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1 Introduction

Recent interest on the equation of motion in general relativity is motivated by the development of gravitational telescope. It is expected that gravitational waves will be directly observed in very near future and will open up a new branch of astronomy, gravitational wave astronomy. However it is obvious that the detailed theoretical understanding for the source and the mechanism of wave generation are indispensable in order to interpret the observed data. This is particularly important for gravitational waves because the signals are extremely small. Without theoretical template no meaningful information can be obtained. This is the reason of renewed interest on the relativistic equation of motion. Although the equation of motion has a long history starting right after the development of general relativity, it has not been fully understood.

For systems whose velocity of typical motion are much less than the speed of light, one can use of the ration between the speed of light and the typical velocity as a smallness parameter \(\epsilon \) to develop the so-called slow motion approximation. If the system in concern is gravitationally bounded and the typical motion is governed by its gravitational potential, the slow-motion expansion is at the same time the expansion with respect to the gravitational potential and thus is called post-Newtonian approximation [1–4]. The post-Newtonian approximation is applied for various astrophysical systems including planetary motion in solar system and relativistic compact binaries with remarkable success [5–8]. The predicted orbital motion for the compact binaries is now used as a tool to determine physical parameters such as masses of component stars. However the usual treatment of the post-Newtonian approximation assumes that the field is everywhere weak including inside the component stars which is not applicable for relativistic binaries. Another difficulty of the post-Newtonian approximation is that the convergence of the expansion series becomes very slow in higher orders. At present 3.5 post-Newtonian approximation has been calculated [9–11], but there is a discussion that 4.5 post-Newtonian calculation will be necessary to make sufficiently accurate theoretical templates for predicting accurate waveform from coalescing binaries. Strong internal gravity in compact binaries becomes problematic in the calculation of higher-order post-Newtonian approximation because of high nonlinearity. The use of singular function such as Dirac delta function in higher order calculation is not satisfactory not only mathematically but also physically. Mathematically one cannot define singular function in nonlinear theory. Physically there is not singular source in general relativity. When one makes a singular source, one ends up a black hole which is regular horizon at least from outside. Although there are some elegant methods to deal with singular source in higher-order post-Newtonian calculation [12–14], it would be better to formulate the post-Newtonian approximation in such a way that it automatically take the strong internal gravity into account without singular sources. This is done by introducing the strong field point particle limit in the post-Newtonian approximation [15, 16].

For systems whose motion is highly relativistic, there are some development recent years and the equation of motion is known [17–20]. However it is still u satisfactory in the sense that practical treatment is possible only for limited type of orbital motions [21–24] (however there have been some progresses in this direction recently [25–27]). Although higher order equation of motion in the relativistic case is not known yet, the use of singular source would cause serious theoretical difficulty. In this sense it is preferable to formulate the equation of motion without singular source. The strong field point particle limit may have a possibility to have such a formulation.

This article is the introduction to the concept of strong field point particle limit and its application to the equation of motion in general relativity. In Sect. 2, we give a formulation of the post-Newtonian approximation on the space of solutions of Einstein equation whose initial data satisfy the Newtonian scaling. Then we introduce the point particle limit introduce and derive the equation of motion in the surface integral form in Sect. 3. In Sect. 4, we apply the strong field point particle limit to derive the equation of motion for a fast moving particle. Although the derivation is still in the stage of formal, we expect to be useful for future practical application. Finally some discussions are given.

2 Newtonian Scaling and Strong Field Point Particle Limit

As discussed above, the strong field point particle limit is introduced in order to take the strong internal gravity of the relativistic binary system in the post-Newtonian approximation. This is done by realizing that the Newtonian scaling with respect to the smallness parameter has a room to allow the strong internal gravity. Thus we first formulate the Newtonian approximation based on the Newtonian scaling. The post-Newtonian approximation is actually the asymptotic approximation to the sequence of solutions of Einstein equation which is defined by Newtonian initial data [28]. In the case of binary system, the Newtonian initial data is characterized by the following scaling relation.

$$\begin{aligned} v^i \sim \epsilon , \, \, m \sim \epsilon ^2 \end{aligned}$$
(1)

where \(v^i\) is the orbital speed, and \(m\) is the mass of component stars. This relation guarantees the Newtonian orbit at least for some short time. Relativistic effects will deviate the scaling. The post-Newtonian approximation becomes accurate as the parameter goes to zero. This means that the limit \(\epsilon \rightarrow 0\) is should be understood, but the point is that the limit taken is along the trajectory on the sequence which connects physically equivalent events at each solution. Namely the time interval is also scaled as \(\epsilon ^{-1}\), thus appropriate time coordinate on the sequence is not the coordinate time but the Newtonian time \(\tau =\epsilon t\). In fact as \(\epsilon \) goes zero, the Newtonian time stay constant and the limit is taken along the trajectory on which \(\tau =\) constant.

