Maximal acceleration in a Lorentz invariant non-commutative space-time

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.

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]

$$\begin{aligned} \Delta E\Delta f(t) \ge \frac{\hslash }{2}\frac{df(t)}{dt} \end{aligned}$$

(A.1)

Using the above relation, we write down uncertainty relation between energy and velocity and that between energy and position as,

$$\begin{aligned} \Delta E\Delta v(t) \ge \frac{\hslash }{2}\frac{dv}{dt} \end{aligned}$$

(A.2)

and

$$\begin{aligned} \Delta E\Delta x(t) \ge \frac{\hslash }{2}\frac{dx}{dt}, \end{aligned}$$

(A.3)

respectively. Multiplying the above uncertainty relations, given in Eqs. (A.2) and (A.3), we get

$$\begin{aligned} \big (\Delta E\big )^2\Delta x\frac{\Delta v}{v}\ge \frac{\hslash ^2}{4}{\mathcal {A}} \end{aligned}$$

(A.4)

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

$$\begin{aligned} \big (\Delta p\big )^2\Delta x\big ({\Delta v}\big )v\ge \frac{\hslash ^2}{4}{\mathcal {A}}. \end{aligned}$$

(A.5)

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

$$\begin{aligned} \big (\Delta x\Delta p\big )^2c^2\ge \frac{\hslash ^2}{4}{\mathcal {A}}\Delta x \end{aligned}$$

(A.6)

After setting $\Delta x$ in the RHS with Compton wavelength, $\lambda _c=\frac{\hslash }{mc}$ we get

$$\begin{aligned} \big (\Delta x\Delta p\big )^2\ge \frac{\hslash ^3}{4mc^3}{\mathcal {A}}. \end{aligned}$$

(A.7)

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

$$\begin{aligned} \big (\Delta X\Delta P)^2 \ge \frac{\hslash ^2}{4}\Big (1-\frac{\lambda ^2a_{\theta }^2\hslash ^2}{2m^2c^6}\Big ). \end{aligned}$$

(A.8)

Thus, we find that the commutation relation between the DFR coordinate and its conjugate momenta gets modified as

$$\begin{aligned}{}[X_i,P_j]=i\hslash \Big (1-\frac{\lambda ^2a_{\theta }^2\hslash ^2}{4m^2c^6}\Big )\delta _{ij}, \text { where }i,j=1\text { to }6. \end{aligned}$$

(A.9)

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

$$\begin{aligned}{}[x_i,p_j]=i\hslash \Big (1-\frac{\lambda ^2a_{\theta }^2\hslash ^2}{4m^2c^6}\Big )\delta _{ij},\text { where }i,j=1,2,3. \end{aligned}$$

(A.10)

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

$$\begin{aligned}{}[x_i,p_j]=i\hslash \Big (1+\frac{\beta l^2_{Pl}p^2}{\hslash ^2}\Big )\delta _{ij}, \text { where }i,j=1,2,3. \end{aligned}$$

(A.11)

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 $\beta =-\frac{a_{\theta }^2\hslash ^4}{4m^2c^6p^2}$.