View source for Demystification of the Spacetime Model of Relativity

{{Infobox paper
| title = Demystification of the Spacetime Model of Relativity
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_6832.pdf Link to paper]
| author = [[Gurcharn S Sandhu]]
| keywords = [[Relativity]], [[Coordinate system]], [[Spacetime continuum]], [[Geodesics]], [[Metric]], [[Manifold]]
| published = 2011
| journal = [[Physics Essays]]
| volume = 24
| number = 1
| num_pages = 10
| pages = 85 - 94
}}

'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_6832.pdf here]

==Abstract==

The geometrical interpretation of gravitation in general theory of relativity imparts certain mystical properties to the spacetime continuum. The mystic connotations associated with this spacetime model may be attributed to the fallacious depiction of spacetime as a physical entity. This paper proves that the spacetime continuum in general relativity is a simple mathematical model and not a physical entity.

==Overview==

G. S. Sandhu's paper does not deny that general relativity works. Its target is narrower and, in its own terms, philosophical: the claim that the four-dimensional spacetime continuum is a ''thing'' — something that can be bent, that can carry stress, that can act on matter. Sandhu's thesis is that spacetime in GR is a '''graphical template''': a coordinate manifold with deliberately non-uniform axis scalings, chosen so that Newtonian trajectories come out looking like straight lines (geodesics). On this reading "mass curves spacetime" describes a bookkeeping device, not a physical deformation, and the mystery evaporates.

The paper departs from the mainstream in two specific ways. First, it insists on a sharp separation between '''coordinate space''' — a human construct carrying a metric tensor — and '''physical space''', the actual void in which matter and fields sit, whose real properties are the permittivity ''ε''<sub>0</sub> = 8.854 &times; 10<sup>&minus;12</sup> C<sup>2</sup>/N·m<sup>2</sup>, the permeability ''μ''<sub>0</sub> = 1.257 &times; 10<sup>&minus;6</sup> N/A<sup>2</sup>, the propagation speed ''c'' = 1/&radic;(''ε''<sub>0</sub>''μ''<sub>0</sub>) = 2.998 &times; 10<sup>8</sup> m/s, and the intrinsic impedance ''Z''<sub>0</sub> = &radic;(''μ''<sub>0</sub>/''ε''<sub>0</sub>) = 376.7 &Omega;. Metric coefficients, he stresses, cannot be measured at a point and are not among these. Second, it argues on grounds of causality for '''presentism''' over the eternalist "block universe", and then constructs an elasticity-theory argument that a physically curved space would have to tear.

==The argument==

===Coordinate space versus physical space===

The opening sections define a metrized ''N''-dimensional manifold as a set of objects in one-to-one correspondence with ordered ''N''-tuples, with the metric tensor supplying the unit scale along each axis. Sandhu's point is that "at any given point ''P'' in space, it is not possible to physically measure the metric tensor components", and that they cannot even be defined without first choosing a coordinate system. Physical space, by contrast, has properties that ''are'' routinely measured — the four electromagnetic constants above, of which only two are independent, since ''ε''<sub>0</sub> may be replaced by ''Z''<sub>0</sub>/''c'' and 1/''μ''<sub>0</sub> by ''cZ''<sub>0</sub>. These, he emphasises, "are not correlated with the metric tensor of the coordinate space". Time is treated in the same spirit: as "a relative measure of change", defined by whichever cyclic process is adopted as reference, and nothing more.

===The log-log demonstration===

The key illustration is elementary and effective. Plot ''y'' = ''ax''<sup>''b''</sup> on linear axes and it is a curve, with ''g''<sub>''xx''</sub> = ''g''<sub>''yy''</sub> = 1. Plot it on log-log axes, substituting ''Y'' = log ''y'', ''X'' = log ''x'', ''A'' = log ''a'', and it becomes the straight line ''Y'' = ''A'' + ''bX''. Since d''y'' = ''e''<sup>''Y''</sup>d''Y'' and d''x'' = ''e''<sup>''X''</sup>d''X'', the same arc element now reads (d''s'')<sup>2</sup> = ''e''<sup>2''X''</sup>(d''X'')<sup>2</sup> + ''e''<sup>2''Y''</sup>(d''Y'')<sup>2</sup>, so the metric coefficients have become ''g''<sub>''XX''</sub> = exp(2''X''), ''g''<sub>''YY''</sub> = exp(2''Y''). "Any single-valued open curve can be represented as a straight line in a suitable coordinate system with appropriate differential scale." The moral Sandhu draws: metric coefficients change how curves ''look'', not what they ''are''.