It should be noticed that it is sufficient for mass not the density to scale as \(\epsilon ^2\) in order to guarantee the Newtonian behavior for the orbital motion. This is important because it allows us to have strong internal gravity for the component stars. Namely one can fix gravitational potential in the course of limit by assuming the size of the component stars shrink as \(\epsilon ^2\). This means the density scales as \(\epsilon ^{-4}\). This scaling suggests the introduction of the body coordinates \((\alpha ^i\) whose relation with the near zone coordinates \((x^i)\) is as follows:

$$\begin{aligned} x^i(\tau )=z^i_A(\tau )+ \epsilon ^2 \alpha ^I(\tau ) \end{aligned}$$
(2)

where we consider a system of compact binary and \(z^i_A\) is the center of mass coordinate of star A (\(A=1,2)\). We have used the Newtonian dynamical time \(\tau \) because we neglect dynamical change of the internal structure. This is guaranteed by taking equilibrium internal configuration for the component stars as their initial data (the tidally induced change will appear at 4th post-Newtonian order). We further define the body zone \(B_A\) whose boundary is defined as

$$\begin{aligned} |{\vec x}-{\vec z}_A| =\epsilon R_A \end{aligned}$$
(3)

where \(R_A\) is some fixed number. This boundary shrinks as \(\epsilon \) in the limit \(\epsilon \rightarrow 0\) in the near zone coordinate, but expands as \(\epsilon ^{-1}\) in the body zone coordinate. Thus the boundary is far zone from the point of view of component star and thus the gravitational field by the component star may be obtained by far zone expansion (multipole expansion).

This property is essential to calculate higher-order post-Newtonian approximation without any singular behavior and also deriving the formal equation of motion for a fast moving particle.

3 Bases of the Post-Newtonian Calculation

In this section we explain how the strong field point particle limit is implemented in the actual calculation of post-Newtonian approximation. This is done by writing equation of motion in the integral form.

We first define the 4-momentum \(P^\mu _A\) of each component star as

$$\begin{aligned} P^\mu _A(\tau )=\int _{B_A} d^3x \Lambda ^{\mu \tau } \end{aligned}$$
(4)

where the integral is over the body zone \(B_A\). \(\Lambda ^{\mu \nu }\) is defined as

$$\begin{aligned} \Lambda ^{\mu \nu }=(-g)\left( T^{\mu \nu }+t^{\mu \nu }_{LL} \right) +\left( h^{\ \mu \nu }h^{\alpha \beta }-h^{\mu \alpha }h^{\nu \beta } \right) _{,\alpha \beta } \end{aligned}$$
(5)

where \(t^{\mu \nu }_{LL}\) is the Landau-Lifshitz pseudotensor [4], and \(h^{\mu \nu }=\eta ^{\mu \nu }-\sqrt{-g}g^{\mu \nu }\) is the Landau-Lifshitz variable. Then we have

$$\begin{aligned} \frac{dP^\mu _A}{d\tau }=-\oint _{\partial B_A} dS_k \Lambda ^{k\mu } +v^k_A(\tau )\oint _{\partial B_A}dS_k \Lambda ^{\tau \mu } \end{aligned}$$
(6)

This is not the equation of motion in the usual sense. We need the relation between the energy \(P^\tau _A\) and mass \(m_A\) as well as the relation between the momentum \(P^i_A\) and the velocity \(v^i_A\). In the Newtonian order these are

$$\begin{aligned} P^\tau _A = m_A \end{aligned}$$
(7)
$$\begin{aligned} P^i_A =M_A \, v^i_A \end{aligned}$$
(8)

These relations suffer from higher-order post-Newtonian corrections. The energy-mass relation is obtained from the time component of the above equation. The momentum-velocity relation is given by

$$\begin{aligned} P^i_A=P^\tau _A v^i_A +Q^1_A +\frac{d D^i_A}{d\tau } \end{aligned}$$
(9)

where \(y^i_A=y^i-z^i_A(\tau )\) and

$$\begin{aligned} Q^i_A= & {} \oint _{\partial B_A} dS_k \left( \Lambda ^{\tau k}-v^k_A \Lambda ^{\tau \tau } \right) \end{aligned}$$
(10)
$$\begin{aligned} D^i_A= & {} \oint _{\partial B_A} d^3y \, \Lambda ^{\tau \tau } y^i_A \end{aligned}$$
(11)

