Geometrical Formulation of Relativistic Mechanics

TL;DR

This paper develops a geometrical approach to relativistic mechanics by deriving covariant equations of motion, a modified Lorentz transformation, and relativistic effects directly from the metric, providing a unified framework.

Contribution

It introduces a metric-based formulation of relativistic mechanics, including covariant equations, a deformed Euler-Lagrange equation, and a modified Lorentz transformation.

Findings

Derived covariant equations of motion from the metric

Formulated a modified local Lorentz transformation

Compared non-relativistic limits with conventional formulations

Abstract

The relativistic Lagrangian in presence of potentials was formulated directly from the metric, with the classical Lagrangian shown embedded within it. Using it we formulated covariant equations of motion, a deformed Euler-Lagrange equation, and relativistic Hamiltonian mechanics. We also formulate a modified local Lorentz transformation, such that the metric at a point is invariant only under the transformation defined at that point, and derive the formulae for time-dilation, length contraction, and gravitational redshift. Then we compare our formulation under non-relativistic approximations to the conventional ad-hoc formulation, and we briefly analyze the relativistic Lienard oscillator and the spacetime it implies.

Equations

$$d{s}^2={\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=c^{2}d{t}^2-|d\mathbf{x}{|}^2.$$

$$d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=g_{00}{c}^{2}d{t}^2+2g_{0i}cdtd{x}^i+g_{ij}d{x}^{i}d{x}^j,i,j=1,2,3.$$

$$L=-m{c}^2\sqrt{1-{\left(\frac{v}{c}\right)}^2}-U.$$

$$g_{\mu \nu }(x)={\eta }_{\mu \nu }+h_{\mu \nu }(x)\underset{x\to \infty }{\lim }{h}_{\mu \nu }(x)=0$$

$$\underset{x\to \infty }{\lim }d{s}^2=\underset{x\to \infty }{\lim }{g}_{\mu \nu }(x)d{x}^{\mu }d{x}^{\nu }={\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$$

$$ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=c^{2}dt^{2}-\big{|}d\bm{x}\big{|}^{2}$$

$$\widetilde{\mathcal{L}}=-mc\sqrt{\left(\frac{ds}{d\tau}\right)^{2}}=-mc\sqrt{c^{2}\dot{t}^{2}-\big{|}\dot{\bm{x}}\big{|}^{2}}=-mc^{2}\dot{t}\sqrt{1-\left(\frac{|\bm{v}|}{c}\right)^{2}}\qquad\text{where }\quad\bm{v}=\dfrac{\dot{\bm{x}}}{\dot{t}}.$$

$$S={\int }_1^{2}d\tau \tilde{\mathcal{L}}=-m{c}^2{\int }_1^{2}dt\sqrt{1-{\beta }^2}={\int }_1^{2}dt\mathcal{L},\text{where }\beta =\frac{|\mathbf{v}|}{c},$$

$$\Rightarrow \mathcal{L}=-mc\sqrt{{\left(\frac{ds}{dt}\right)}^2}=-m{c}^2\sqrt{1-{\beta }^2}.$$

$$p_{\mu }=\frac{\partial \tilde{\mathcal{L}}}{\partial {\dot{x}}^{\mu }}=\{\begin{array}{l}\mathbf{p}=\frac{\partial \tilde{\mathcal{L}}}{\partial \dot{\mathbf{x}}}=\frac{m\mathbf{v}}{\sqrt{1-{\beta }^2}}=m\mathbf{v}\gamma \\ \mathcal{E}=-\frac{\partial \tilde{\mathcal{L}}}{\partial \dot{t}}=\frac{m{c}^2}{\sqrt{1-{\beta }^2}}=m{c}^2\gamma =\frac{\mathbf{p}.\dot{\mathbf{x}}-\tilde{\mathcal{L}}}{\dot{t}}=\mathbf{p}.\mathbf{v}-\mathcal{L}\end{array}$$

$$\gamma =\frac{1}{\sqrt{1-{\beta }^2}}.$$

$${\mathcal{E}}^2={\eta }^{\mu \nu }{p}_{\mu }{p}_{\nu }=|p{|}^2{c}^2+m^2{c}^4.$$

$$ds^{2}=g_{00}(\bm{x})c^{2}dt^{2}-\big{|}d\bm{x}\big{|}^{2}\qquad\qquad\text{where}\quad g_{00}(\bm{x})=1+\frac{2U({\bm{x}})}{mc^{2}}.$$

$$S={\int }_1^{2}d\tau \mathcal{L}\mathcal{L}=-mc\sqrt{{\left(\frac{ds}{d\tau }\right)}^2}=-mc\sqrt{g_{00}(\mathbf{x})c^2{\dot{t}}^2-{\left|\dot{\mathbf{x}}\right|}^2}.$$

$$\mathcal{L}=-m{c}^2\sqrt{\left(1+\frac{2U(\mathbf{x})}{m{c}^2}\right){\dot{t}}^2-{\left(\frac{|\dot{\mathbf{x}}|}{c}\right)}^2}=-m{c}^2\dot{t}\sqrt{1-\frac{2}{m{c}^2}\left(\frac{m|\mathbf{v}{|}^2}{2}-U(\mathbf{x})\right)}.$$

$$L=T-U=\frac{m|\mathbf{v}{|}^2}{2}-U(\mathbf{x}),T=\frac{m|\mathbf{v}{|}^2}{2}.$$

$$\mathcal{L}=-mc\sqrt{{\left(\frac{ds}{d\tau }\right)}^2}=-m{c}^2\sqrt{1-\frac{2L}{m{c}^2}}.$$

