Coordinate transformations in general relativity

Question

Let's assume a non-rotating point mass with mass $M$. A non-massive object travels with constant velocity $\mathbf{v}_t$, with respect to the point mass, in the vicinity of the point mass. A non-massive observer, with constant velocity $\mathbf{v}_o\neq\mathbf{v}_t$, with respect to the point mass, is observing the target.

Without the point mass special-relativistic Lorentz transformations can be applied to perform a coordinate transformation. The question is how the coordinate transformation looks like in general-relativistic case, i.e. by considering the effect of the point mass?

In principle, the transformation should contain the Lorentz transformation as a limiting case for $M\rightarrow 0$.

Usually, the Schwarzschild metric is cited for a point mass potential $${\displaystyle \mathrm {d} s^{2}=-\left(1-{\frac {2M}{r}}\right)\mathrm {d} t^{2}+{\frac {1}{1-{\frac {2M}{r}}}}\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}(\theta )\;\mathrm {d} \phi ^{2}},$$

which for $M\rightarrow 0$ gives

$${\displaystyle \mathrm {d} s^{2}=-\mathrm {d} t^{2}+\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}(\theta )\;\mathrm {d} \phi ^{2}},$$

i.e. the classical special relativistic metric in spherical coordinates. But how to derive the transformation from the Schwarzschild metric?

Answer 1

The Schwarzschild metric $G$ is:

$$G=\left[ \begin {array}{cccc} -1+2\,{\frac {M}{r}}&0&0&0 \\ 0& \left( 1-2\,{\frac {M}{r}} \right) ^{-1}&0&0 \\ 0&0&{r}^{2}&0\\ 0&0&0&{r}^{2} \left( \sin \left( \theta \right) \right) ^{2}\end {array} \right] $$

we first transformed the metric $G$ to $\eta$

$$G'=T_1\,G\,T_1=\eta= \left[ \begin {array}{cccc} {\frac {1}{\sqrt {1-2\,{\frac {M}{r}}}}}&0 &0&0\\ 0&{\frac {1}{\sqrt { \left( 1-2\,{\frac {M}{r }} \right) ^{-1}}}}&0&0\\ 0&0&{r}^{-1}&0 \\ 0&0&0&{\frac {1}{r\sin \left( \theta \right) }} \end {array} \right] \left[ \begin {array}{cccc} -1+2\,{\frac {M}{r}}&0&0&0 \\ 0& \left( 1-2\,{\frac {M}{r}} \right) ^{-1}&0&0 \\ 0&0&{r}^{2}&0\\ 0&0&0&{r}^{2} \left( \sin \left( \theta \right) \right) ^{2}\end {array} \right] \left[ \begin {array}{cccc} {\frac {1}{\sqrt {1-2\,{\frac {M}{r}}}}}&0 &0&0\\ 0&{\frac {1}{\sqrt { \left( 1-2\,{\frac {M}{r }} \right) ^{-1}}}}&0&0\\ 0&0&{r}^{-1}&0 \\ 0&0&0&{\frac {1}{r\sin \left( \theta \right) }} \end {array} \right] = \left[ \begin {array}{cccc} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0\\ 0&0&0&1\end {array} \right] $$

Then we transformed $G'=\eta$ to spherical coordinates

$$T_2\,\eta\,T_2= \left[ \begin {array}{cccc} 1&0&0&0\\ 0&1&0&0 \\ 0&0&r&0\\ 0&0&0&r\sin \left( \theta \right) \end {array} \right] \left[ \begin {array}{cccc} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0\\ 0&0&0&1\end {array} \right] \left[ \begin {array}{cccc} 1&0&0&0\\ 0&1&0&0 \\ 0&0&r&0\\ 0&0&0&r\sin \left( \theta \right) \end {array} \right]= \left[ \begin {array}{cccc} -1&0&0&0\\ 0&1&0&0 \\ 0&0&{r}^{2}&0\\ 0&0&0&{r}^{2} \left( \sin \left( \theta \right) \right) ^{2}\end {array} \right] $$

so the transformation matrix to bring the Schwarzschild metric $G$ to a spherical coordinates (metric $G_s\quad$) is:

$$T=T_2\,T_1=\left[ \begin {array}{cccc} {\frac {1}{\sqrt {1-2\,{\frac {M}{r}}}}}&0 &0&0\\ 0&{\frac {1}{\sqrt { \left( 1-2\,{\frac {M}{r }} \right) ^{-1}}}}&0&0\\ 0&0&1&0 \\ 0&0&0&1\end {array} \right] $$

