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I cannot completely understand what is a regular method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time-independence.

E.g. how can one derive Schwarzschild metric starting from arbitrary coordinates $x^0, x^1, x^2, x^3$? I don't even understand the stress-energy tensor form in such a case - obviously it should be proportional to $\delta(x - x_0(s))$, where $x_0(s)$ is a parametrized particle's world-line, but if the metric is unknown in advance how do I get $x_0(s)$ without any a priori assumptions?

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  • $\begingroup$ There is the perturbation method leading to the weak field metric. $\endgroup$ – Torsten Hĕrculĕ Cärlemän Dec 29 '13 at 19:33
  • $\begingroup$ @TorstenHĕrculĕCärlemän, what to do if the field is not weak? $\endgroup$ – xaxa Dec 29 '13 at 19:56
  • $\begingroup$ Well, you can always see the field equations as a system of partial differential equations and solve them numerically. Of course, there would be simplifications like the Bianchi identites etc,. $\endgroup$ – Torsten Hĕrculĕ Cärlemän Dec 29 '13 at 20:55
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    $\begingroup$ Not my field, but I believe "singularities" like point masses are replaced with analytical solutions (Schwartzschild, and so on) near the mass and the numerical simulation is done in the discretised spacetime outside this reagion: the analytical solution sets the boundary conditions at the chosen bounding surface that "excises" the mass. $\endgroup$ – WetSavannaAnimal Dec 30 '13 at 0:29
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    $\begingroup$ Look at the ADM formalism for a 3+1 decomposition with explicit Cauchy problem formulation/evolution equations for the metric. Version more suitable for numeric calculations is called BSSN formalism. $\endgroup$ – user23660 Dec 30 '13 at 6:18
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First, there is no mechanical algorithm to solve a general differential equation. Einstein's equations are obviously no exception – in fact, they belong among the more complicated and less "solvable" equations among those one may learn about. Analytically writable solutions only exist in very special, simple, and/or symmetric cases (simple enough equations describing simple enough physical situations).

Second, Einstein's equations don't determine the metric uniquely. Even with well-defined initial/boundary conditions, they only determine the solution (metric tensor field) up to a general coordinate transformation (which may be determined by 4 functions $X^\mu(x^\nu)$ of the old coordinates). It means that out of the 10 components of the symmetric metric tensor, only 6 functions are really independently physical. When we impose 4 "gauge-fixing" conditions on the metric tensor field, we effectively define the "right" coordinates and we are left with 6 independent equations for the remaining 6 functions that determine the metric tensor as a function of the coordinates. Einstein's equations are superficially 10 equations but 4 of them (more precisely 4 equations constructed out of the derivatives of these equations and the equations themselves), the covariant divergence $\nabla_\mu (G^{\mu\nu} - K\cdot T_{\mu\nu})=0$, are obeyed identically so they don't constrain the metric.

Third, general relativity may also contain point masses, the point-like sources of the gravitational field that indeed add a delta-function of a sort to the metric tensor. If that's so, general relativity is a coupled system of mutually interacting Einstein's partial differential equations and ordinary differential equations for the world lines which may be parameterized e.g. by $t(x^i)$ or otherwise (e.g. using an auxiliary time parameter along the world line – which requires us to deal with a one-dimensional coordinate transformation redundancy analogous to the four-dimensional above). Alternatively, matter may be described by electromagnetic, Klein-Gordon, Dirac, and other fields. In that case, we deal with a coupled system of many partial differential equations – Einstein's equations plus Maxwell's equations plus the Dirac equation(s) and Klein-Gordon equation(s) with various source terms.

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  • $\begingroup$ I'm not expecting to find an analytic solution - I don't quite understand how to state the problem so that it would be a complete system of equations + boundary conditions. Having chosen 4 constrains on $g_{\mu\nu}$ how do I proceed to connect coordinates with $T_{\mu\nu}$? $\endgroup$ – xaxa Dec 29 '13 at 21:35
  • $\begingroup$ Dear xaxa, $g_{\mu\nu}$ and $R_{\mu\nu}$ and $T_{\mu\nu}$ are just tensors i.e. packages of 10 functions of the four coordinates $x^\lambda$; the curvature tensors are expressed in terms of the metric tensor and their derivatives using the standard formulae. So Einstein's equations are sets of partial differential equations like any other set. The tensors' being collections of functions of coordinates is how they are "connected" with the coordinates - any other "connection" you are thinking about probably means that you don't understand the concept of a differential equation. $\endgroup$ – Luboš Motl Dec 30 '13 at 17:20
  • $\begingroup$ That is all very fine, but if the meaning of coordinates is unknown how do I find $T_{\mu\nu}(x)$? This is a question of connection between physics and math. For example, in Schwarzschild coordinates $r$ is "distance", however under the horizon it turns into "time". So initial physical statement that point particle is at rest in the origin is not actually correct. But in this particular case there is a "guiding" principle of spherical symmetry. What is a procedure in general case? $\endgroup$ – xaxa Dec 30 '13 at 19:15
  • $\begingroup$ Dear xaxa, $T_{\mu\nu}$ is just another tensor field, a set of functions of x. It is determined in terms of other, more fundamental degrees of freedom. For the electromagnetic field, it is expressed using the $FF$ products, and similarly for other fields. For idealized point masses, $T_{\mu\nu}$ has the form of the mass times the delta-function localized at the right place $x^\mu(\tau)$, and so on. The more fundamental degrees of freedom such as $F_{\mu\nu}(x^\lambda)$ or $x^\mu(\tau)$ are functions that are constrained by the (Newton or Maxwell or Dirac analogous) differential equations, too! $\endgroup$ – Luboš Motl Dec 31 '13 at 11:05
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    $\begingroup$ Dear @Luboš, HNY to you! It seems you don't quite get my question. I understand what a tensor is and how $G_{\mu\nu}$ is built from $g_{\mu\nu}$. My question is more about dependence of $T_{\mu\nu}$ on $x$. Even if we have a plain EM-field, so that $T$ is constructed from $F$, still somewhere there are currents $j$ and/or boundary conditions. Example problem: Two wires are separated by distance $a$ and constant current $j_0$ floats through them. Question: find grav. field. $\endgroup$ – xaxa Jan 3 '14 at 16:43
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Please read the following with a grain of salt. I did not read the following from a textbook but just mentally constructed it.