$$\begin{split}{\bm{v}}=0\quad\\ \quad U({\bm{x}})=0\end{split}\Bigg{\}}\quad\Rightarrow\quad L=0\quad\Rightarrow\quad{\mathcal{L}}=L_{0}=-mc^{2}.$$

$$\mathcal{L}=L_0\sqrt{1+2\frac{L}{L_0}}.$$

$$\frac{ds}{dt}=c\sqrt{1+2\frac{L}{L_0}}\Rightarrow {\Gamma }^{-1}=\frac{d\tilde{t}}{dt}=\sqrt{1+2\frac{L}{L_0}}\stackrel{U=0}{\to }{\gamma }^{-1}=\sqrt{1-{\left(\frac{|\mathbf{v}|}{c}\right)}^2}.$$

$$p_{\mu }=\frac{\partial \mathcal{L}}{\partial {\dot{x}}^{\mu }}=\{\begin{array}{l}\mathbf{p}=\frac{\partial \mathcal{L}}{\partial \dot{\mathbf{x}}}=m\mathbf{v}\Gamma \text{vskip}3.0ptplus1.0ptminus1.0pt\\ \mathcal{E}=-\frac{\partial \mathcal{L}}{\partial \dot{t}}=m{c}^2{g}_{00}(\mathbf{x})\Gamma =\frac{\mathbf{p}.\dot{\mathbf{x}}-\tilde{\mathcal{L}}}{\dot{t}}\end{array}$$

$$1+2\frac{L}{L_0}=0\Rightarrow g_{00}(\mathbf{x})c^2{\dot{t}}^2-{\left|\dot{\mathbf{x}}\right|}^2=0\Rightarrow \left|\mathbf{v}\right|=\left|\frac{d\mathbf{x}}{dt}\right|=c\sqrt{g_{00}(\mathbf{x})}.$$

$$\mathcal{L}\stackrel{L<<m{c}^2=-L_0}{\to }{L}_0\left(1+\frac{L}{L_0}\right)=L+L_0.$$

$$\mathcal{L}\stackrel{L<<m{c}^2=-L_0}{\to }-m{c}^2\left(1-\frac{L}{m{c}^2}\right)=L-m{c}^2=-\frac{m}{2}{\left(\frac{ds}{dt}\right)}^2-\frac{m{c}^2}{2},$$

$$L_{eff}=-\frac{m}{2}\left(\frac{ds}{dt}\right)^{2}=\frac{m}{2}\left(\big{|}\dot{\bm{x}}\big{|}^{2}-c^{2}g_{00}(\bm{x})\right)\qquad\qquad g_{00}(\bm{x})=1+\frac{2U(\bm{x})}{mc^{2}}.$$

$$\frac{d}{d\tau }\left(\frac{\partial \mathcal{L}}{\partial {\dot{x}}^i}\right)=\frac{\partial \mathcal{L}}{\partial x^i}.$$

$$\frac{d}{dt}\left(m\Gamma \mathbf{v}\right)=-\Gamma \nabla U(\mathbf{x}).\frac{d}{dt}\left(m\Gamma \mathbf{v}\right)=-\Gamma \nabla U(\mathbf{x}).$$

$$m\frac{d^2\mathbf{x}}{d{\tilde{t}}^2}=\Gamma \frac{d}{dt}\left(m\Gamma \mathbf{v}\right)=-{\Gamma }^2\nabla U(\mathbf{x}).$$

$${\Gamma }^2={\left(1+2\frac{L}{L_0}\right)}^{-1}=1-2\frac{L}{L_0}+(-2{)}^2{\left(\frac{L}{L_0}\right)}^2+....$$

$$\therefore m\frac{d^2\mathbf{x}}{d{\tilde{t}}^2}=-\nabla U(\mathbf{x})+2\frac{L}{L_0}\nabla U(\mathbf{x})-.....$$

Full text

1* S.N. Bose National Centre for Basic Sciences*

*JD Block, Sector-3, Salt Lake,

Kolkata-700098, INDIA.*

2* Instituto de Física de São Carlos; IFSC/USP*

Universidade de São Paulo Caixa Postal 369,

CEP 13560-970, São Carlos-SP, Brazil

sumanto12@bose.res.in , partha@bose.res.in

PACS classification : 03.30.+p, 95.30.Sf.

Keywords : Relativistic mechanics, local Lorentz transformation, gravitational redshift, Semi-relativistic

formulation, relativistic Kepler, Bohlin-Arnold duality, relativistic Chiellini condition, Lienard equation.

1. 1 Introduction

2. 2 Preliminary: Relativistic Mechanics

3. 2.1 Flat space

4. 2.2 Curved space for single scalar potential

5. 2.2.1 Relativistic equations of motion

6. 2.2.2 Conserved Quantity

7. 2.3 Relativistic Hamiltonian mechanics in curved spaces

8. 3 A modified Local Lorentz Transformation

9. 3.1 Time-dilation and length contraction

10. 3.2 Gravitational redshift

11. 4 Approximations and Limits

12. 4.1 Weak potential limit

13. 4.2 Semi-relativistic formulation

14. 4.3 Hamiltonian formulation under weak potential

15. 5 Bohlin-Arnold Duality

16. 5.1 Relativistic 2D Isotropic Oscillator and Kepler systems

17. 5.2 Semi-Relativistic Kepler-Hooke duality

18. 6 Relativistic Lienard-type oscillator

19. 6.1 Integrability and Relativistic Chiellini condition

20. 6.2 Metric and Lagrangian

21. 7 Conclusion

1 Introduction

In relativistic mechanics, we describe a geometric perspective of dynamics. This means that we start by describing pseudo-Riemannian spaces via metrics $(M,g)$ by which we shall measure infinitesimal arc lengths in such spaces. Dynamical trajectories or geodesics between any two chosen fixed points are the shortest path in terms of integrated length in between. We are familiar with the usage of infinitesimal arc length in special relativity for flat spaces.

