source distribution is not enough. A common boundary conditions in E&M is "zero potential at infinity", in gravity a rougly similar assumption is "asymptotically flat space-time" which says that the metric "at infinity" is Minkowskian. The metric coefficeints g_ij obey second order differential equations in GR, so they are formally similar to the potentials in E&M, which also obey second order differential equations. Thus specifying the metric at infintiy is very similar to specfiying the E&M potential at infinity.
In GR, the stress-energy tensor is the density of energy and momentum. It is somewhat similar to the charge density in E&M. In E&M, one can interpret the "source" of electromagnetism as a scalar, charge, which has a density given by a rank-1 tensor, the four-current.
In GR, one can interpret the "source" of gravity as energy and momentum (a four-vector) which has a density given by a rank-2 tensor, the stress-energy tensor.
Thus the four-current, multipled by a vector representing a volume, gives the charge density (a scalar) contained in that volume in E&M. The stress-energy tensor, a rank 2 tensor, mutliplied by a vector representing a volume, gives the energy-momentum (a four-vector) contained in that volume.
In order to solve either E&M, one needs more than the distribution of charge. One also needs additional boundary conditions. E&M is linear, so that one can add any homogeneous solution of Maxwell's equations to an inhomogenoeous solution, and the result will still be a solution of Maxwell's equations. In E&M, it's usually more convenient to deal with the electromagnetic potential rather than the fields. In terms of the potential, one talks for instance about the solution to Poisson's equations (in a region where there is charge) or Laplace's equation (in a region that is charge free), but one needs boundary conditions, for instance Direchlett boundary conditions (see http://en.wikipedia.org/wiki/Boundary_condition) to specify a unique solution.
Gravity is not linear, so it's not as simple as it is for E&M, but one still needs both the stress-energy tensor AND the boundary conditions. Just knowing the source distribution is not enough. A common boundary conditions in E&M is "zero potential at infinity", in gravity a rougly similar assumption is "asymptotically flat space-time" which says that the metric "at infinity" is Minkowskian. The metric coefficeints g_ij obey second order differential equations in GR, so they are formally similar to the potentials in E&M, which also obey second order differential equations. Thus specifying the metric at infintiy is very similar to specfiying the E&M potential at infinity.