\(D^i_A\) is the dipole moment of the star A which we can choose arbitrary value. The above relation is obtained from the identity

$$\begin{aligned} \Theta ^{\tau i}=\left( \Lambda ^{\tau \tau }y^i_A\right) _{,\tau }+\left( \Lambda ^{\tau j} y^i_A \right) _{, j} +v^i_A \Lambda ^{\tau \tau } \end{aligned}$$
(12)

This is the consequence of the conservation law \(\Lambda ^{\mu \nu }_{\, \, ,\nu }=0\).

Then we have the following equation of motion in the integral form.

$$\begin{aligned} P^\tau _A \frac{d v^i_A}{d\tau }= & {} -\oint _{\partial B_A} dS_k \Lambda ^{ki} + v^i_A \oint dS_k \Lambda ^{\tau i} \end{aligned}$$
(13)
$$\begin{aligned}+ & {} v^i_A \left( \oint _{\partial B_A}dS_k \Lambda ^{k\tau }-v^i_A \oint _{\partial B_A} dS_k \Lambda ^{\tau \tau } \right) -\frac{dQ^i_A}{d\tau }- \frac{d^2 D^i_A}{d\tau ^2} \end{aligned}$$
(14)

As mentioned above the energy mass relation is obtained from the time component of the Eq. (5).

The first step of the calculation is to write down the formal solution of the Einstein equation.

$$\begin{aligned} h^{\mu \nu }(\tau , {\vec x}) =4\int d^3y \frac{\Lambda ^{\mu \nu }(\tau -\epsilon |{\vec x}-{\vec y}|, {\vec y})}{|{\vec x}-{\vec y}|} \end{aligned}$$
(15)

This integral is divided into two parts, one from the body zone and the other from the other region except the body zone.

$$\begin{aligned} h_B(\tau , {\vec x})= & {} 4\sum _{A=1,2} \int _{B_A} d^3\alpha \frac{\Lambda ^{\mu \nu }(\tau -\epsilon |{\vec r}_A-\epsilon ^2{\vec \alpha }_A|, {\vec z}_A+{\epsilon ^2 {\vec \alpha }_A)}}{|{\vec r}_A-\epsilon ^2{\vec \alpha }_A|} \end{aligned}$$
(16)
$$\begin{aligned} h^{\mu \nu }(\tau , {\vec x})= & {} 4\int _{N/B} d^3y \frac{\Lambda ^{\mu \nu }(\tau -\epsilon |{\vec x}-{\vec y}|, {\vec y})}{|{\vec x}-{\vec y}|} \end{aligned}$$
(17)

where \({\vec r}_A={\vec y}-{\vec z}_A\) and \({N{/}B}\) means the near zone region except the body zones. One can show by explicit calculation that the artificial boundary does not contribute to the final result [29].

In the surface integral approach we only need the gravitational field on the body zone boundary \(\partial B_A\). Since the body zone boundary is the far zone in the body zone coordinate the field \(h_B\) is calculated by multipole expansion. On the other hand the field \(h_{B/N}\) is calculated by using the following formula.

$$\begin{aligned} \int _{N/B} d^3 y \frac{f({\vec y})}{|{\vec x}-{\vec y}|} =-4\pi g({\vec x})+\oint _{\partial (B/N)} dS_k \left[ \frac{1}{|{\vec x}-{\vec y}|}\frac{\partial g({\vec y})}{\partial y^k} -g({\vec y})\frac{\partial }{\partial y^k}\left( \frac{1}{|{\vec x}-{\vec y}|}\right) \right] \end{aligned}$$
(18)

where \(g\) is the super potential of \(f\), namely

$$\begin{aligned} \Delta g({\vec x})=f({\vec x}) \end{aligned}$$
(19)

For example, if \(f=1/(r_1 r_2)\), then

$$\begin{aligned} g=\ln (r_1+r_2+r_{12}) \end{aligned}$$
(20)

Up to 2.5 post-Newtonian order all the necessary super-potentials have been known by the work of Jaranowski and Schäfer [30, 31]. In higher-order calculation we have integrals in which super-potential cannot be found. Even in these cases, we can evaluate the field at the body zone boundary [11].