===Presentism and the causality objection===

Section IV builds the philosophical case. A particle circling in the ''XY'' plane traces a helix in ''XYT'' space, but "the helical trace does not physically exist anywhere at any time". Consider a thin metal sheet in the ''XY'' plane of an ''XYT'' manifold, with ''t''<sub>p</sub> marking the present. Taking a "mental snapshot" of the whole time axis, the sheet is found only at ''t'' = ''t''<sub>p</sub>; all other sections are physically empty. That is presentism. The eternalist alternative populates every section, which means the state of all matter and fields is "predetermined at all future locations" — and a predetermined future "does not permit a causal evolution of the physical state with progression in time" and so "violates the fundamental principle of cause and effect". The same argument is then run for the solar system in a 4D ''XYZ''–''T'' manifold, with the conclusion that the spacetime continuum "is not a physical entity but just an abstract mathematical notion which can neither influence any physical phenomenon nor can its geometry be influenced by any physical phenomenon."

===The elasticity argument: Schwarzschild strain is incompatible===

Section V is the paper's technical core, and it is worth following. Sandhu treats curved space as a deformed elastic continuum and asks whether the deformation is kinematically possible. Taking flat spherical polars ''g''<sub>''rr''</sub> = 1, ''g''<sub>''θθ''</sub> = ''r''<sup>2</sup>, ''g''<sub>''φφ''</sub> = ''r''<sup>2</sup>sin<sup>2</sup>''θ'' as the undeformed state, and the spatial Schwarzschild metric ''h''<sub>''rr''</sub> = 1/(1 &minus; 2''GM''/''c''<sup>2</sup>''r''), ''h''<sub>''θθ''</sub> = ''r''<sup>2</sup>, ''h''<sub>''φφ''</sub> = ''r''<sup>2</sup>sin<sup>2</sup>''θ'' as the deformed state, he forms the strain from 2''e''<sub>''ij''</sub> = ''h''<sub>''ij''</sub> &minus; ''g''<sub>''ij''</sub>. For 2''GM''/''c''<sup>2</sup>''r'' &lt;&lt; 1 this gives

:''e''<sub>''rr''</sub> = ''GM''/''c''<sup>2</sup>''r'', &nbsp;&nbsp; ''e''<sub>''θθ''</sub> = ''e''<sub>''φφ''</sub> = 0, &nbsp;&nbsp; all shear components zero

Now, for a purely radial displacement ''u''<sub>''r''</sub>, elasticity gives ''e''<sub>''rr''</sub> = ∂''u''<sub>''r''</sub>/∂''r'' and ''e''<sub>''θθ''</sub> = ''e''<sub>''φφ''</sub> = ''u''<sub>''r''</sub>/''r''. A non-zero ''e''<sub>''rr''</sub> therefore forces a non-zero ''u''<sub>''r''</sub>, which in turn forces non-zero tangential strains — contradicting ''e''<sub>''θθ''</sub> = ''e''<sub>''φφ''</sub> = 0. Sandhu concludes that "the specification of metric coefficients as per the Schwarzschild solution is physically invalid and unacceptable on the grounds of incompatible induced strain components."

He then generalises. Saint-Venant's compatibility conditions ''e''<sub>''ij,kl''</sub> + ''e''<sub>''kl,ij''</sub> &minus; ''e''<sub>''ik,jl''</sub> &minus; ''e''<sub>''jl,ik''</sub> = 0 are equivalent, in the finite-strain case, to the vanishing of the Riemann–Christoffel tensor built from ''e''<sub>''ij''</sub>. That can hold only if both ''g''<sub>''ij''</sub> and ''h''<sub>''ij''</sub> are Euclidean — which contradicts the premise of curvature. Hence any pseudo-Riemannian metric from the field equations produces a strain field that fails compatibility, "leading to discontinuities in the induced displacements". If spacetime were a physical continuum, it would tear.

