Chen, Carley, Fang, and Sadeh

# PURPOSE IN THE MACHINE: DO TRAFFIC SIMULATORS PRODUCE DISTRIBUTIONALLY
EQUIVALENT OUTCOMES FOR REINFORCEMENT LEARNING APPLICATIONS?

Rex Chen
Kathleen M. Carley
Fei Fang
Norman Sadeh
School of Computer Science
Carnegie Mellon University
4665 Forbes Avenue
Pittsburgh  PA 15213  USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

Traffic simulators are used to generate data for learning in intelligent
transportation systems (ITSs). A key question is to what extent their
modelling assumptions affect the capabilities of ITSs to adapt to various
scenarios when deployed in the real world. This work focuses on two simulators
commonly used to train reinforcement learning (RL) agents for traffic
applications, CityFlow and SUMO. A controlled virtual experiment varying
driver behavior and simulation scale finds evidence against distributional
equivalence in RL-relevant measures from these simulators, with the root mean
squared error and KL divergence being significantly greater than 0 for all
assessed measures. While granular real-world validation generally remains
infeasible, these findings suggest that traffic simulators are not a deus ex
machina for RL training: understanding the impacts of inter-simulator
differences is necessary to train and deploy RL-based ITSs.

##  1 Introduction

Transportation efficiency is becoming an increasingly critical challenge due
to continual growth in the volume of people and objects that need to be
transported. The 2021 Urban Mobility Report Schrank2021 projected that, while
the COVID-19 pandemic alleviated congestion, traffic levels in the US will
quickly rebound in areas with expanding populations and job markets to produce
the most rapid congestion growth since 1982. The increased traffic will stress
existing infrastructure and result in social, economic, and environmental
costs Schrank2021, thus making the development and deployment of intelligent
transportation systems (ITSs) a critical priority. At the same time, advances
in computational algorithms and roadway infrastructure made in response to
these challenges provide opportunities to enhance ITS learning. For example,
novel traffic signal control technologies based on reinforcement learning
(RL), which learn adaptive signaling policies from simulations generated using
real-world traffic data, have already achieved performance on par with and
even exceeding traditional control methods Chen2020.

However, collecting data for ITS learning remains a nontrivial task. First,
many state-of-the-art deep learning approaches for ITSs inherently require
large quantities of data for training Garg2019, but they also need to be able
to learn from a heterogeneous set of experiences so that they can achieve
robust performance in the real world. Second, once they are trained, ITSs will
need to be able to interact with many human users, including both end-users
(i.e., drivers) and decision-makers (i.e., traffic engineers and city
planners). End-users have unique needs and make unique choices that must be
addressed individually and equitably. Decision-makers also have complex and
potentially conflicting requirements Goh2012 that necessitate repeated
iteration of the design of ITSs. Thus, data collection for ITS learning must
be large-scale and continuously-occurring. However, performing data collection
from real-world transportation systems in this manner may result in
significant efficiency and safety costs Garg2019.

Due to these issues, traffic simulators are commonly used to substitute or
complement real-world data collection. They serve as safe sandboxes that can
generate realistic data for both ITS learning Barcelo2004 and real-world
decision-making AlvarezLopez2018,Fries2007. This work narrowly focuses on
open-source traffic simulators that have been developed for the training of
ITSs based on RL methods, including systems for traffic signal control,
autonomous driving, ramp metering, and tolling Haydari2022. RL algorithms are
particularly demanding in terms of the number of environmental interactions
that they require, so traffic simulators for RL applications must be able to
quickly generate high-fidelity, high-granularity data Zhang2019.

Released in 2001, SUMO AlvarezLopez2018 is the traffic simulator that has been
most commonly used to train deep RL-based ITSs Noaeen2022. It supports an
expansive framework for simulation definitions, as well as efficient
programmatic API access. However, it is single-threaded and thus scales less
well to very large road networks. Motivated by this limitation, Zhang2019
introduced CityFlow, a traffic simulator designed for applications of RL to
traffic signal control. It is multi-threaded, and consequently was reported to
achieve a speedup of $>$20x over SUMO. However, its framework for simulation
definitions is more restricted and focuses on aspects which are essential for
RL training. Its adoption is limited but increasing Noaeen2022.

