There are some famous corollaries of the equivalence principle:
1. Unruh effect is locally equivalent to Hawking radiation near the horizon, therefore "static" observer outside a blackhole will see radiation, while a free-fall observer will not.
2. Firewall, i.e., a free-fall observer will be evaporated at the horizon, is forbidden by the equivalence principle.
However, there is no physical law saying that no inertial observer will see thermal radiation or Firewall (it is bizarre, but not forbidden) in flat spacetime. The problem is the equivalence principle only restricts the evolution rule of the system but has no hint on boundary conditions. When people are using the above corollaries, are they assuming some particular boundary conditions?
How can we apply the equivalence principle in general?
Hawking radiation is an illustration of the Unruh effect, that, in turn, just describes the fact that, in curved spacetime, the vacuum can't be globally defined.
In flat spacetime the vacuum can be globally defined, that's why an inertial observer can't detect radiation, i.e. a state that's not the vacuum, by definition.
The equivalence principle doesn't have anything to do with this discussion, since whether curved spacetime is exclusively defined by the metric or not doesn't enter the picture. Whether it does or not the vacuum isn't globally defined and that's all that matters.
Stam Nicolis Thank you for your reply. I see there is no unique vacuum in curved spacetime, but the problem can be rephrased in such way: why do we assume the free-fall observer is in his vacuum? Particularly, different geodesic observers may have different vacuum state in general.
So is there a simple way to decide the state (boundary condition) in curved spacetime?
The two terms ``free fall observer'' and ``vacuum state'' are equivalent, they're two expressions of the same physical concept.
The geodesic equation has a unique solution, which is another way of saying that all inertial observers are equivalent.
So it's just not possible to have more than one inertial observer.
What happens in curved spacetime, that doesn't happen in flat spacetime is, howver, that the observers at different points of the geodesic can't define equivalent inertial frames-that's what spacetime curvature expresses.
In curved spacetime it's always possible to define an inertial frame and, therefore, the vacuum state at any given point-but there doesn't exist a global transformation that maps the vacuum state in one point to the vacuum state in another. Otherwise spacetime would have been flat.
The boundary conditions, of course, play a role. But, once more, in curved spacetime, in general, the space of states depends on the point in spacetime and can't be globally defined. Depending on the boundary conditions, it can be possible to map the spacetime geometry to properties of the boundary; this was known, already, in general relativity and has been a topic of considerable activity within the so-called holographic correspondence, which shows what properties of the bulk can be defined on the boundary.
These might be useful: Article The World as a Hologram
Article Dimensional Reduction in Quantum Gravity
They were the harbingers of the holographic correspondence, that's, still, work in progress.
A more recent, introductory, presentation of the correspondence may be found here: Article Introduction to AdS-CFT
Draw a curve and notice that it looks flat when observed through a microscope. In this flat domain, general relativity becomes teleparallel equivalent general relativity. Down here, the covariant spin connection is used to regularize quantum general relativity and the renormalized action becomes just the integral of the mass over time. Einstein chose to use curvature, and not torsion, to define general relativity for the sole purpose of elevating the equivalence principle. Down here we separate gravity from inertia and quantize just gravity, using the covariant spin connection as a regulator. Down here the equivalence principle is replaced by a new boson, dubbed the Lorentz boson, that couples to fermion number at gravitational strength. Down here Bell’s theorem is broken by torsion. Down here becomes macroscopic, and macroscopic entanglement is produced, before fermions localize and couple to Lorentz bosons, forcing entanglement entropy to become causal. That all happens down here. Just have a look through the microscope.
Are you sure they won't see any radiation in free fall? The radiation is being pulled in to the black hole after all. At the event horizon, the observer will stretch as if they where on ground that has heavy gravity. If space and time are being twisted then the observer is bound to be twisted as well once passed the event horizon, then ultimately get ripped apart.
“How to apply the equivalence principle?”
- for weak Gravity fields the equivalence principle in sense that, despite that they are fundamentally different, inertial and gravitational masses are equivalent, was, and is till now, well applicable in Newtonian Gravity law; whereas that
“…There are some famous corollaries of the equivalence principle:
1. Unruh effect is locally equivalent to Hawking radiation near the horizon, therefore "static" observer outside a blackhole will see radiation, while a free-fall observer will not.
2. Firewall, i.e., a free-fall observer will be evaporated at the horizon, is forbidden by the equivalence principle.…..”, etc.
- are examples when the equivalence principle is used fundamentally erroneously, though in this sense it is postulated in the GR, where, because of the fundamental phenomena/notions, first of all in this case “Matter”– and so everything in Matter, i.e. “particles”, “fields”, etc., “Consciousness”, “Space”, “Time”, “Energy”, were fundamentally completely transcendent/uncertain/irrational for the author – and for mainstream physicists till now,
- and so in every case, when some authors develop some fundamental theory, the result completely obligatorily logically is nothing else than some transcendent fantastic mental constructions.
Including in framework of the GR really fundamentally impossible transcendent material objects/effects/events/processes are quite legitimate, besides in the quote above an recent example see in SS post in https://www.researchgate.net/post/Is-it-true-that-physicists-do-not-want-to-understand-quantum-mechanics/190.
Gravity fundamentally is nothing else than some fundamental Nature force, which is in a few traits similar to the fundamental Nature Electric force, which – as, of course, Electric Force also, in the Matter’s fundamentally absolute, fundamentally flat, and fundamentally “Cartesian”, [5]4D spacetime with metrics (cτ,X,Y,Z,ct), which fundamentally cannot be “contracted”, “dilated”, “curved”, “warped”, etc.,
more see the Shevchenko-Tokarevsky’s informational physical model
https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics , section 2.9 “Mediation of the forces in complex systems”,
- about what really happens at the “event horizon” see
https://www.researchgate.net/publication/355361749_The_informational_physical_model_and_fundamental_problems_in_physics, section “Cosmology”.
More see the links above, SS posts in https://www.researchgate.net/post/What_is_the_physical_origin_of_equality_between_the_inertial_mass_and_gravitational_mass/13are relevant to this thread question as well.
Cheers
The equivalence principle in a standard model gauge theory says that aikyons minimally couple to fermion number at gravitational strength. See (PDF) Testable Quantum Gravity Theory (researchgate.net).