Re: Theory of Relativity
- From: "Harry" <harald.vanlintel@xxxxxxx>
- Date: Fri, 29 Apr 2005 10:22:22 +0200
"Ken S. Tucker" <dynamics@xxxxxxxxxxxx> wrote in message
news:1114717067.202370.203430@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> In this post, it's my intention to show the
> extraordinary beauty and simplicity I find
> in the Theory of Relativity.
> It employs the ingenious logic developed
> by mathematicians called tensor analysis.
>
> The fundamental assumption of relativity is
> *absolute _spatial_ motion does not exist*,
That assumption has been definitely disproved by Sagnac and Michelson-Gale,
as they detect a component of absolute spatial motion.
..
Harald
> however placing that principle into a succint
> mathematical form seems to be difficult. So
> that's what I'll try to do.
>
> Beginning with the well known
>
> ds^2 = g_uv dx^u dx^v , u,v,w={0,1,2,3}.
>
> We can use association provided the covariant
> derivative,
>
> g_uv;w = 0.
>
> Then by association,
>
> ds^2 = dx_u dx^u.
>
> Expanding to time and space gives,
>
> ds^2 = dx_0 dx^0 + dx_i dx^i , i,j={1,2,3}.
>
> The absolute spatial motion I'll define by
>
> dx_i dx^i = Absolute spatial motion.
>
> Absolute spatial motion cannot exist, IOW's
> it vanishes, hence,
>
> dx_i dx^i =0.
>
> However, relative spatial motion cannot vanish,
> it is always possible, so I'll select dx^i to
> be relative spatial motion, so that dx^i >0 generally
> and then require
>
> dx_i =0 always,
>
> to insure
>
> dx_i dx^i=0 always,
>
> and is the mathematical description of the
> Principle of Relativity. More formally it's
> expressed using the covariant 3-velocity,
>
> U_i = dx_i/ds =0.
>
> By using tensor algebra we obtain from that,
>
> g_0i = - g_ij dx^j/dx^0,
>
> and generally,
>
> ds^2 = g_00 dx^0 dx^0 - g_ij dx^i dx^j , (always).
>
> For an SR application, sub the metric values,
>
> g_00 = g_11 = g_22 = g_33 =1,
>
> g_ij =0 when i =/= j and
>
> g_0i = -dx^i/dx^0,
>
> and find by algebra,
>
> ds^2 = g_00 dx^0 dx^0 - g_ij dx^i dx^j
>
> == dt^2 - dx^2 - dy^2 - dz^2.
>
> The succinct U_i=0 provides Minkowski spacetime,
> which embodies the Lorentz transformation, but
> done Generally Covariantly.
>
> Moving to General Relativity, the following absolute
> derivative vanishes (because U_i=0),
>
> DU_i = U_i;w dx^w =0 .
>
> Using association,
>
> U_i = g_iu U^u =0
>
> therefore,
>
> DU_i = g_iu DU^u =0
>
> and thus,
>
> DU^u =0, aka the geodesic equation.
>
> We have arrived at the equation for the geodesic
> (for ref see Weinberg's, Grav&Cosmo, Eq.(5.1.6))
> using the Principle of Relativity given by U_i =0
> and g_uv;w =0.
>
> The g_uv;w=0 is the mathematical expression for
> the Principle of Equivalence, and as is obvious,
> is required to get DU^u=0 from U_i=0.
>
> The geodesic equation is expanded to,
>
> DU^u/ds = dU^u/ds + GAMMA^u_ab U^a U^b = 0
>
> (ref, see Weinberg's Eq. (5.1.7)), and is the equation of
> motion in General Relativity.
>
> Up to this point we've used two assumptions
>
> 1) U_i=0
>
> 2) g_uv;w=0
>
> where (1) is a statement of the law of Relativity that excludes
> "absolute motion", and (2) is a statement that excludes absolute
> acceleration, and at the same time is the Principle of Equivalence,
> used to derive the General Relativity geodesic.
>
> I use (1) and (2) in relativity mathematics, is there a reason not to?
>
> TIA
> Ken S. Tucker
>
.
- References:
- Theory of Relativity
- From: Ken S. Tucker
- Theory of Relativity
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