If traffic simulators are to serve as training environments for RL algorithms,
their modelling assumptions must be sufficiently realistic that RL-based ITSs
can learn to adapt to a variety of scenarios during deployment. Thus, real-
world validation is crucial. However, granular validation with real-world data
is usually not possible due to the aforementioned challenges of data
collection. Comparisons between simulators take a partial step towards this
goal by verifying that different simulators lead to equivalent outcomes. This
work compares CityFlow against the more granular SUMO. Zhang2019 included a
preliminary comparison of CityFlow to SUMO using the expected travel time of
vehicles in a road network. However, travel time is a long-term, system-level
outcome, whereas RL algorithms use instantaneous, local features for learning.
A more comprehensive comparison was performed to answer the following research
questions:

  1. RQ1

Do the low-level simulation outcomes of CityFlow and SUMO have a statistically
significant level of distributional equivalence?

  2. RQ2

How is this distributional equivalence affected by incorporating variation
between vehicles in terms of different driver behavioral models?

  3. RQ3

How is this distributional equivalence affected by the scale of the
simulation, in terms of traffic demand and road network size?

##  2 Related Work

###  2.1 Validating Traffic Simulators

Validation is an important yet challenging aspect of the development of
traffic simulators that ensures their fidelity to the real world. A multitude
of road networks have been used to validate SUMO itself
Bedogni2015,AlvarezLopez2018 and to calibrate its car-following models
Krajzewicz2002 in comparison to detector data. For instance, Lobo2020 compared
traces for vehicle counts and crossing times between SUMO and detector data
for a road network in Ingolstadt, Germany; their simulation is used in this
work. Meanwhile, CityFlow was validated by comparing average travel times
under various traffic volumes to that of SUMO Zhang2019.

A number of studies have compared outcomes from multiple simulators; this work
falls in this setting. Maciejewski2010 compared vehicle counts under different
traffic demand and driver behavior settings for SUMO, the commercial simulator
VISSIM, and TRANSIMS. Several studies involved SUMO and the commercial
simulator AIMSUN. Leksono2012 compared travel times and queue lengths in these
simulators under different traffic management interventions for a roundabout
in Norrköping, Sweden. Ronaldo2012 used $t$-tests to compare flows and speeds
in these simulators to observations from a highway interchange in Stockholm,
Sweden. BazaSolares2022 applied $t$-tests to vehicle counts from the two
simulators for a road network from Bucaramanga, Colombia. This work differs
from prior approaches in that: (1) the measures used relate to the
distribution of outcomes across the network, not just at individual points;
and (2) these measures were evaluated across multiple road network scales.

###  2.2 Modelling Driver Behavior

There has been a significant body of literature on building realistic models
of driver behavior. Car-following behavior was the first to be modelled, with
the early Gazis-Herman-Rothery model being over 50 years old. Subsequent work
has yielded optimal velocity, fuzzy logic, collision avoidance, action point,
and cellular automaton models Li2012,Saifuzzaman2014. Collision avoidance and
action point models were implemented for this work. Lane-changing behavior has
been modelled by a newer, separate line of work Moridpour2010a,Zheng2014,
which has produced models based on rules, discrete choice, game theory, and
cellular automata. Some work has built unified models of car-following, lane-
changing, and other driver behavior Toledo2007,Markkula2012. Here, car-
following and lane-changing models were considered separately but co-varied in
virtual experiments to elucidate the effects of their interactions. The most
relevant prior work is CapelaDias2013, who analyzed the impact of driver
behavior on system-level travel time in SUMO; by contrast, this work focuses
on distributional equivalence between two simulators in terms of lower-level
outcomes.

##  3 Traffic Simulators for RL

This section reviews the use of traffic simulators for RL-based ITS training.

