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Recorded: 9/10/2017 Released: 4/14/2018

Jim talks to Randy about the different ways in which the equivalence principle of general relativity can be formulated. More than just the equivalence of accelerations, the different possible meanings of the equivalence principle mean different things about how gravity works. From weak to strong, from Einstein's equivalence principle to Schiff's conjecture, the implications of these theories are explored.

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Notes:

1. The papers we read for this program:

- Sotiriou, T., Faraoni, V., and Liberati, S, "Theory of Gravitation Theories: A No Progress Report."Intl J Mod Phys D 17, 399 (2008). [arXiv]
- Di Casola, E. and Liberati, S, "Nonequivalence of Equivalence Principles." Am J Phys 83, 39 (2015). [arXiv

2. My review of Will's book, which I talk about a little too much in this podcast.

3. Related Episodes of Physics Frontiers:

- Physics Frontiers 33: The Positive Energy Theorem
- Physics Frontiers 31: The Parameterized Post-Newtonian Formalism
- Physics Frontiers 29: Gravitational Alternatives to Dark Energy
- Physics Frontiers 23: Dark Energy
- Physics Frontiers 10: Requirements for Gravitational Theories
- Physics Frontiers 9: f(R) Theories of Gravity
- Physics Frontiers 1: G4V: The Gravitational 4-Vector Formulation of Gravity

4. Our subreddit.

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A review of "Theory of Gravitation Theories" from the arXiv_plus subreddit:

This delightfully named paper explains the relationship between the equivalence principles – the three equivalence principles listed in Will’s book: weak, Einstein, and strong – and the modified gravities that serve as competitors to general relativity (GR) with a cosmological constant in the quest to explain dark energy. The meat of the text centers around showing that a theory of gravity is a cluster of mathematical representations of that theory that are linked in an analogous way as different gauges in electromagnetism, although no general method of transformation is given, and that in order to satisfy the intermediate level of equivalence only one of these representations must satisfy the metric postulates. This ambiguity in representation is reflected in the way in which the additional fields of modified gravities are represented: whether the field is additional “matter” or a coupling that changes the “geometry” is a matter of the representation, not the theory itself.

The metric postulates simply ask a theory to (1) produce a metric g

^{mn}that describes the geometry of space-time and (2) admit only stress-energy tensors whose covariant derivative is zero. This is what you have in GR. It has been shown that the weakest version of the equivalence principle, which in its simplest form means that gravitational provides a preferred set of trajectories for small, uncharged particles to take without reference to their mass, is not, in itself, sufficient to establish these metric postulates. Instead, two other postulates are required, both self-explanatory: Local Lorentz Invariance and Local Position Invariance. These three postulates together form the Einstein equivalence principle and, along with some reasonable limitations on the mathematics, are sufficient to create a general class of metric theories of gravity with an additional scalar field (scalar-tensor theories). Each theory itself will be a cluster of the mathematical representations of the same.

One of the take-aways from this paper is that common procedures for determining if a theory satisfies EEP have made an error of checking single mathematical representations in those clusters against the metric postulates or Local Lorentz Invariance and ignoring the equivalent representations. The main example in the text compares the Jordan and Einstein frames in scalar-tensor theories, showing that they are equivalent, but the Einstein representation does not satisfy the metric postulates. However, since the Jordan frame representation of a scalar-tensor theory does satisfy the metric postulates, the theory itself does. This does, however show an issue with the frames that do not satisfy the postulates: changes need to be made in interpreting their metric tensors or geodesic equations, or even the connection in that frame. For example, in moving to the Einstein frame, you add an additional term in the geodesic equations that can be interpreted as:

(1) The gradient of the scalar field

(2) The variation of the masses of elementary particles in space

(3) Changes in the “unit of mass” along the trajectory

Although I’m not sure if (2) and (3) are the same or that the relative masses also change, making a coherent standard impossible.

Unfortunately, according to the authors, only individual mathematical representations of modified gravities exist in the literature, not more general theories that encompass the all of the representations of those theories, and so miscategorizations are likely. More unfortunately, a way forward towards a more abstract definition of individual theory clusters is not given in the text.

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