[TABLE]

In general or curved spaces, the general infinitesimal arc length element is given by:

[TABLE]

which becomes flat when $g_{\mu\nu}=\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$. Thus, we must derive mechanical formulation from (1.2) to correctly describe relativistic mechanics in general on curved spaces. Spacetime for the usual problems dealt with in classical mechanics simply involve a 4-potential $A^{\mu}=(U,-{\bm{A}})$, for which, we shall have the spatial terms of the metric are flat ($g_{ij}=-\delta_{ij}$).

One important rule for a metric to abide by is that it must be invariant under the Lorentz transformation, which is easily formulated in special relativity for free particles travelling at constant velocity. However, the familiar Lorentz transformation does not preserve metrics describing trajectories for particles in the presence of potential fields. Thus we must define the Lorentz transformation in such a way that it locally preserves such metrics, meaning that the metric at a point is invariant only under the transformation rule defined at that same point. Its local nature means that position co-ordinates cannot be transformed as in special relativity.

There is an alternate ad hoc approach taken to formulate the relativistic Lagrangian, employed in publications by Harvey [1] and Babusci et al [2], to describe the dynamics of the relativistic oscillator and Kepler. It essentially draws from the conventional formulation of the Lagrangian provided by Goldstein in [3]

[TABLE]

where the kinetic energy term is kept seperate from the potential term. Although (1.3) gives the correct relativistic answers for practical problems, this Lagrangian is not Lorentz covariant. One may suspect that under some approximations, the formulation born from the metric will transform into the conventional Lagrangian based approach.

The conventional gravitational field yields conserved dynamics in a central force field without drag. However, motion involving drag is a significant topic in the study of dynamical systems, describing realistic situations, with practical applications mostly in engineering. It would naturally be very interesting to see what kinds of spacetimes and gravitational fields produce dynamics involving drag. Such systems are not always integrable unless certain conditions are fulfilled by the drag co-efficient functions and the force-field functions. The relativistic generalizations of Lagrangian/Hamiltonian systems with position-dependent mass [4, 5, 6] could be treated within this formalism. Such systems are in some cases equivalent to constant mass motion on curved spaces, and some nonlinear oscillators can be interpreted in this setting.

This paper is organized as follows: Section 2 is devoted to the preliminaries on the formulation of special relativistic mechanics in flat spacetime, followed by static curved spaces dealt with in classical mechanics. There we deduce the relativistic deformation of the Euler-Lagrange equation, and a conserved quantity related to such mechanics. We also describe relativistic Hamiltonian mechanics in curved spaces.

Section 3 deals with the modification of the Lorentz transformation under which such metrics are invariant. This is necessary since the regular Lorentz transformation, designed to work for free particles in the case of special relativity, will not suffice for particles accelerating under the influence of a potential field. Then, we will deduce the formulas for time-dilation, length contraction, and gravitational redshift from the modified Lorentz transformation formula.

Section 4 will list the various approximations that can be made and how they affect our formulations. Here we will verify if the relativistic Lagrangian under any of these approximations transforms into the conventional one employed by Harvey and Babusci et al.

Section 5 will cover formulation of the relativistic 2D oscillator using our approach. Here we will verify if the Bohlin-Arnold-Vasiliev duality between relativistic Kepler and Hooke systems holds in such non-classical settings, and what approximations, if any, are required. Such dualities are observable in Bertrand Spacetime metrics which correspond either to oscillator or to Kepler systems on the associated three dimensional curved spatial manifold [9], which could be studied from this persepective.

Section 6, finally, will study relativistic damped mechanical systems and redefine the Chiellini integrability condition in relativistic form. Then we will deduce the related metric for damped systems and define its contact Hamiltonian structure.

2 Preliminary: Relativistic Mechanics

Here, we shall describe relativistic formulation first for free particle, then for classical static curved spaces with scalar potentials. The general spacetime metric for curved spaces in the locality of potential sources can be formulated by defining (1.2) as a perturbation around the flat spacetime metric (1.1) that asymptotically vanishes towards infinity:

[TABLE]

This is necessary to ensure that the metric is asymptotically flat at large distances from the potential field source.

[TABLE]

After that, we will discuss the Hamiltonian mechanical description. Then we shall explore various approximations and the modified results that follow.

2.1 Flat space

In special relativity, we are discussing free particle mechanics $U({\bm{x}})=0$. This means that the metric describes flat space (1.1):

[TABLE]

from which the Lagrangian derived is:

[TABLE]

Alternatively, we can define the reparameterised Lagrangian $\mathcal{L}$ from the geometric action as:

[TABLE]

[TABLE]

This results in the relativistic momentum and energy given as:

[TABLE]

where, we have designated a reparameterising factor:

[TABLE]

from which, we have the familiar equations for relativistic energy:

[TABLE]

Furthermore, the singularity that occurs when $\left|\bm{v}\right|=c$ in the denominator in (2.1.1), ensures that the speed of light is never exceeded, establishing is as the physical upper limit of velocity in flat spaces as elaborated in [7, 8]. This concludes the basics of flat space. From here on, we will use the alternative convention involving mechanical systems parameterized wrt time. Next we shall look at curved spaces with scalar potentials.