###  3.1 RL for Transportation

RL algorithms learn policies to maximize their expected cumulative reward
through repeated interactions with an environment, which for ITSs is the
traffic simulator. Examples of cumulative reward objectives include travel
time minimization for traffic signal control and speed limit control
Noaeen2022,Wu2020, and revenue maximization for tolling Pandey2020. However,
cumulative rewards are not useful for learning because they are system-level
measures calculated over longer timespans. Instead, RL algorithms observe
instantaneous vehicle-level state features, use them to select actions, and
then receive instantaneous rewards Haydari2022.

Formulations of the observation and instantaneous reward spaces vary between
papers, but they are rarely aligned directly with the cumulative reward.
Observations usually consist of counts (e.g., queue lengths), positions,
speeds, or other aggregated properties of individual vehicles
Noaeen2022,Pandey2020,Wu2020. Rewards are more problem-dependent. For traffic
signal control, queue length, waiting time, and speed are common factors
Koohy2022; speed limit control may use proxies for travel time, crash
probability, or emissions Wu2020; and tolling may optimize a combination of
vehicle count, travel time on segments, and collected tolls Pandey2020.
Regardless of these different formulations, it is evident that system-level
measures involving the movement of vehicles through the whole road network are
not directly used for RL training.

###  3.2 Comparing CityFlow and SUMO

Microscopic traffic simulators are most often used for RL in transportation
Haydari2022. In contrast to macroscopic and mesoscopic simulators, microscopic
simulators are primarily agent-based, as they model the behavior of individual
vehicles Barcelo2010. This is helpful for gathering the types of vehicle-level
observations introduced in subsection 3.1. Both CityFlow and SUMO are
microscopic. They represent road networks as graphs, with intersections as
nodes (“junctions” in SUMO) and roads as edges between nodes. Vehicles are
generated by flows; all vehicles within a flow share similar attributes,
including behavioral parameters and routes. Routes are defined as sequences of
edges. Where edges are joined at an intersection, vehicles select a pair of
lanes (“roadlinks” in CityFlow, “connections” in SUMO) to traverse the
intersection. However, key differences exist in finer details. Unlike
CityFlow, SUMO supports definitions for more complex types of lanes (e.g.,
sidewalks, bicycle lanes, and ramps) and vehicle classes (e.g., public transit
and emergency vehicles, pedestrians, and cyclists). It also supports
stochastic generation of routes and vehicle types, and more sophisticated
traffic signalling, including yellow lights.

Where possible, the virtual experiments in this work were designed to control
or co-vary differences between the simulators. In particular, flows in both
CityFlow and SUMO were converted to be single-vehicle and deterministic, and
to share the same routes. However, differences remain, particularly aspects of
driver behavior (section 4) driven by randomness in SUMO; by contrast,
CityFlow only uses random generation for vehicle priority. Such uncontrolled
factors may account for their differences.

##  4 Varying Driver Behavior

Addressing RQ2 requires distributional equivalence to be assessed under
different variations of driver behavior; as established in subsection 2.2,
these are implemented through car-following and lane-changing models in
traffic simulators. In both CityFlow and SUMO, car-following logic is used to
maintain a safe gap to the leading vehicle, while lane changes are used to
switch vehicles between lanes so that they can take appropriate roadlinks at
intersections to continue their routes. However, in CityFlow, car-following
and lane-changing are handled in separate threads; in SUMO, they are handled
in sequence, with car-following logic being executed before lane-changing
logic.

###  4.1 Car-Following Models

Car-following models determine the speeds at which vehicles travel
unobstructed (free speed), follow behind a lead vehicle (following speed), and
stop at an obstacle (stopping speed). Two types of numerical integration can
be used to calculate vehicle speeds: a Euler update, which solves for the
speed at discrete timesteps, and a ballistic update, which solves for the
acceleration at discrete timesteps and applies it to the speed. Both
simulators were controlled to use ballistic updates.