$$T\,G\,T=G_s$$

Answer 2

Lorentz transformations should be performed for the Schwarzschild metric in rectangular coordinates. For weak gravity metric has the form $$d{s^2} = {c^2}\left( {1 - \frac{\alpha }{{r}}} \right)d{t^2} - \left( {1 + \frac{\alpha }{{r}}} \right)(d{x^2} + d{y^2} + d{z^2}).$$ At low velocity, the accelerations of a material particle along the coordinates follow from the geodesic equations are $$ \frac{d^2r}{ds^{2}}=-\frac{\alpha }{2r^2} \left(\frac{d(ct)}{ds}\right)^{2},$$ $$ \frac{d^2t}{ds^{2}}=-\frac{\alpha }{r^2}\frac{dt}{ds}\frac{dr}{ds}.$$ After Lorentz transformations, when the observer moves along the coordinate x, metric takes the form $$ds^2=c^2\left(1-\frac{1+{{\beta}}^2}{1-{{\beta}}^2}\frac{\alpha}{r^\prime}\right)d{t^\prime}^2-\frac{4{v}}{1-{{\beta}}^2}\frac{\alpha}{r^\prime}dt^\prime dx^\prime-\left(1+\frac{1+{{\beta}}^2}{1-{{\beta}}^2}\frac{\alpha}{r^\prime}\right)d{x^\prime}^2-\left(1+\frac{\alpha}{r^\prime}\right)(d{y^\prime}^2+ d{z^\prime}^2)$$ with $\beta =v/c$. The expressions for the accelerations along the space coordinates and the flow of time for $\beta <<1,\ \alpha /r<<1$ will be $$ \frac{d^2x^\prime}{ds^{2}}=-x^\prime\frac{\alpha }{2{r^\prime}^3} \left(\frac{d(ct^\prime)}{ds}\right)^{2},$$ $$ \frac{d^2y^\prime}{ds^{2}}=-y^\prime\frac{\alpha }{2{r^\prime}^3} \left(\frac{d(ct^\prime)}{ds}\right)^{2},$$ $$ \frac{d^2z^\prime}{ds^{2}}=-z^\prime\frac{\alpha }{2{r^\prime}^3} \left(\frac{d(ct^\prime)}{ds}\right)^{2},$$
$$ \frac{d^2t^\prime}{ds^{2}}=-\frac{\alpha x}{{r^\prime}^3}\frac{dt^\prime}{ds}\frac{dx^\prime}{ds}-\frac{\alpha y}{{r^\prime}^3}\frac{dt^\prime}{ds}\frac{dy^\prime}{ds}-\frac{\alpha z}{{r^\prime}^3}\frac{dt^\prime}{ds}\frac{dz^\prime}{ds}$$ without small quantities of higher orders.

This result can explain the annual variations in the additional acceleration of Pioneer 10, see 2, 3. Let us consider the expected additional acceleration of Pioneer 10, determined using the Doppler effect. It has a periodic component with an amplitude $ (2.9−2.4)×10^{−8} cm^2/s$ at a distance of 40 AU and $ (1.3−0.8 )×10^{−8} cm^2/s$ for 60 AU. Accelerations and distances are determined approximately from the graph and flight diagram 3, p.5. The total additional acceleration was determined from the formula relating the calculated frequency of the received signal to the observed one (https://arxiv.org/pdf/gr-qc/9903024 ) $$ \nu _{obs} = \nu _{model} ×( 1− a_P× t/c)$$ under the assumption that this is caused by the acceleration of the spacecraft itself. However, the same effect will be produced by time dilation calculated by the formula $$\nu _{obs} = \nu _{model} ×( 1−с\int_{s}^{s_0} \frac{d^2t}{ds^{2}}ds)$$ for $ s \approx ct $.

Let us assume that the X axis is directed from the Sun to the Pioneer 10 apparatus. The velocity $u= \frac{dx^\prime}{ds}$ is calculated as follows: $$ u =u_P+ u_Ecos(\omega t^\prime + \phi _0),$$ where $u_P$ is the velocity of the Pioneer relative to the Sun, $ u_E$ is orbital velocity of the Earth, $\omega$ is the period of the Earth's revolution around the Sun, and $\phi _0$ is the initial angle. Calculations using formula for $\frac{d^2t^\prime}{ds^{2}}$, when converted to acceleration for the periodic component, yield $ 3.7×10^{−8} cm^2/s$ at a distance of 40 AE and $1.6×10^{−8} cm^2/s$ for 60 AE.These values ​​are close to those observed.

Using the Lorentz transformations to the linearized Schwarzschild metric, the active gravitational mass of a cloud of rarefied relativistic particles is found 4, AIP Conf. Proc. 2872, 100001 (2023).