Upon reading the comments section, I could glean that one of the major concerns of @xaxa was how to define and use the coordinates. Only when one can define the space-time coordinates $x$, he/she can start writing down $T(x)$. I think this is where local inertial frames (LIFs) come to our rescue.

Let's take a very specific problem. We have a heavy gaseous matter swirling around. Let's choose two point particles in that gaseous matter ($P_1$ and $P_2$). The question I aim to answer is: After what proper time (in the frame of $P_1$) does $P_2$ collide with it (assuming that we know a priori that they collide)? I am assuming it's totally OK if I can't come up with an analytic, closed form solution. A numerical solution (involving numerical integration on a computer) is equally fine. Even a three-body problem in Newtonian mechanics has no closed form solution.

To answer this problem, here is the ritual. Set up a large number of LIFs between $P_1$ and $P_2$, denoted by $F_1, F_2....F_n~$, so that $F_{1}/F_{n}$ is stationed at $P_{1}/P_{2}$. Also, let's arrange the frames $F_1, F_2....F_n~$ ($n \rightarrow \infty $), so that any two LIFs, say $F_m$ and $F_{m+1}$ are very close, both spatially and temporally. Hence, wrt. $F_{m}$, $F_{m+1}$ is obeying special relativistic laws. Also, in $F_{m}$, both $F_{m}$ and $F_{m+1}$ are Lorentz frame. This means we can easily transform any four-vector (including any particle's differential four-displacement ($dt,dx,dy,dz$)) between $F_{m}$ and $F_{m+1}$. This above setup of frames is the key to transform the spacetime coordinates of $P_2$ in the frame of $P_1$ (which we call $F_1$).

Let's work in $P_2$'s LIF, i.e. $F_{n}$. In this LIF, special relativistic definition of $T^{\mu \nu}$ applies

$T^{\mu \nu} = (\rho + p)U^{\mu}U^{\nu}+ p \eta^{\mu \nu} $.

With this definition, $T^{\mu \nu}$ of the fluid mass in the close vicinity of $P_{2}$ can be computed. Note that all the quantities occurring in the definition of $T^{\mu \nu}$ are measurable since in a LIF, we measure quantities in the special-relativistic way. Einstein's field equation (EFE) then gives us the metric deformation at $P_2$ in the LIF $F_n$. Since $P_2$ moves on a geodesic, we can compute the differential displacement $(dx, dy,dz)$ in time $dt$ in the frame $F_n$. With the differential four-displacement $(dt, dx,dy,dz)_{Fn}$ in frame $F_n$ we can obtain the differential four-displacement $(dt, dx,dy,dz)_{F1}$ in frame $F_1$ attaced with particle $P_1$ via infinite number of compounded Lorentz transformations: $F_n \rightarrow F_{n-1} \rightarrow F_{n-2} \rightarrow......F_2 \rightarrow F_{1} $. Note that the transformation $F_n \rightarrow F_1$ is not a Lorentz one but the transformation $F_{m} \rightarrow F_{m-1}/F_{m+1}$ is, since $ F_{m-1}/F_{m+1}$ are within a very small time-space neighborhood of $F_{m}$ (where physics is special-relativistic). All in all, with this procedure we have found out the infinitesimally small four displacement of $P_2$ $(dt, dx,dy,dz)_{F1}$ in frame $F_1$.

We repeat the above procedure for each little journey of $P_2$ and add all of them up to get $(\Delta t, \Delta x,\Delta y,\Delta z)_{F1}$. $P_1$ and $P_2$ colliding means that we will at some instant end up with $(\Delta t_s, 0,0, 0)_{F1}$ for the four-displacement of $P_2$ in frame $F_1$ (subscript 's' stands for this 'special' epoch when the $P_1$ and $P_2$ collide). This $\Delta t_s $ is the sought after answer: the time (in frame $F_1$) after which $P_1$ and $P_2$ collide.

Please read chapter 5 of this online book for more clarification

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