Table 1: Parameter settings for six aggressiveness types based on Capela Dias et al. (2013). Maximum emergency deceleration was uniformly set to $-9.0\mathrm{\nicefrac{{m}}{{s^{2}}}}$, as they did not specify this parameter. Type |  Max. accel. |  Max. decel. |  Max. emerg. decel. |  Min. gap |  Min. headway
---|---|---|---|---|---
Aggressive young |  $3.1\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-5.5\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-9.0\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  1.2 m |  1.0 s
Courteous young |  $2.5\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-4.5\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-9.0\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  2.5 m |  1.0 s
Aggressive middle-aged |  $2.9\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-5.0\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-9.0\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  2.0 m |  1.3 s
Courteous middle-aged |  $2.4\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-4.1\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-9.0\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  2.5 m |  1.5 s
Aggressive old |  $2.6\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-4.5\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-9.0\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  2.0 m |  1.7 s
Courteous old |  $2.3\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-3.8\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  $-9.0\ \mathrm{\nicefrac{{m}}{{s^{2}}}}$ |  2.5 m |  1.9 s

Several shared parameters have varying effects on different car-following
models. CityFlow assumes that vehicles have usual and maximum accelerations
and decelerations; SUMO assumes that vehicles have a maximum possible
acceleration and deceleration (which are used as the usual values), and a
maximum emergency deceleration. The latter formulation was used here, although
this may have resulted in more aggressive behavior than if lower usual
accelerations and decelerations were used. Additionally, both simulators model
vehicles as having minimum desired following distances in terms of space
(i.e., the minimum gap) and time (i.e., the minimum headway). CapelaDias2013
co-varied these parameters to model the effect of driver aggressiveness on
travel time. This work adopts their taxonomy of aggressiveness types, but
excludes gender effects due to conflicting conclusions regarding their
significance in the literature TaiebMaimon2001,Tasca2000. This gave six
parameter settings (Table 1).

The default car-following models in CityFlow and SUMO are both modified from
the collision avoidance model of Krauss1998. For free speed, CityFlow’s
implementation uses the maximum speed, while SUMO’s implementation modulates
this by the visible lookahead distance. For stopping speed, both simulators
solve somewhat different quadratic equations to determine the deceleration
needed to stop within a fixed distance. For following speed, a target distance
is maintained, which is computed by the desired minimum headway as well as the
speed and maximum deceleration of the lead vehicle. In this work, SUMO’s
variant of the Krauß model was added to CityFlow. SUMO implements several
other car-following models, which were also re-implemented in CityFlow to
introduce variation in driver behavior:

  * •

The collision avoidance/action point model of Wagner2008, which
probabilistically combines Krauß’s model with the action point model of
Todosiev1963.

  * •

The collision avoidance model of Wiedemann1974, which is a behavioral model
that varies between free, approaching, following, and emergency modes based on
the gap to the lead vehicle.

  * •

The adaptive cruise control (ACC) model of Milanes2014, which determines
acceleration using a “speed control” method if the gap to the lead vehicle is
large and using a “gap control” method if the gap is small.

###  4.2 Lane-Changing Models

In both CityFlow and SUMO, lane changes are initiated by signaling the lead
and lag vehicles on the destination lane, and the vehicle is instantly moved
to the other lane upon completion of the lane change. (SUMO also supports a
sublane model that explicitly models lateral movement, which was excluded as a
control.) However, the implementation details of lane-changing likewise differ
between these simulators. Vehicles in CityFlow only perform lane changes to
follow their routes, and lane-changing is based on the explicit insertion of a
copy of the vehicle (the “shadow vehicle”) on the target lane that the lag
vehicle will follow. Vehicles in SUMO follow a parameterized hierarchy of
motivations, which includes strategic lane changes to follow routes as well as
tactical lane changes for overtaking.