2.2 Curved space for single scalar potential

When considering mechanics on classical static curved spaces, where the source of spacetime curvature is one scalar potential source $U({\bm{x}})\neq 0$, the metric based arc length can also be used for curved spaces by including the potential $U({\bm{x}})$ into the metric as a specific version of (1.2) following Gibbons’ prescription [10, 11] as shown below:

[TABLE]

where the potential $U({\bm{x}})$ is a factor in the only non-zero perturbative term $h_{00}(x)$ of the temporal metric term $g_{00}({\bm{x}})$ that vanishes asymptotically, while $h_{ij}(\bm{x})=0\ \forall\ i,j=1,2,3$. The relativistic action $S$ and Lagranigan $\mathcal{L}$ are:

[TABLE]

If we include the potential linearly as a perturbation into the metric, then we have

[TABLE]

Now, we are familiar with the traditional non-relativistic or classical Lagrangian

[TABLE]

Thus, the relativistic Lagrangian $\mathcal{L}$ is given by:

[TABLE]

Under the circumstances that we are dealing with a stationary free particle, we can define the ground-state relativistic Lagrangian as follows:

[TABLE]

Thus, we can re-write the relativistic Lagrangian as follows:

[TABLE]

showing that the classical Lagrangian $L$ is embedded within the relativistic Lagrangian $\mathcal{L}$. Furthermore, we can say that we recover flat space (2.1.2) when $g_{00}(\bm{x})=1$ ie. $(U(\bm{x})=0)$.

[TABLE]

while the relativistic momenta are clearly Lorentz-covariant, with $\widetilde{t}$ being the proper time in the particle frame.

[TABLE]

From the momenta (2.2.6), we can see that the singularity that occurs when $\Gamma^{-1}=0$ establishes a different physical upper limit for the velocity of a particle:

[TABLE]

which is less than the speed of light $c$ since $g_{00}(\bm{x})<1$, ($U(\bm{x})\leq 0$ for gravitational potentials). This deformation of the relativistic 4-momentum as a consequence of the change of the speed limit, is due to the change in length contraction and time- dilation due to the gravitational potential under a local Lorentz transformation.

Under the approximation $L<<L_{0}$, binomially expanding (2.2.4) :

[TABLE]

Thus, in the non-relativistic limit, $L$ given by (2.2.2) will suffice to produce the equations of motion. One can alternatively say that the effective classical Lagrangian directly derived from the metric (2.2.1) is

[TABLE]

[TABLE]

We shall now proceed to analyze the relativistic equations of motion.

2.2.1 Relativistic equations of motion

Now we shall turn our attention to formulating of the equations of motion. The Euler-Lagrange equation is given by:

[TABLE]

when applied to (2.2.4), we get the relativistic equation of motion:

[TABLE]

then we can write using (2.2.5):  $\widetilde{\bm{v}}=\dfrac{d\bm{x}}{d\widetilde{t}}=\dfrac{dt}{d\widetilde{t}}{\bm{v}}=\Gamma{\bm{v}}$, making (2.2.9) into

[TABLE]

If we expand $\Gamma^{2}$ in this equation, we will get:

[TABLE]

,

[TABLE]

which is the equation of motion with additional terms associated with the force function. Alternatively, we can say that on applying the Euler-Lagrange equation to the relativistic Lagrangian (2.2.4), we will get:

[TABLE]

[TABLE]

Recalling (2.2.5), and writing $\Gamma=\dfrac{L_{0}}{\mathcal{L}}$ according to (2.2.4), we get from the Euler- Lagrange equation $\frac{d\ }{dt}\left(\frac{\partial\mathcal{L}}{\partial v^{i}}\right)-\frac{\partial\mathcal{L}}{\partial x^{i}}=0$

[TABLE]

[TABLE]

Thus, we have a relativistic deformation (2.2.11) of the Euler-Lagrange equation for the classical Lagrangian $L$ derived by applying the original Euler-Lagrange equation to the relativistic Lagrangian $\mathcal{L}$.

Under non-relativistic limits, $\Gamma\approx 1$, (2.2.10) becomes the more familiar form of the equation of motion given below that directly derive from Euler-Lagrange equations applied upon (2.2.2):

[TABLE]

Similarly, the Euler-Lagrange equation that derives from the effective Lagrangian (2.2.8) is:

[TABLE]

which is equivalent to (2.2.12). Under the circumstances that one has either the relativistic or classical equations of motion, there is an algorithm to deduce the spacetime metric that generates such dynamics:

Convert the relativistic equation to the non-relativistic version (2.2.12) (with $\gamma=1$). 2. 2.

Deduce $U(\bm{x})$ from the non-relativistic equation (2.2.12). 3. 3.

Use $U$ to reproduce the spacetime metric according to (2.2.8).

This algorithm allows us to formulate a curved spacetime metric that reproduces the dynamics described by the equation of motion considered. If the force is not gravitational in nature, it essentially produces the equivalent curved spacetime that can act as a gravitational substitute for the mechanical system that imitates its observed motion classically.

2.2.2 Conserved Quantity

We shall now regard a conserved quantity deduced from (2.2.10) comparable to the Hamiltonian and the geodesic equation of motion. The conserved quantity in question is:

[TABLE]

where $K$ is essentially a multiple of the relativistic energy given by (2.2.6). Applying (2.2.14) to (2.2.5), we can say that

[TABLE]

Along a geodesic, the equation of motion in the particle frame (2.2.10) can be re-written as

[TABLE]

[TABLE]

which is one way of writing the relativistic equation of motion. If we are consider Lorentz transformations of space-time event intervals, and $g_{00}=g_{00}(\bm{x})$, we can say that

[TABLE]

which lets us write the Lorentz-covariant equation of motion:

[TABLE]

Now we shall reproduce the known and familiar relativistic phenomena of time dilation and gravitational red-shift from this formulation as a test.

2.3 Relativistic Hamiltonian mechanics in curved spaces

The Hamiltonian formulation of classical mechanics is very useful, not just for its geometrical properties, but also for enabling extension of the classical theory into the quantum context via standard quantization. Having described a relativistic Lagrangian formulation for mechanics in the presence of a scalar potential, it is natural to also consider the Hamiltonian formulation.

