# Special Relativistic approximation to GR

Some time ago I was talking to a professor in college about some of the fundamental aspects and origin of General Relativity. I was surprised to learn, in fact, that a pretty good approximation to GR can be achieved simply by using Special Relativity and applying a series of (infinitesimal?) Lorentz boosts to simulate acceleration/a gravitational field. In fact, I believe this is precisely the approach used by GPS system for accurate triangulation/location, as mathematically and computationally it is a somewhat simpler process than applying the Einstein field equations.

So my question is: has anyone else heard of this? An explanation of how exactly this approximation works, and where/why it fails in comparison to true GR (perhaps the strong gravitational field case?) would be much appreciated.

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 Maybe relativistic Newton would be useful to you? – voix Jan 15 '11 at 18:21 @voix: That's interesting, but not really pertinent to my question in this case. – Noldorin Jan 15 '11 at 20:43

The modeling used in the GPS is based the static weak field metric $$ds^2 = -(1+2\Phi)dt^2 + (1-2\Phi)dS^2,$$ where $dS^2$ is the Euclidean metric and $\Phi$ is the gravitational potential of the Earth, though only the monopole and quadrupole terms are used. The proper time of a clock is therefore determined by $$d\tau = dt\sqrt{1+2\Phi-(1-2\Phi)\frac{dS^2}{dt^2}}\approx\left[1+\Phi-\frac{v^2}{2}\right]dt.$$ The above metric the Newtonian limit of linearized GTR for a stationary source, which in turn assumes the metric is a perturbation of the usual flat Minkowski one. (For the specific case of a spherically symmetric potential, the metric is also equiv/alent to the Schwarzschild geometry in isotropic coordinates with $Φ = -GM/r$ and the higher-order terms dropped.)

The idea of approximating gravity as acceleration found by infinitesimal Lorentz boosts in a flat background is not at all crazy by itself. A fairly canonical example is that of an observer undergoing a constant proper acceleration in the z direction in Minkowski spacetime: $$ds^2 = -a^2z^2dt^2 + dS^2,$$ which can be thought of as performing the same infinitesimal Lorentz boost at each instant of proper time, producing a hyperbolic worldline. The Rindler chart looks a lot like Schwarzschild near the horizon at the equator and the zero azimuthal angle, under the substition $x = 2M(\theta-\pi/2)$, $y = 2M\phi$, $z = 4M(1-2M/r)$: $$ds^2 = - \left(\frac{z}{4M}\right)^2dt^2 + \left[1 + {\mathscr{O}}\left(\frac{z}{4M},\frac{x}{2M}\right)^2\right]dS^2,$$ again after dropping the higher-order terms. But I don't see any obvious of doing so in the case of GPS, at least not without making the situation much more complicated than the very simple approximation given above.

But at least that particular case is a bit more literal about "special relativistic approximation to GTR" than a perturbation of the Minkowski metric. How far can we get by assuming that the gravitational field is a field on Minkowski background? Some symmetric rank-2 tensor field $h_{\mu\nu}$ is a good candidate for the job, since it has the same number of local degrees as the metric in GTR, although we can do a scalar, vector, or what-have-you as well. It turns out that scalar and vector fields don't predict the gravitational deflection of light at all, and while $h_{\mu\nu}$ gets the right answer, it gets the precession of Mercury too high by a third. The symmetric rank-2 field also turns out to be formally identical to the usual perturbative method of linearized GTR that assumes $g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$, so in that case it's not anything particularly different.

In general, this approach of a field on a Minkowski background is bound to fail eventually, not just because of the pitfalls of approximations, but of a more fundamental incompatibility. For suppose A and B are at different elevations in a static, non-uniform gravitational field, and A sends a monochromatic light pulse at B, of some set number of oscillations (or photons), taking some amount of A's proper time t to send and B's proper time t' to receive. Due to gravitational redshift, the frequencies must be different, but the number of oscillations must remain the same: ft = f't', so $t\not=t'$. Therefore, gravitational redshift is incompatible with special relativity.