One of SUMO’s parameters, the gap tolerance factor, was introduced to CityFlow
in this work. When vehicles are changing lanes, the minimum gap on the
destination lane required for a vehicle to initiate a lane change is given by
the necessary gap for collision avoidance divided by this constant factor. A
factor of 1.0 represents the default behavior. The virtual experiments in this
work varied this factor between 0.5, 0.82, 1.0, 1.18, and 1.5, representing
decreases/increases of 18% and 50% in tolerance. This is based on Sun2010, who
observed the mean gaps for forced, cooperative, and free lane change maneuvers
to be 45 ft, 53 ft, and 109 ft in a video dataset.

##  5 Experimental Setup

To address the research questions, two virtual experiments were designed. Both
address RQ1 and RQ2 by assessing CityFlow and SUMO’s distributional
equivalence under different settings of driver behavior. The two experiments
also address separate aspects of RQ3: Experiment 1 assesses distributional
equivalence while varying the simulation scale by traffic demand, and
Experiment 2 while varying it by road network size. This was accomplished
using road networks from the benchmark dataset of Ault2021, as described in
the following subsections. All of these networks were initially defined using
SUMO syntax; the SUMO-to-CityFlow network converter of Zhang2019 was used to
generate their CityFlow counterparts, and a flow converter was created to map
between vehicular flows in the two simulators.

Four independent variables were considered in both virtual experiments: the
car-following model (subsection 4.1, 5 levels), the car-following
aggressiveness parameters (subsection 4.1, 6 levels), the lane-changing gap
tolerance (subsection 4.2, 5 levels), and the road network (2 levels). For the
car-following model, the default and SUMO-based implementations of the Krauß
model in CityFlow were compared to Krauß model in SUMO, while the other
models’ CityFlow implementations were compared to their respective SUMO
counterparts. The fundamental lane-changing models for the two simulators were
held constant, as was the traffic signal program — a simple fixed-time program
retained from the original dataset.

Eight instantiations of two types of measures were used to assess
distributional equivalence. First, the root mean squared error (RMSE)
quantifies the point-to-point difference in individual outcome measures. This
was computed as the mean RMSE of the total travel time and waiting time
(defined as time that a vehicle spends queued with a speed
$<0.1\mathrm{\nicefrac{{m}}{{s}}}$) over all vehicles and timesteps; the mean
RMSE of per-lane total vehicle counts and queued vehicle counts over all lanes
and timesteps; and the mean RMSE of the speed and acceleration over all
individual vehicles and timesteps. Second, the Kullback-Liebler (KL)
divergence measures the difference in the distribution of outcomes over the
entire road network. This was computed for the distributions of vehicle counts
and queued vehicle counts as the mean over all timesteps.

###  5.1 Experiment 1: Traffic Demand

For this experiment, the road networks arterial4x4 Ma2020 and grid4x4 Chen2020
(Figure 1) from Ault2021’s RESCO benchmark were used. Both are synthetic grid
networks with similar topologies, but they vary in the level of congestion.
grid4x4 consists of six-lane roads with a uniformly distributed demand of
1,473 vehicles. arterial4x4 consists of major (four-lane) and minor (two-lane)
roads, and a demand of 2,484 vehicles that alternates between major and minor
roads. arterial4x4’s traffic pattern leads to congestion and degraded RL
performance in simulations Ault2021.

(a) arterial4x4

(b) grid4x4

Figure 1: Screenshots in CityFlow of the arterial4x4 and grid4x4 road
networks.

The following power analysis uses these variable names: $C$ for car-following
models, $A$ for car-following aggressiveness, $L$ for lane-changing gap
tolerance, and $R$ for road network. The second-order linear multiple
regression included 5 ($C$) + 6 ($A$) + 1 ($L$) + 1 ($L^{2}$) + 2 ($R$) + 10
($C\cdot R$) + 12 ($A\cdot R$) + 2 ($L\cdot R$) + 30 ($C\cdot A$) + 5 ($C\cdot
L$) + 6 ($A\cdot L$) = 80 variables. Using G*Power 3.1.9.7’s power calculation
for ordinary linear multiple regression with a fixed model and $R^{2}$
increase, a small effect size of 0.02, and $\alpha=\beta=0.95$, the total
necessary sample size was computed as 2,646. Divided by the number of cells,
$5\cdot 6\cdot 5\cdot 2=300$, the number of replications per cell was computed
as $\lceil\frac{2,646}{300}\rceil=9$. All replications were executed using
Python 3.9.16, SUMO 1.12.0, and a modified version of CityFlow 0.1 on a shared
server with four cores, two 4.2GHz Intel i7-7700K processors per core, and 62
GiB of RAM.