Referring to (2.2.6), the relativistic energy for curved spaces is:

[TABLE]

This would effectively make the Hamiltonian $\mathcal{H}$:

[TABLE]

and the Hamilton’s equations of motion for $g_{00}(\bm{x})=1+\dfrac{2U(\bm{x})}{mc^{2}}$:

[TABLE]

Such formulation has been applied to study the relativistic version of the Quantum Harmonic Oscillator [14]. In the following section, we will elaborate on the Lorentz transformation operation for such metrics.

3 A modified Local Lorentz Transformation

An important issue that emerges is the invariance of such metrics under Lorentz transformations. The Lorentz transformation we are familiar with applies only to special relativity, where we deal with free particles.

The conventional Lorentz boost of co-ordinates $x$ of frame $F$ to $\widetilde{x}$ of frame $\widetilde{F}$ is:

[TABLE]

where $v$ is the constant velocity of the particle in frame $F$, and $V_{\widetilde{F}F}$ is the speed of frame $\widetilde{F}$ with respect to frame $F$. A better way to write (3.1) this locally is to replace: $x^{\mu}\rightarrow dx^{\mu}$ under which the spacetime metric is invariant.

[TABLE]

[TABLE]

The scenario we are dealing with in this case involves a particle under the influence of a potential field. The metric is easily invariant under rotations in the presence of spherically symmetric potentials, which leaves only boosts to be considered. Due to the presence of a potential, we are required to use a modified Lorentz boost operation, which we shall briefly derive here. The Lorentz boost equations are:

[TABLE]

If we consider a Lorentz transformation to the particle frame, we should have $d\widetilde{x}=0$, which means that from the second equation of (3.3), we have:

[TABLE]

Furthermore, the determinant of the matrix $\Lambda$ must be unity to preserve volume elements spanned by 4-vectors.

[TABLE]

Now, demanding that the metric be invariant under the transformation gives us another rule:

[TABLE]

[TABLE]

[TABLE]

Combining the equations (3.4) and (3.6) into (3.5) gives us:

[TABLE]

[TABLE]

Thus, using (3.7), the modified Lorentz boost matrix (3.3) is given by:

[TABLE]

[TABLE]

Thus, we have a modified local Lorentz transformtion that preserves the metric. For a $3+1$ spacetime, the modified local Lorentz boost matrix between one co-ordinate and time would be written as:

[TABLE]

The local nature of this transformation should not be surprising since according to the Equivalence principle, at any point on a curved manifold, there always exists a diffeomorphism that transforms it locally into a flat manifold. This means that alternatively, within the locality of that point in the local inertial frame, we can transform the problem into special relativity and apply the regular Lorentz transformation (3.2).

Naturally, it is not possible to perform a global co-ordinate transformation like (3.1). In fact, we must understand that (3.1) derives from (3.2) via integration, and not the other way around via differentiation. This is because special relativity, as the name implies, describes a special case where spacetime is isotropic due to the absence of any potentials and any global co-ordinates are described as $x^{\mu}=\int_{1}^{2}d\tau\ \dot{x}^{\mu}=\dot{x}^{\mu}\tau$.

3.1 Time-dilation and length contraction

One of the reasons a speed limit exists is the Minkowsian signature of spacetime, which allows null geodesics for non-null spacetime intervals. This results in phenomena such as time-dilation and length contraction. The inclusion of a gravitational potential results in a modification of such phenomena, just as it has resulted in modification of the Lorentz transformation.

Since we are discussing a modified local Lorentz transformation (3.9), let us define the local co-ordinates in the neighbourhood of $x$ where $g_{00}$ is roughly constant:

[TABLE]

Now consider a problem where in a frame $S$, a stationary particle lies at a position $x=L$ along the $x$-axis, and $S$ moves at a velocity $v=\beta c$ along the $x$-axis wrt an observer in frame $\widetilde{S}$. Using (3.1.1) the local Lorentz transformation equations from $S$ to $\widetilde{S}$ are:

[TABLE]

Since the particle is stationary in $S$ ($d\chi=0$), the first equation of (3.1.2) gives us:

[TABLE]

Integration of (3.1.2) including constants of integration yields:

[TABLE]

Since $\int d\chi=0$ in the particle frame $S$, we set the following boundary conditions

[TABLE]

and using (3.1.5), the 2nd equation of (3.1.4) gives us:

[TABLE]

Now since the particle is in motion in $\widetilde{S}$, it moves from its starting position $\widetilde{\chi}_{0}=\widetilde{L}$ with the velocity of frame $S$ wrt $\widetilde{S}$. Thus, using (3.1.3), the contracted length $\widetilde{L}$ is:

[TABLE]

[TABLE]

[TABLE]

For flat space, we set $g_{00}=1$, and $\Gamma\longrightarrow\gamma$, which should restore the original time-dilation and length-contraction rules of special relativity for a free particle.

Due to the lowered speed limit in comparison to the speed of light for special relativity, time intervals shall dilate and length intervals shall contract further in presence of a gravitational potential field.

3.2 Gravitational redshift

An alternative way to arrive at the time-dilation formula (3.1.3) is to write the metric in its two equivalent forms in the two frames $\widetilde{S}$ and $S$:

[TABLE]

Thus, we can see from (2.2.5) that time dilation will occur under circumstances of either motion, or presence in a potential field, or due to both. Comparison to an equivalent metric in a frame in flat space without motion, we have

[TABLE]

If we consider free particle motion, we can see that:

[TABLE]

On the other hand, for stationary observation, time dilation is caused by potential fields:

[TABLE]

This is confirmed by the theories of gravitational redshift that occurs as monochromatic light of a certain frequency in free space enters a gravitational field. If the time period of the light frequency in presence of a gravitational field is given by $T$, then according to (3.2.2)

[TABLE]

[TABLE]

[TABLE]

If we define the Event Horizon radius as $r_{0}=\frac{GM}{c^{2}}$, then the new frequency in the presence of a gravitational field according to (3.2.3) is given by:

[TABLE]

[TABLE]

Under a weak potential limit $2U<<mc^{2}$ or $r_{0}<<r$, we will have:

[TABLE]

Furthermore, we will also have:

[TABLE]

which is confirmed in any literature on the topic of general relativity [13]. Next, we shall briefly summarize the formulation of relativistic Hamiltonian mechanics on curved spaces.