The surface integral expression for the equation of motion allows us to calculate the higher-order equation of motion step by step starting from only non-vanishing Newtonian field.

$$\begin{aligned} h^{\tau \tau }=4\epsilon ^4 \sum _A \frac{m_A}{r_A} \end{aligned}$$
(21)

with

$$\begin{aligned} m_A=\lim _{\epsilon \rightarrow 0} P^\tau _A \end{aligned}$$
(22)
$$\begin{aligned} P^i_A=m_A v^i_A \end{aligned}$$
(23)

Detailed calculation are presented in the references [16, 29, 32].

4 Application to the Fast Motion Equation of Motion

We shall apply the point particle limit to the equation of motion for a fast moving small body in an arbitrary background field. Our starting point is the integral form of equation of motion derived in the previous section.

$$\begin{aligned} \frac{dP^\mu _A}{d\tau } \simeq \lim _{\epsilon \rightarrow 0} \oint _{\partial B_A} dS n_\nu \Theta ^{\mu \nu } =-\lim _{\epsilon \rightarrow 0} \epsilon ^2 \int d\Omega n_\nu (1+\epsilon a\cdot n ) t^{\mu \nu }_{LL}|_{\partial B} \end{aligned}$$
(24)

Here we used \(\Theta =(-g)(T+t_{LL})\) instead of \(\Lambda \) in the integrand because the essential contribution comes from this part,and \(n^\mu \) is the unit normal vector to the surface and \(a^\mu = Du^\mu /d\tau \) the 4-acceleration.

Let suppose we have a background field \(g_b\) and write the deviation from the background as \(h\) in this section (do not confuse \(h\) in the previous section). The the deviation field contains the singular field at the position of the particle if it is really a point. For the irrotational extended source the field which fall off as \(r^{-1}\) corresponds to the singular field. We shall write down the monopole field as \(\delta g\), then the whole field at the body boundary may be written as follows.

$$\begin{aligned} g=g_b+h=g_s+\delta g \end{aligned}$$
(25)

where \(g_s\) is the sum of the background and the regular part of the self field. There is no ambiguity of separating the regular and singular field because the separation is done on the body zone boundary. Then the Landau-Lifshitz pseudotensor at the body zone boundary may be symbolically written as follows

$$\begin{aligned} \Theta |_{\partial B} \simeq t_{LL}|_{\partial B} \simeq \Gamma (g_s+\delta g)\Gamma (g_s+\delta g) \\ \simeq \Gamma (g_s) \partial (g_s) +\Gamma (g_s) \partial (\delta g) +\Gamma (\delta g)\partial (\delta g) \end{aligned}$$

where \(\Gamma \) represents Christoffel symbol. Each term has different \(\epsilon \) dependences because \(g_s \sim O(\epsilon ^0), \, \, \delta g \sim \epsilon ^{-1}\), and thus the second term \(\Gamma (g_s)\partial (\delta g)\) cancels another \(\epsilon ^2\) dependence coming from surface element and gives nonzero contribution. The third term vanishes by the angular integral.

Let us make the above argument more explicit. The singular part of the self field may be written as follows.

$$\begin{aligned} \delta g^{\mu \nu }(x) = 2\int d^4y \sqrt{-g} u^{\mu \nu }_{\, \, \, \, \alpha \beta }(x, y)\delta (\sigma (x,y))T^{\alpha \beta } \end{aligned}$$
(26)

where \(\sigma \) is the world function and the field point is supposed to be on the body zone boundary, and \(u^{\mu \nu }_{\, \, \, \, \alpha \beta }\) is the coefficient of singular part in the retarded tensor Greens function [20, 33]. In the above expression we do not consider the spinning particle. Note that our field point \(x\) is on the body zone boundary which is in the far zone of the star. This allows us to make use of the multipole expansion. For this purpose we choose our coordinate as

$$\begin{aligned} y^\alpha =z^\alpha (\tau )+\delta y^\alpha , \, \, z^0(\tau )=\tau , \, \, \delta y^0=0 \end{aligned}$$
(27)

where \(z(\tau )\) is the world line of the center of the star. The center is assumed always to be inside the body in the point particle limit, and thus there is no ambiguity regarding the choice of the center. Thus we have

$$\begin{aligned} \delta g^{\mu \nu }(x)=\frac{2m}{{\dot{\sigma }}(x, z(\tau _z))} u^{\mu \nu }_{\, \, \, \, \alpha \beta }(x, z(\tau _z)){\dot{z}}^\alpha (\tau _z){\dot{z}}^\beta (\tau _z) \end{aligned}$$
(28)

where \(\tau _z\) is the retarded time at the center given by \(\sigma (x, z(\tau _z))=0\), and

$$\begin{aligned} m{\dot{z}}^\alpha (\tau _z){\dot{z}}^\beta (\tau _z) \equiv \lim _{\epsilon \rightarrow 0} \int \delta ^3 y \sqrt{-g} T^{\alpha \beta } \end{aligned}$$
(29)

