REVIEW
A Generalized Uncertainty Principle in Quantum Gravity
High Energy Physics - Theory · Submitted 17 Jan 1993 · 12 references
Pith's verdict: The generalized uncertainty principle was already known from strings; what's new here is a model-independent black-hole-microscope route to it, and the paper is honest about the parts it can't pin down.
T0 · AI review
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Referee report
The paper
The paper claims
Maggiore argues that measuring a black hole's horizon with Hawking radiation forces a minimum length of order the Planck length, and that the resulting uncertainty relation matches the one string theory predicts.
The reading
What carries the argument
The load-bearing device is a two-error decomposition of a single measurement. The first error, $Δx_{(1)} ∼ λ$ (more precisely $λ sin θ$), is ordinary optics. The second, $Δx_{(2)} ∼ GΔM ∼ L_{Pl}^2/λ$, comes from the horizon's own motion during emission; for a general Reissner–Nordström hole it is bounded below by $2GΔM$ and is saturated only by the Schwarzschild case, so the fact that a mass- and charge-independent minimum survives is a nontrivial feature of the geometry. Combining the two linearly and using $λ/sin θ ≥ λ$ to eliminate the apparatus angle gives a relation free of experimental detail, at the acknowledged cost of losing control of the numerical coefficient. The paper then packages the result as an interpolating function $R_*(M)$ that behaves as $ℏ/M$ for $M ≪ M_{Pl}$ and as $2GM$ for $M ≫ M_{Pl}$ and has a minimum of order $L_{Pl}$ in between, writing the generalized principle compactly as $Δx ≥ R_*(Δp)$.
Extended reading notes
Core claim
The paper's claim is that a generalized uncertainty principle is not a peculiarity of string theory but a consequence of any theory in which black holes radiate. The argument is a Gedanken experiment: an extremal Reissner–Nordström black hole is probed by a photon of wavelength $λ$ and observed through the Hawking photon it re-emits at right angles. Besides the standard microscope error $Δx_{(1)} ∼ λ$, the horizon radius changes discontinuously as the hole's mass drops from $M + ΔM$ back to $M$, and the paper argues that no operational meaning attaches to whether the outgoing photon carries information about the hole before, during, or after that change. That second error scales as $GΔM = L_{Pl}^2/λ$, so combining the two errors linearly gives $Δx ≳ λ + L_{Pl}^2/λ$, minimized at the Planck length. In momentum variables the same statement becomes $Δx ≳ ℏ/Δp + cGΔp$, agreeing in form with the string-theory result. From this Maggiore draws a strong corollary: the horizon of a black hole with mass below the Planck mass has no operational definition, and the Compton radius $ℏ/M$ and the Schwarzschild radius $2GM$ should be treated as mutually exclusive descriptions of an object, valid in the limits $M ≪ M_{Pl}$ and $M ≫ M_{Pl}$ respectively.
Load-bearing premise
The central claim depends on adding the two independent measurement errors linearly rather than in quadrature; the paper chooses this 'for the sake of definiteness', and the term linear in momentum, hence the agreement with string theory, exists only under that choice.
Editorial extensions
If this is right
- A minimum measurable length of order the Planck length follows from Hawking radiation plus classical horizon geometry, without assuming strings or any specific quantum gravity.
- The horizon radius relation $R_h = GM[1 + (1 - Q^2/GM^2)^{1/2}]$ ceases to be operationally testable below Planckian precision, so it is a definition rather than a falsifiable prediction at those scales.
- There is no operational notion of a black hole with mass below the Planck mass, and no reason to expect gravitational substructure at the Schwarzschild radius of an elementary particle.
- The Compton radius and the Schwarzschild radius are mutually exclusive attributes of an object of mass $M$, each meaningful only in its own limit.
- Because the linear-in-momentum term survives the pointlike limit $α' → 0$ of string theory, the result suggests a consistent quantum gravity cannot be built from pointlike objects.
Reading between the lines
- The claimed agreement with string theory rests on the choice to add the two errors linearly; adding them in quadrature would give $\sqrt{λ^2 + L_{Pl}^4/λ^2}$, which has no term linear in momentum and therefore no string-theory form. The paper flags the linear choice as made 'for the sake of definiteness' but does not explore how much of the agreement depends on it.