4 Approximations and Limits

When dealing with a relativistically described system, we often are required to apply approximations and limits to comform with the established non-relativistic formulation or the conventional relativistic formulation which uses $\gamma$ instead of $\Gamma$. This will confirm if we are on the right track in our analysis. Special relativity was formulated entirely for free particles. Now we shall explore various ways of applying a weak potential.

4.1 Weak potential limit

Under the weak potential approximation, we shall consider the case where

[TABLE]

Directly from (2.2.10), we can write that

[TABLE]

Applying weak potential approximation (4.1.1), we can say from (2.2.5) that

[TABLE]

[TABLE]

and since we are considering Lorentz transformations of space-time event intervals, and $g_{00}=g_{00}(\bm{x})$, we can say that

[TABLE]

[TABLE]

which lets us write the Lorentz-covariant equation of motion:

[TABLE]

Now, we shall look at an alternate formulation with weak potentials.

4.2 Semi-relativistic formulation

Another way to describe the non-relativistic approximation $U(\bm{x})<<mc^{2},|\bm{v}|<<c$, of the metric (2.2.4) using the expression of $\gamma$ from (2.2.5) is:

[TABLE]

Binomially expanding the expression within the square-root gives us:

[TABLE]

[TABLE]

which is different from the form of the Lagrangian employed by Goldstein [3]. However, we must keep in mind that upon applying it into Euler-Lagrange equations, we get the semi-relativistic equation of motion:

[TABLE]

which is the same equation of motion given by (4.1.2). Another approximation we can employ here is based on the comparison of the magnitude of factors paired with $\gamma^{-1}$ and $\gamma$ in (4.2.1) upon binomial expansion. In simple words, if $\gamma^{-1}$ and $\gamma$ in (4.2.1) are expanded to 1st order:

[TABLE]

we can see that higher order terms from expansion will become significant contributors depending on the factor multiplied to it. Now we can see that for weak potentials

[TABLE]

This means that at least 1st order contribution from $\gamma^{-1}$ will be significant in $mc^{2}\gamma^{-1}$. On the other hand, the 1st order contribution from$\gamma$ will be insignificant in $U\gamma$. This analysis is elaborated as shown below:

[TABLE]

Clearly, we can see that in (4.2.3), the existence of the $\beta^{2}$ term in the potential energy part of (4.2.1) allows us to safely omit a part of of the Lagrangian for the limit $\beta<<1$. This means that we can say:

[TABLE]

Thus, we can say that the semi-relativistic Lagrangian for low velocities can be written as:

[TABLE]

Now this matches the form of the semi-relativistic Lagrangian (1.3) employed by Goldstein [3]. We can further proceed to say that if the same pattern of approximation is applied to the semi-relativistic equation of motion (4.2.2), we shall have:

[TABLE]

Using the same approximation rule (4.2.3), we can ignore the term with $\beta^{2}$ to write:

[TABLE]

[TABLE]

This equation is thus nearly the same as the usual equation of motion known classically, except for the usage of proper time $\widetilde{t}$ in the particle frame instead of $t$. From this equation, we can write a conserved quantity given as:

[TABLE]

As stated, this formulation shall only apply in the low velocity limit for weak potentials. Now we shall look at the relativistic Hamiltonian formulation under weak potentials.

4.3 Hamiltonian formulation under weak potential

Under circumstances of a weak potential ($2U(\bm{x})<<mc^{2}$) and low momentum $\left(\dfrac{|\bm{p}|}{mc}\approx 0\right)$, another way to write the relativistic energy (2.3.1) is:

[TABLE]

[TABLE]

[TABLE]

Thus, the Hamilton’s equations of motion (2.3.2) evolve into the form presented in [2]:

[TABLE]

Now we shall proceed to study the relativistic oscillator and its duality with the Kepler system.

5 Bohlin-Arnold Duality

Now that we have properly described the relativistic formulation for classical mechanical systems in a general scalar potential field, we shall now apply this formulation to two important mechanical systems frequently discussed in classical mechanics. They are the Hooke oscillator and Kepler systems. These systems are also dual to each other via a conformal transformation known as the Bohlin-Arnold-Vasiliev transformation.

5.1 Relativistic 2D Isotropic Oscillator and Kepler systems

One may ask how such a duality is a matter of concern here in the analysis of Newtonian gravity. While the Kepler potential is known to describe Newtonian gravity, it doesn’t seem possible to find a Hooke’s law potential that can be described as a result of curved space. The answer lies in the way Hooke’s oscillator mechanics are applied in physics; around equilibrium points. In the study of planetary motion, one encounters equilibrium points known as Lagrange points that allow planets to maintain stable orbits. It is locally around these points that one will find single particles to exhibit Hooke oscillatory motion, whose potential function can be described as the curvature of the local spacetime.