===Spacetime as a graphical template===

Section VI assembles the positive account. Plot the free-fall trajectory of a body on a ''Y''–''T'' graph and it is a parabola; rescale the axes appropriately and it becomes a straight line. Impose the Minkowski-style constraint

:(d''S'')<sup>2</sup> = ''g''<sub>''tt''</sub>(''c'' d''t'')<sup>2</sup> &minus; ''g''<sub>''xx''</sub>d''x''<sup>2</sup> &minus; ''g''<sub>''yy''</sub>d''y''<sup>2</sup> &minus; ''g''<sub>''zz''</sub>d''z''<sup>2</sup>

and the rescaling becomes unique, and also imposes an upper speed limit ''c''. Extend this to four dimensions and you have a template manifold whose metric coefficients, tuned to the mass ''M'', turn Newtonian trajectories into geodesics — after which any other body's path can be obtained by setting its initial position and velocity and computing the geodesic. "This is precisely what has been attempted through Einstein Field Equations." Sandhu holds that the correlation between mass-energy density and metric coefficients embodied in the field equations "is essentially an empirical correlation", not deduced from any established law, and that consequently "only those solutions of the EFE can be regarded as of any practical significance which can accurately simulate the particle trajectories with geodesic curves in a Newtonian gravitational field. All other solutions of EFE may be regarded as speculative."

Three "misleading connotations" are listed in closing: the block view (invalid by causality), physical curvature (invalid by the discontinuity argument), and gravitational modification of clock rates (invalid because it violates "the fundamental notion of time, as a relative measure of change"). He concedes one point to GR: incorporating the speed limit ''c'' "may be regarded as an advancement over the Newtonian model of gravitation."

==Assessment==

There is real value in the paper's first half. The coordinate-space/physical-space distinction is well drawn and worth making; the log-log demonstration is a clean and honest piece of pedagogy; and the reminder that metric coefficients are not directly measurable at a point is correct. The quoted electromagnetic constants are accurate and the relations among them are stated correctly — ''c'' = 1/&radic;(''ε''<sub>0</sub>''μ''<sub>0</sub>) and ''Z''<sub>0</sub> = &radic;(''μ''<sub>0</sub>/''ε''<sub>0</sub>) do leave only two independent. The Schwarzschild strain arithmetic is right too: 1/(1 &minus; 2''GM''/''c''<sup>2</sup>''r'') &minus; 1 &asymp; 2''GM''/''c''<sup>2</sup>''r'', so ''e''<sub>''rr''</sub> = ''GM''/''c''<sup>2</sup>''r'' as stated, and it is true that ''h''<sub>''θθ''</sub> = ''g''<sub>''θθ''</sub> = ''r''<sup>2</sup> in Schwarzschild coordinates. The final observation that ''R''<sub>''jkli''</sub> built from a compatible strain must vanish is a correct statement of finite-strain compatibility.

The incompatibility argument nevertheless does not work, and the reason is instructive: it is an artifact of the coordinate choice. Schwarzschild's ''r'' is defined as an ''areal'' radius, fixed by the requirement that a sphere at ''r'' have area 4π''r''<sup>2</sup> — which is exactly why ''h''<sub>''θθ''</sub> = ''r''<sup>2</sup> with no correction, and hence why Sandhu finds ''e''<sub>''θθ''</sub> = 0 while ''e''<sub>''rr''</sub> ≠ 0. Write the same geometry in isotropic coordinates and the spatial metric becomes conformally flat, ''h''<sub>''ij''</sub> = (1 + ''GM''/2''c''<sup>2</sup>''ρ'')<sup>4</sup>''δ''<sub>''ij''</sub>, whereupon radial and tangential "strains" are equal and a radial displacement field reproduces them without contradiction. The same geometry thus passes or fails Sandhu's test depending on which chart one writes it in — which means the test is not testing the geometry. A quantity that changes when you relabel points is not a physical strain.