Since we are going to take the point particle limit, the world function is simply approximated as \(1/2 \eta _{\mu \nu }(x-z)^\mu (x-z)^\nu \) and \(u^{\mu \nu }_{\, \, \alpha \beta } \simeq \delta ^{(\mu }_\alpha \delta ^{\nu )}_\beta \), we have

$$\begin{aligned} \delta g^{\alpha \beta } \simeq \frac{4m u^\alpha u^\beta }{u_\mu r^\mu } \end{aligned}$$
(30)

where \(u^\mu ={\dot{z}}, \, \, r^\mu =x^\mu -z^\mu \) and the \(z\) and \(u\) are evaluated at the retarded time. Thus its derivative may be evaluated at the body zone boundary as follows

$$\begin{aligned} \partial _\alpha \left( \delta g^{\mu \nu }\right) \simeq \frac{4m \delta ^\mu _\alpha u^\nu }{\epsilon ^2} +O\left( \epsilon ^{-1}\right) \end{aligned}$$
(31)

Using the above expressions in the LL tensor, we have the following result in the point particle limit:

$$\begin{aligned} \frac{dP^\mu }{d\tau }=-m\Gamma ^\mu _{\alpha \beta }(g_s)u^\alpha u^\beta -\frac{1}{2}m\Gamma ^\alpha _{\alpha \beta }(g_s)u^\beta u^\mu \end{aligned}$$
(32)

Here we need the relation between 4-momentum and the mass. Higher-order post-Newtonian result indicates that the ADM mass is related to the 4-momentum as

$$\begin{aligned} P^\mu =\sqrt{-g_s}m u^\mu \end{aligned}$$
(33)

Finally we have the equation of motion for a fast moving particle.

$$\begin{aligned} \frac{du^\mu }{d\tau }+\Gamma ^\mu _{\alpha \beta }(g_s) u^\alpha u^\beta =0 \end{aligned}$$
(34)

This is the geodesic equation but not in the background geometry, but on the smooth part of the geometry which is the sum of the background and the smooth part of the self field.

5 Conclusions

We have reviewed the strong field point particle limit which allows one to take the strong internal gravity into account in the equation of motion without any artificial and unphysical singularities. Although the concept is developed within the framework of post-Newtonian approximation, it can be applicable to the equation of motion for a fast moving small body in an arbitrary background spacetime as long as the size of the body is much smaller than the typical scale of background geometry. It has been shown that the strong field point particle limit can be applicable to the equation of motion up to 3.5 post-Newtonian order without any mathematical difficulty. It is interesting to see if it is applicable to ,ore higher order calculation. In particularly there will be tidal effect at the 4th order and a simple point particle limit should be modified. Since our limit allows us to calculate higher-order moments of the component star as a simple expansion with respect to \(\epsilon \), the tidal effect should be calculated without difficulty. In fact it is straightforward to formally calculate all the effect of Newtonian mass multipole moments in the equation of motion in the surface integral approach.

$$\begin{aligned} m_1 \frac{d^2 z^i_1}{d\tau ^2}=\sum _p \sum _q \frac{(-1)^{p+1}(2p+2q+1)!!}{p!q!} \frac{I^{<M_p>}_1 I^{<N_q>}}{r^{p+q+2}_{12}}N^{<iM_pN_q>} \end{aligned}$$
(35)

where \({\vec r}_{12} ={\vec z}_1-{\vec z}_2, \, N_i = r^i_{12}/r_{12}, \, M_p = m_1 m_2 \ldots m_p\) is a corrective index,\( <\cdots >\) denotes the symmetric-tracefree operation on the indices between the brackets, and \(I^{\, M_p}_A\) are the Newtonian mass multipole moments of order 2p.

Unfortunately the present derivation of equation of motion for a fast moving small body is still on a formal level, the strong field point particle limit dose not have any ambiguity for the separation of the singular part in the self field, there might be some way of deriving more tractable expression for the equation of motion.