In a manner similar to (2.2.1), the metric of the relativistic gravitational oscillator can be given by:

[TABLE]

The Lagrangian corresponding to (5.1.1) according to (2.2.4) would be

[TABLE]

For planar motion $\theta=\frac{\pi}{2}$, the momenta are given by:

[TABLE]

So the relativistic radial equation would be given by:

[TABLE]

while the angular equation is given by:

[TABLE]

If we choose the proper time $\widetilde{t}$ as parameter

[TABLE]

then we can modify equation (5.1.4) in accordance with (2.2.10) to:

[TABLE]

For small oscillations, according to (4.2.1), the relativistic Lagrangian is:

[TABLE]

The Euler-Lagrange equation of motion that we can derive from (5.1.6) are:

[TABLE]

which does not match the form presented in [1]. However, for small oscillations during which the maximum velocities achieved are relatively small compared to the speed of light, we shall have according to (4.2.4) and (4.2.5):

[TABLE]

[TABLE]

which matches the form presented in [1].

Again, according to (2.2.1), the relativistic Kepler system can be given by:

[TABLE]

The Lagrangian corresponding to (5.1.10) according to (2.2.4) would be

[TABLE]

For planar motion $\theta=\frac{\pi}{2}$, the momenta are given by:

[TABLE]

So the relativistic radial equation would be given by:

[TABLE]

while the angular equation is given by:

[TABLE]

showing that the angular momentum is conserved for radial forces. If we choose the proper time $\widetilde{t}$ as parameter

[TABLE]

then we can modify equation (5.1.13) in accordance with (2.2.10) to:

[TABLE]

For small oscillations, according to (4.2.1), the relativistic Lagrangian is:

[TABLE]

The Euler-Lagrange equation of motion that we can derive from (5.1.15) are:

[TABLE]

Again, for relatively low velocities, we shall have according to (4.2.5):

[TABLE]

[TABLE]

We shall now briefly turn our attention to the Bohlin-Arnold duality to demonstrate semi- relativistic Kepler-Hooke duality.

5.2 Semi-Relativistic Kepler-Hooke duality

The Kepler-Hooke duality established by Bohlin, Arnold and Vasiliev [15, 16] is a connection between the two mechanical systems which according to Bertrand’s Theorem are the only two that are possible with closed, periodic orbits. This duality is established for the classical cases, but are not possible for the relativistic versions of the mechanical systems due to the $\Gamma$ factors involved in the equations of motion. Here, we show that it is again possible for the semi-relativistic equations of motion given by (5.1.9) and (5.1.18).

The Bohlin transformation is a conformal map given by:

[TABLE]

Now we must note that another Noether invariant, the angular momentum will change form under this transformation. We re-parametrize to preserve the form of angular momentum.

[TABLE]

[TABLE]

The velocity and acceleration transformation can be given using (5.2.1) and (5.2.2):

[TABLE]

Thus, the semi-relativistic equation of motion for oscillators (5.1.9) eventually becomes:

[TABLE]

[TABLE]

Using the conserved quantity $H$ from (4.2.6) for the oscillator system

[TABLE]

we can complete the transformation (5.2.3) using $\left(\frac{|\xi^{\prime}|^{2}}{2}+\frac{2k}{m}\right)=\frac{4H}{m}\frac{1}{|\xi|}=\kappa\frac{1}{|\xi|}$:

[TABLE]

showing that the transformation restores the central force nature of the system and produces the complex version of the Kepler equation (5.1.18).

6 Relativistic Lienard-type oscillator

In the study of dynamical systems, the Lienard system is a 2nd order differential equation named after French physicist Alfred-Marie Liénard [17]. It is a very generalized way to describe 1-dimensional motion under the influence of scalar potential and damping effects. Such differential equations were used to model oscillating circuits for applications in radios and vaccum tubes. For oscillatory systems, Liénard’s Theorem, under certain assumptions assures uniqueness and existence of a limit cycle for the system.

The equation of a damped 1-dimensional relativistic harmonic oscillator is:

[TABLE]

In contrast, the damped relativistic Lienard equation is:

[TABLE]

Under reparametrization $t\longrightarrow d\widetilde{t}=\gamma^{-1}dt$, we have $x^{\prime}=\frac{dx}{d\widetilde{t}}=\gamma\frac{dx}{dt}=\gamma\dot{x}$, letting us write:

[TABLE]

[TABLE]

Thus, under reparametrization, (6.1) becomes

[TABLE]

Remember that

[TABLE]

Thus, when $\dfrac{x^{\prime}}{c}\longrightarrow 0$, we will have $\gamma\longrightarrow 1$, which is the same as when $\dfrac{\dot{x}}{c}\longrightarrow 0$.

6.1 Integrability and Relativistic Chiellini condition

Now, the relativistic Chiellini condition is given by:

[TABLE]

Thus, using (6.1.1), and using the integrating factor $\Omega=\int^{\widetilde{t}}d\tau\ \gamma f(x)$, we can rewrite (6.2) as:

[TABLE]

[TABLE]

[TABLE]

Now, using the Chiellini condition (6.1.1), and referring to (6.2) we can see that

[TABLE]

[TABLE]

Thus, we have a conserved quantity given by

[TABLE]

Thus, we have a conserved quantity for a relativistic Liénard system. To solve it for the damped oscillator, we write $f(x)=\kappa$, $g(x)=x$.

6.2 Metric and Lagrangian

To deduce the metric from the relativistic equation of motion for a damped system (6.1), we shall execute a more elaborate 5-step procedure than the 3-step procedure for undamped systems, given by:

Convert the equation (6.1) to the non-relativistic version (with $\gamma=1$).

[TABLE] 2. 2.

Determine the reparametrization factor $\text{e}^{\Omega}$ by converting (6.2.1) to the form (2.2.12).

[TABLE]

[TABLE] 3. 3.

Deduce the undamped potential $U$ from (6.2.2) by factoring out $\text{e}^{2\Omega}$. 4. 4.