The general Saint-Venant argument has a deeper version of the same problem, and it also proves less than it appears. Setting ''e''<sub>''ij''</sub> = ½(''h''<sub>''ij''</sub> &minus; ''g''<sub>''ij''</sub>) presupposes that the curved metric and a flat metric are both defined on the same manifold in the same chart, so that their difference means something — that is, it presupposes that curved space is a deformation of a flat background. GR denies exactly that. The conclusion Sandhu reaches — that compatibility forces both metrics to be Euclidean, so a genuinely curved space is not a compatible deformation of a flat one — is a correct theorem of elasticity, and it is precisely what a general relativist would say: intrinsically curved geometry is ''not'' obtained by straining a flat continuum. Presented as a refutation, it is a standard result restated. It refutes the aether-elasticity picture of gravity, not GR.

The presentism argument targets an interpretation rather than the theory. GR's field equations are indifferent to the metaphysics; the ADM or 3+1 formulation writes them explicitly as the evolution of a three-geometry through a foliation, and every numerical-relativity code — including those that produced the waveform templates matched to the LIGO detections — treats spacetime that way in practice. A presentist can therefore hold GR without contradiction, and the causality objection loses its bite. It should also be said that determinism and the block universe are separable questions: Newtonian mechanics is deterministic too, and no one takes that to abolish cause and effect.

Two further points bear on the physics rather than the philosophy. First, Sandhu argues that a solution with ''g''<sub>''tt''</sub> ≠ 1 and ''g''<sub>''xx''</sub> = ''g''<sub>''yy''</sub> = ''g''<sub>''zz''</sub> = 1 "will represent the speed of light propagation to be different from ''c''". That is the coordinate speed of light, which is indeed generally not ''c'' in a curved spacetime and is not supposed to be; the locally measured speed, in a freely falling frame, is always ''c''. The Shapiro time delay — measured to about one part in 10<sup>5</sup> by the Cassini spacecraft in 2003 — is precisely this coordinate effect, and it is observed.

Second, and decisively, the "template" reading understates what the field equations do. It is true that axis rescaling can straighten any ''one'' curve. What is not trivial, and what no graphical device delivers, is that a ''single'' metric field determined by the source makes ''every'' test body's trajectory a geodesic simultaneously, irrespective of its mass or composition — the content of the [[Equivalence Principle|equivalence principle]], now verified by the MICROSCOPE satellite to about one part in 10<sup>15</sup>. And the criterion Sandhu proposes for taking a solution seriously — that it reproduce Newtonian trajectories — would discard exactly the results that confirmed the theory: the 43 arcseconds per century of [[Perihelion Precession of Mercury|Mercury's perihelion advance]] beyond the Newtonian value, the 1.75-arcsecond light deflection (twice the Newtonian figure, and now confirmed by VLBI to better than 0.02%), and the orbital decay of the Hulse–Taylor binary pulsar matching the quadrupole [[Gravitational Waves|gravitational-wave]] prediction to within 0.2%.

Finally, the dismissal of gravitational time dilation as violating "the fundamental notion of time" cannot be sustained against measurement. Pound and Rebka detected the 2.5 &times; 10<sup>&minus;15</sup> fractional frequency shift over 22.5 metres in 1960; optical lattice clocks now resolve a height difference of a few centimetres; and the [[GPS|GPS]] constellation's satellite clocks are offset before launch by 38 microseconds per day for exactly this reason. Whether time "is" a relative measure of change is a definitional matter; that identically constructed clocks at different gravitational potentials accumulate different elapsed readings is an observation, and any account of gravity has to reproduce it.

Read for what it does rather than what it claims, the paper is a careful statement of an anti-substantivalist position on spacetime — a live position in the philosophy of physics — supported by one instructive analogy and one elasticity calculation that does not survive a change of coordinates.

==See also==

* [[Gurcharn S Sandhu]]
* [[Equivalence Principle]]
* [[Simultaneity]]
* [[Arrow of Time]]
* [[Time Dilation]]
* [[Perihelion Precession of Mercury]]
* [[Gravitational Lensing]]
* [[Gravitational Waves]]
* [[Black Hole]]
* [[Vacuum]]
* [[GPS]]

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