(If this is unclear, imagine following up with an identical pulse immediately thereafter, so that the signal trajectories are identical, forming a 'parallelogram' $A_{\text{send1}}B_{\text{receive1}}B_{\text{receive2}}A_{\text{send2}}$. If spacetime is flat, $t = A_{\text{send1}}A_{\text{send2}}$ should be equal to $t' = B_{\text{receive1}}B_{\text{receive2}}$, regardless of whether the signal trajectories are straight, so long as they're congruent. But gravitational redshift forces them to be different. This argument was originally made by A. Schild.)

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first, I don't know exactly whta "sequence of infinitesimal Lorentz boosts" you're referring to. A boost of the flat Minkowski space gives you a Minkowski space back, so you can't get a curved space by any sequence of boosts that act globally on the spacetime.

Also, the adjective "infinitesimal" could be pretty much inconsequential. Transformations in physics are often represented as the product of infinitely many infinitesimal transformations.

On the other hand, the statement that the world - the Solar System - may be approximated by special relativity is obviously true. The curvature of spacetime is small and may be treated perturbatively.

In pretty much any reference system you choose, and you may even choose the geocentric system, all the gravitational effects including the general relativistic effects may be incorporated as small corrections.

In particular, you may always define some operational coordinates in which the spacetime is nearly flat, so the metric is given by the flat Minkowski metric. The actual metric dictated by general relativity, whatever it is, may be expressed as a small correction to the Minkowski metric.

In principle, such a correction may be calculated at arbitrary accuracy - not just the leading level - by perturbative expansions. This approach to general relativity is usually referred to as linearized gravity - even if one goes beyond the linearized level - and it is not only essential for the actual calculations of observable phenomena but also for a proper conceptual understanding of general relativity (and quantum gravity - the linearization is necessary to understand that there are gravitons, and what their mass, spin and physical polarizations are).

Einstein himself calculated the effects of a point mass - e.g. the Sun - on other bodies - such as the Mercury - in the perturbative scheme. It was before Schwarzschild gave an exact solution for Einstein's equations in the spherically symmetric case (when extrapolated everywhere in space, it is now referred to as the Schwarzschild black hole).

The effects that the GPS has to take into account include, of course, the standard classical mechanics forces such as the centripetal force or the Coriolis force, as well as some special relativistic and general relativistic effects. But of course, when things are calculated in practice, people are de facto assuming that the whole world takes place on a Minkowski background and all the gravitational effects are small corrections that occur on the Minkowski arena.

Even if Einstein hadn't discovered GR by methods of an ingenious physicists, engineers working for GPS would later find the right corrections in the process of adjusting their system and removing the discrepancies. They could experimentally measure the dependence on the daytime, latitude, and longitude of any effect that matters and that made their previous GPS plus software inaccurate. They wouldn't appreciate the beauty of GR right away, but of course, they could guess the right form of the correction terms from the experimental deviations.

Again, to answer your last question, the perturbative approach to GR, expanding around an SR background, doesn't have to fail anywhere. It may be exact just like $\exp(x)$ may be exactly expressed by the Taylor expansion, $1+x+x^2/2+x^3/3!+\dots$. Of course, the perturbative approach becomes awkward, to say the least, for causally nontrivial setups such as black holes - including the interior; wormholes and other topologically nontrivial shapes of space; or some very global questions in cosmology. But when the spacetime is topologically trivial, perturbative expansions work. The flatter the space is, the more useful the perturbative expansions become.

Best wishes Lubos

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 Thanks Lubos. I'll read over this more fully when I get a bit more time. – Noldorin Jan 15 '11 at 16:30

Special relativity is a good local approximation to general relativity. Special relativity or any succession of Lorentz transformations will reveal nothing about the global aspects of gravitating systems.

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Ah, but Lorentz transformations can be used to simulate acceleration! – Noldorin Jan 15 '11 at 17:14
Correct. The local acceleration can be described using Lorentz transformations. But you can not build a black hole (or even a weakly gravitating system or tidal acceleration for that matter) with Lorentz transformations alone. – Johannes Jan 15 '11 at 17:35
Okay - thanks for the clarification. – Noldorin Jan 15 '11 at 20:42