Abstract
In this paper, we derive the non-commutative corrections to the maximal acceleration in the Doplicher–Fredenhagen–Roberts (DFR) space-time and show that the effect of the non-commutativity is to decrease the magnitude of the value of the maximal acceleration in the commutative limit. We also obtain an upper bound on the acceleration along the non-commutative coordinates using the positivity condition on the magnitude of the maximal acceleration in the commutative space-time. From the Newtonian limit of the geodesic equation and Einstein’s equation for linearised gravity, we derive the explicit form of Newton’s potential in DFR space-time. By expressing the non-commutative correction term of the maximal acceleration in terms of Newton’s potential and applying the positivity condition, we obtain a lower bound on the radial distance between two particles under the gravitational attraction in DFR space-time. We also derive modified uncertainty relation and commutation relation between coordinates and its conjugate, due to the existence of maximal acceleration.
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Notes
Note that here we have absorbed the factors of \(\pi \) into the definition of M.
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Acknowledgements
S.K.P. thanks UGC, India, for the support through JRF scheme (id.191620059604). V.R. thanks Government of India, for support through DST-INSPIRE/IF170622.
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Appendix A
Appendix A
In this appendix, we derive a length-scale-dependent commutation relation between commutative coordinate and its conjugate using the expression for the maximal acceleration obtained in the DFR space-time.
The uncertainty relation between energy and an arbitrary function of time f(t) can be expressed as [36,37,38,39]
Using the above relation, we write down uncertainty relation between energy and velocity and that between energy and position as,
and
respectively. Multiplying the above uncertainty relations, given in Eqs. (A.2) and (A.3), we get
where we have used the definitions \(v=\frac{dx}{dt}\) and \({\mathcal {A}}=\frac{dv}{dt}\). Now we express \(\Delta E\) as \(\Delta E=v\Delta p\), and we get
From the special theory of relativity, we infer that the uncertainty in velocity of the particle cannot exceed the velocity of light, i.e. \((\Delta v)^2=<v^2>-<v>^2\le v_{max}^2\le c^2\), so we take \((\Delta v)v\le c^2\). After using this relation in the above equation, we find that
After setting \(\Delta x\) in the RHS with Compton wavelength, \(\lambda _c=\frac{\hslash }{mc}\) we get
It has been shown in [36,37,38,39] that one obtains the maximal acceleration in the commutative space-time, by using the uncertainty relation between the spatial coordinate and its conjugate momenta in the above expression.
Now we generalise Eq. (A.7) into DFR space-time by replacing the commutative spatial coordinate \(x_i\) and its conjugate \(p_i\) with DFR spatial coordinate \(X_i\) (where \(X_i=(x_i,\theta _i/\lambda )\)) and its conjugate \(P_i\) (where \(P_i=(p_i,\lambda k_i)\)). We also rewrite \({\mathcal {A}}\) using \({\mathcal {A}}_{max}\) expression given in Eq. (3.9). Thus, Eq. (A.7) becomes
Thus, we find that the commutation relation between the DFR coordinate and its conjugate momenta gets modified as
Note that Eq. (A.9) gives a modified commutation relation between the DFR spatial coordinates and their conjugates. Thus, we get a modified commutation relation between \(x_i\) and \(p_i\) apart from that between \(\theta _i\) and \(k_i\). Therefore, we have
This shows that the existence of maximal acceleration implies a minimal length-scale-dependent modification to the commutation relation between coordinate and its conjugate. In [68], a length-scale-dependent commutation relation known as GUP, between coordinate and its conjugate given by
has been analysed and corresponding uncertainty relation between x and p was obtained. Comparing Eq. (A.10) with Eq. (A.11) and setting \(\lambda =l_{Pl}\), we find the generalised uncertainty principle parameter \(\beta \) in terms of the \(a_{\theta }\) as \(\beta =-\frac{a_{\theta }^2\hslash ^4}{4m^2c^6p^2}\).
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Harikumar, E., Panja, S.K. & Rajagopal, V. Maximal acceleration in a Lorentz invariant non-commutative space-time. Eur. Phys. J. Plus 137, 966 (2022). https://doi.org/10.1140/epjp/s13360-022-03195-4
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DOI: https://doi.org/10.1140/epjp/s13360-022-03195-4