Minimum Length in Quantum Gravity
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Quantum gravity is a theory in the making. It aims to unify the theories of quantum mechanics and general relativity. A “virtual” particle can be considered as a particle which exists in space-time for a very short amount of time. As the scale of space-time decreases, the energy of “virtual” particles increases. These virtual particles come into existence because the uncertainty principle dictates that as the scale of space-time decreases, energy and “virtual” particles come into existence for a short amount of time and then annihilate. According to the theory of general relativity, energy curves the fabric of space-time. This further implies that at small scales the energy fluctuations would be significant enough to cause a reverberation on large scales where an observable might exist.
In the study of quantum gravity a virtual black hole is said to be a black hole which exists for a short amount of time. The manner in which such a phenomenon occurs is due to a quantum fluctuation of space-time. Theory suggests that some properties of a virtual black hole need to include quantities such as a Planck mass and a total lifetime of a Planck time. A Planck mass is just a unit of mass and is given by:
= = 2.176 × 10-8 kg (1)
A Planck time is defined as the time it would take for a single photon of light to travel the distance of a Planck length (1.616 x 10-35 m). This quantity is given by:
= 5.391 x 10-44 s (2)
Now a virtual black hole can be thought of as an example of quantum foam. Quantum foam is a theoretical description for the foundation of the fabric of the entire universe. This concept is effective when theorizing about virtual black holes since it provides a qualitative description of the turbulence encountered on scales of the Planck length.
Moreover, when considering a generalized time-energy uncertainty relation in quantum gravity there is a prediction which exemplifies a paradox. This prediction states that when energy is created at a Planck scale this excitation can exist for a very, very long period of time. The reason why this does not make sense is because for the excitation to occur at a Planck scale the mass of the virtual object has to have a Planck mass and since the Planck mass is so small the virtual object cannot exist for a long time at all. It has to exist at a timescale of a Planck time. This shakes the foundation of particle physics and also is inconsistent with phenomena occurring in the universe. The generalized time-energy relation can safely be applied to classical macroscopic objects; this would make sense because classical macroscopic objects have a mass that is much larger than that of the Planck mass.
The real paradox comes into play when considering virtual black holes. Since virtual black holes are only a concept of a theory one would expect for the theory to be valid that there would be some sort of evidence or mass spectrum of them. This is still undetermined. However, the argument that virtual black holes present themselves within the quantum foam structure still holds. Any general quantum theory states that at zero temperature the contribution of virtual states to the system is defined by the time that they exist for. So when considering the generalized uncertainty principle there is a contribution of virtual black holes that are larger than the Plank mass limit (normal virtual black holes). Since they are larger they will have a larger life time. These slightly larger black holes are known as extremal black holes. These black holes are said to emit no Hawking radiation and are stable. This is weird because the cut-off length (Planck length) determines that space-time is filled with Planck mass black holes which have a life time of a Planck time.
Proton decay is a theoretical concept of radioactive decay in the domain in which a proton decays into smaller and lighter particles. This phenomenon has not yet been observed but the concept is valid when talking about the theory of quantum gravity. When gravity is taken into consideration in a quantum theory the virtual Planck mass black holes are said to provoke proton decay. Virtual particles that are of order of magnitude 103MPl (where MPl is the Planck mass) will induce proton decay. This fact is questionable since it would raise eyebrows about the validity of the theory.
Since virtual black holes cannot be observed directly by experiment we have to remember that they will just be “virtual”. The manner in which their existence can be verified or observed is by observing their contribution or interaction with lower energy physical events. There is no solidified theory of quantum gravity just yet but with the discussion of certain quantities there is a still a lot of room for improvement and development. Although the implication of virtual black holes that have a mass that is larger than that of the Planck mass causes for some discrepancy it should still be considered as a basis for the theory. The standard model of particle physics predicts the fraction of particles which decay individually with respect to the total number of particles that decay is not altered by the presence of virtual black holes that are much heavier than the limit of the Planck mass. This fact still needs to be resolved and explained in detail. Moreover, such phenomenona is expected when constructing a valid theory of virtual black holes.
In conclusion, in order to construct or derive a strong theory of quantum gravity which is valid at higher energy scales there needs to be an assumption that the theory will be simple in order for it to be compared to existing theories for ideas that will make it branch the various existing theories. If this theory can be made concrete there will be many fundamental questions that will be answered; fundamental questions involving problems associated with black holes and the origins of the universe.
 C. Bambi and K. Freese, “Dangerous Implications of a minimal length in quantum gravity”