- The second error presumes the excited hole returns to extremality by emitting a single photon of the same wavelength; the paper notes that near-extremal holes are not thermal, so the emission spectrum is an assumption rather than a derived input.
- If the mechanism is right, the same two-error bookkeeping should apply to any horizon whose radius depends linearly on the energy absorbed, which suggests a broader class of probes (acoustic or analogue horizons, for instance) where an analogous minimum resolution could be computed and compared.
- The interpolating function $R_*(M)$ is presented as suggestive rather than derived; a specific quantum gravity that fixes the coefficient $c$ would turn the whole argument into a quantitative prediction rather than a consistency check.
Original abstract
We discuss a Gedanken experiment for the measurement of the area of the apparent horizon of a black hole in quantum gravity. Using rather general and model-independent considerations we find a generalized uncertainty principle which agrees with a similar result obtained in the framework of string theories. The result indicates that a minimum length of the order of the Planck length emerges naturally from any quantum theory of gravity, and that the concept of black hole is not operationally defined if the mass is smaller than the Planck mass.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We discuss a Gedanken experiment for the measurement of the area of the apparent horizon of a black hole in quantum gravity. Using rather general and model-independent considerations we find a generalized uncertainty principle which agrees with a similar result obtained in the framework of string theories. The result indicates that a minimum length of the order of the Planck length emerges naturally from any quantum theory of gravity, and that the concept of black hole is not operationally defined if the mass is smaller than the Planck mass.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We discuss a Gedanken experiment for the measurement of the area of the apparent horizon of a black hole in quantum gravity. | contribution 0.76 | 0.70 | 0.03 | asserted 0.71 |
| s1 | Using rather general and model-independent considerations we find a generalized uncertainty principle which agrees with a similar result obtained in the framework of string theories. | result 0.74 | 0.94 | 0.25 | asserted 1.00 |
| s2 | The result indicates that a minimum length of the order of the Planck length emerges naturally from any quantum theory of gravity, and that the concept of black hole is not operationally defined if the mass is smaller than the Planck mass. | interpretation 0.97 | 0.79 | 0.83 | hedged possibility 0.97 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:hep-th/9301067 by the authors.
References
How the literature uses it
156 arXiv papers cite this in 214 sentences
Papers citing it · 118 sentences without a confident label
Compares with it 3 papers
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Accordingly, a generalization of the uncertainty principle has been concluded, which agrees well with the one deduced from the string theories [ref] .
Generalized Uncertainty Principle: Approaches and Applications · Black hole physics · 0.90
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Table [ref] summarizes a short comparison between quadratic [ref] and linear [ref] GUP approaches in different aspects, namely Heisenberg algebra, minimal length uncertainty and maximal moment uncertainty and maximal measured moment.
Impacts of Generalized Uncertainty Principle on Black Hole Thermodynamics and Salecker-Wigner Inequalities · Black Hole Lifetime · 0.81
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The same relation [ref] and the same conclusions were reobtained by Maggiore [ref] , with a “model independent” argument quite close to that of Mead; probably Maggiore was not aware of Mead's analysis.
Quantum Spacetime: a Disambiguation · Planck length and probes · 0.90
Cites it as support 2 papers
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In fact, the MLUR can be motivated by fairly elementary general relativistic considerations independent of string theory, which suggests that it is a universal feature of quantum gravity [ref] .
On the Minimal Length Uncertainty Relation and the Foundations of String Theory · Introduction · 1.00
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Thus, there are strong indications that there exists a minimum measurable length scale for the spacetime [ref] .
Inflationary universe in the presence of a minimal measurable length · Introduction · 0.97
Cites it as background 81 papers
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The quadratic GUP approach was introduced in Ref. [ref]
Impacts of Generalized Uncertainty Principle on Black Hole Thermodynamics and Salecker-Wigner Inequalities · Generalized Uncertainty Principle · 0.93
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Furthermore, we will study the feature that most QG theories predict, namely a minimum measurable length and the related Generalized Uncertainty Principle (GUP) [ref] .
Test of Quantum Gravity in Statistical Mechanics · Introduction · 0.85
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One of the phenomenological models, which describes such discreteness is the Generalized Uncertainty Principle (GUP), which has been studied in great detail [ref] .
Baryon Asymmetry from the Generalized Uncertainty Principle · Introduction · 0.96
Discussion
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