Abstract
Since the pioneering works on isolated black hole thermodynamics in the 1970s, the so-called first law of mechanics has gradually been extended to a variety of binary systems, such as black hole binaries, magnetized neutron stars or spinning point particles. They have also been generalized to account for the wide variety of relativistic orbits: quasi-circular, eccentric, and black hole geodesics. The aim of this chapter is to present these first laws, explain what they are about, and understand why they have become a central tool in the theory of relativistic mechanics and gravitational waves.
Yesterday I received from Miss Noether a very interesting paper on the generation of invariants. I am impressed by the fact that these things can be understood from so general a point of view. It would have done the Old Guard of Göttingen no harm to be sent back to school under Miss Noether. She really seems to know her trade!
A. Einstein,
Letter to D. Hilbert (1918)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It is possible to give another justification of the term surface gravity, by identifying \(\kappa \) as the force exerted by an operator at infinity necessary to keep a unit mass at rest close to \(\mathscr {H}\), see [355].
- 2.
“Mechanics” because it holds for any BH spacetime, even for those with disconnected components, for example binary BHs.
- 3.
If there are two BHs, then this horizon \(\mathscr {H}\) has two disconnected components \((\mathscr {H}_1,\mathscr {H}_2)\) and \(k^a\), when evaluated at one component \(\mathscr {H}_{\textrm{i}}\) is tangent to its respective generators.
- 4.
In the case of a nonisentropic, but otherwise irrotational flow (vanishing vorticity) only the third term vanishes. Other fluid configurations are discussed in [382].
- 5.
Their work also includes of logarithmic contributions of 4PN and 5PN orders, which we omit here for clarity, but which are also checked to verify the first law in their paper.
- 6.
We limit the present discussion to 3PN, even though this Hamiltonian is known at 4PN [462, 463]. However, at that level it brings new, non-local effects, which are precisely the content of the first law derived in the next paragraph.
- 7.
This action is defined in the same way in Newtonian mechanics, and we shall use it extensively in the second part of this thesis, in particular in Chap. 10.
- 8.
These are nothing but the Poincaré invariants, used already in the ADM context, for the Hamiltonian first laws. They will also be used in the second part of this thesis, cf. Chap. 10. However, see [378] for details on such a construction of actions in a relativistic context.
- 9.
Even though \(\hat{E}_{\text {SF}}\) may appear like a first-order term (multiplied by q), it does contains all second-order perturbations through the masses (4.43).
- 10.
Strictly speaking, they stumbled upon integral that could not be computed with available GSF codes and data at the time. This integral provided a key information between the asymptotic quantities and the local properties of the particle, which is precisely the type of information the first law provides, by good fortune, to quote the authors of [482].
- 11.
These ambiguity parameters find their origin in the use of regularization schemes for the calculation. Typically, the so-called “UV” divergences arise from the use of point-particle models (introducing a singularity), while “IR” divergences come from the infinite temporal range of the tail effects, discussed in Sect. 4.1.3.
- 12.
To reach this high PN order, their method uses at once PN, post-Minkowskian, multipolar-post-Minkowskian, GSF and EOB, and is therefore coined the tutti-frutti method.
- 13.
By Cartan’s magic formula we mean \(\mathcal {L}_{\xi }\boldsymbol{L}=\textrm{d}(\xi \cdot \boldsymbol{L}) + \xi \cdot \textrm{d}\boldsymbol{L}\). It is due to Élie Cartan (in [499]) and not his son Henri, as it is sometimes claimed in textbooks.
- 14.
If \(\xi ^a\) Lie derives all the dynamical fields in the background (\(\mathcal {L}_{\xi }\phi = 0\)) and not merely the metric \(g_{ab}\), then the right-hand side of (4.54) and (4.63) vanish identically, yielding \(\textrm{d}\bigl ( \delta \boldsymbol{Q}_{\textsf {g}}[\xi ] - \xi \cdot \boldsymbol{\Theta }_{\textsf {g}} \bigr ) = - \textrm{d}\bigl ( \delta \boldsymbol{Q}_{\textsf {m}}[\xi ] - \xi \cdot \boldsymbol{\Theta }_{\textsf {m}} \bigr )\). This condition is stronger than merely equating the right-hand sides of Eqs. (4.54) and (4.63), but it gives rise to the same identity (4.64).
- 15.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Ramond, P. (2023). First Laws of Mechanics. In: The First Law of Mechanics in General Relativity & Isochrone Orbits in Newtonian Gravity. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-031-17964-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-031-17964-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-17963-1
Online ISBN: 978-3-031-17964-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)