Comparing (6.2.2) to (2.2.13), we can formulate the damped effective Lagrangian $L_{d}$ for reparameterized time $\tau$, just as (2.2.8) can be derived from (2.2.13).

[TABLE]

[TABLE] 5. 5.

Deduce the classical damped effective Lagrangian $L_{ed}$ from (6.2.3) by multiplying $\text{e}^{-\Omega}$, and write the damped metric $ds^{2}_{d}$ from it.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Looking back at (6.1), we can take the non-relativistic version of the equation and say that

[TABLE]

If we reparametrize as $d\tau=\text{e}^{-\Omega}dt$ then we will have using the non-relativistic version of Chiellini condition (6.1.1)

[TABLE]

[TABLE]

[TABLE]

Arguing that $x=x(t)$ and from (6.2.5) we have the undamped potential $U({\bm{x}})$ and $g_{00}({\bm{x}})$:

[TABLE]

Using (6.2.6), the damped effective Lagrangian according to (6.2.3), is given from (6.2.5) as:

[TABLE]

Now, we can rewrite this as

[TABLE]

and finally, the metric with drag factor from (6.2.7) is (6.2.4) given by:

[TABLE]

where we can restore the integrating factor $\Omega$ back to the relativistic version $\Omega=\int^{t}d\widetilde{t}\ \gamma f(x)$.

This further goes to show that damped mechanical systems can be described by a metric that is spatially isotropic with a non-unity co-efficient. Furthermore, if it is a solution of Einstein’s equations, then we should have:

[TABLE]

so from (6.2.8) we get the metric:

[TABLE]

This topic will be elaborated upon in greater detail in another project where such spaces are studied as damped mechanical systems. 111Contact Hamiltonian mechanics [18] extend symplectic Hamiltonian mechanics [19], geometrically describing non-dissipative and dissipative systems, eg.: thermodynamics [20], mesoscopic dissipative mechanical systems [21], and mechanical systems drawing energy from a reservoir. The Hamiltonian is effectively provided by $I$ in (6.1.3). If we define a new variable $s$, as in [22], ignoring the decay-countering factor $\text{e}^{\Omega}$ gives the decaying Hamiltonian embedded in (6.1.3):

$s:=-\frac{1}{\alpha\left(\alpha+1\right)}\left(\frac{g}{f}\right)\gamma{\dot{x}},\qquad H=(\gamma\dot{x})^{2}-\frac{1}{\alpha\left(1+\alpha\right)}\left(\frac{g}{f}\right)^{2}+s.$

From the relativistic non-decaying Lagrangian of (6.2.8), the relativistic momentum under weak potential approximates to $p\approx\gamma\dot{x}$, which we can replace in $s$ and $H$ defined above to rewrite them, from which we can see that

$\dot{s}=f(x)\left[\frac{\partial H}{\partial p}p-H\right].$

However, as stated in [22], unless $f(x)=const$, we cannot recast the Liénard equation in contact form, which is simply the damped oscillator.

7 Conclusion

The form of relativistic Lagrangian and equations of motion in [1] also used by Goldstein [3], do not match with the formulations derived directly from solving for the geodesic from the spacetime metric. The latter formulation clearly reproduces the time dilation effects in a potential field as observed in the phenomena of gravitational redshifts [13], while the former does not. The Hamiltonian formulation shown in [2] applies only for weak potentials and low momentum. We effectively managed to write a Lorentz-covariant form of equations of motion in (2.2.16) for geodesics that does not match the classic non-relativistic form of such equations and describe a relativistic deformation of the Euler-Lagrange equation.

Since the familiar Lorentz transformation was designed for the special case of special relativity, where we deal with free particles without any potentials, it will not suffice in a more general case where a particle accelerates under the influence of potentials. The solution was to formulate a modified Lorentz transformation that will locally leave the metric invariant. The fact that such a transformation is limited to work locally is not surprising, given that according to the Equivalence principle, a curved space is locally diffeomorphically equivalent to a flat space, where a regular Lorentz transformation would be valid.

The Bohlin-Arnold duality [15, 16] is not valid for the relativistic oscillator under this direct formulation due to the $\Gamma$ factors in the equation of motion. However, it is possible to use a semi-relativistic approximation for weak potential and low velocity to produce the type of semi-relativistic Lagrangian in [1, 3]. Using such approximation, and replacing the time with the proper time in the particle frame, the semi-relativistic equation of motion takes the classical form, allowing the Bohlin-Arnold duality to be valid under this approximation.

We have written the relativistic version of the Liénard equation, and deduced its conserved quantity, Lagrangian and metric. The conserved quantity was derived after first redefining the Chiellini integrability condition relativistically. The metric derived for the Liénard oscillator implies metrics that are spatially isotropic with non-unity co-efficient function. Thus, if the Schwarzschild, and most solutions to Einstein’s equations are rewritten to spatially isotropic co-ordinates, they shall be found to exhibit damped motion. This implies that the dynamics of such solutions can be examined as some form of the relativistic Liénard mechanical system.

So far, this article helps to revise and generalize our fundamental formulation of relativity, while showing how familiar results can still be reproduced under suitable approximations. The results of this article can be applied in future along the direction of canonical ADM gravity and relativistic quantum mechanics as covered in articles like [23, 24] respectively.

Acknowledgement

We thank Professors Ángel Ballesteros and Alexei A. Deriglazov for their correspondence and advice, which were invauable in developing this article, and Prof. Gary Gibbons furthermore for his collaboration in [11, 12] and encouragement. Furthermore, we also wish to express special gratitude to the anonymous referee whose valuable advice and questions helped improve and expand the content of this manuscript. The research of Partha Guha is supported by FAPESP through Instituto de Fisica de São Carlos, Universidade de Sao Paulo with grant number 2016/06560-6.