###  5.2 Experiment 2: Network Scale

For this experiment, the road networks ingolstadt1 and ingolstadt7 (Figure 2)
from Ault2021’s RESCO benchmark were used. Both are subsets of the Ingolstadt
road network that was simulated by Lobo2020. They respectively contain 1 and 7
signalized intersections, representing a single busy intersection and a larger
arterial road; ingolstadt7 is a superset of ingolstadt1. The total demands of
the two road networks are respectively 1,716 and 3,031 vehicles.

(a) ingolstadt1

(b) ingolstadt7

Figure 2: Screenshots in CityFlow of the ingolstadt1 and ingolstadt7 road
networks.

Based on the analysis in subsection 5.1, the number of replications per cell
was computed as $\lceil\frac{2,646}{300}\rceil=9$. All replications were
executed using Python 3.9.16, SUMO 1.12.0, and a modified version of CityFlow
0.1 on a shared server with four cores, two 4.2GHz Intel i7-7700K processors
per core, and 62 GiB of RAM.

##  6 Experimental Results

One-sample $t$-tests indicated that all of the RMSE and KL divergence measures
were significantly different from 0, with a $p$-value $\ll 0.001$ for all
cells in Experiments 1 and 2. This suggests a lack of distributional
equivalence between CityFlow and SUMO. The following subsections explore the
results of second-order linear multiple regression for each of the measures.
Notably, coefficients involving the difference between the SUMO and CityFlow
Krauß model implementations were generally not significant.

###  6.1 Experiment 1: Traffic Demand

For total time and waiting time RMSEs, the road network, car-following model,
and aggressiveness generally had significant effects, as did various pairwise
interactions between them. The uncongested grid4x4 network had significantly
lower RMSEs (coefficients: $-890.61$/$-1035.46$) than the congested
arterial4x4 network (intercepts: $1718.61$/$1815.46$), suggesting that
congestion worsened the distributional equivalence of these measures. In
arterial4x4, the Wagner (coefficients relative to SUMO Krauß:
$546.8$/$466.65$) and Wiedemann models (coefficients: $100.96$/$208.99$) had
significantly higher RMSEs; these differences were smaller for grid4x4
(coefficients relative to SUMO Krauß: $49.5$/$33.75$ for Wagner;
$48.91$/$34.85$ for Wiedemann).

For total and queued vehicle count RMSEs and KL divergences, the road network,
car-following model, aggressiveness, and gap tolerance generally had
significant effects, as did various pairwise interactions between them. The
RMSEs and KL divergences showed distinct patterns: the RMSEs were much lower
for grid4x4 than arterial4x4 (coefficients: $-5.15$/$-4.93$), but the KL
divergences had less variation (coefficients: $-0.621$/$-0.013$). Yet, the KL
divergences had low enough standard deviations that the road network’s effects
remained significant. High aggressiveness in arterial4x4 generally yielded
higher RMSEs (coefficients for aggressive young relative to aggressive middle-
aged: $2.16$/$2.46$) and KL divergences (coefficients: $0.228$/$0.139$). While
the same was true for the RMSEs in grid4x4, its KL divergences were lower for
more aggressive settings (coefficients: $-0.204$/$-0.122$). Despite higher
time measures, the Wagner model led to lower measures for total vehicle count
(coefficients relative to SUMO Krauß: $-1.16$/$-0.193$).

For vehicle speed and acceleration RMSEs, the road network, car-following
model, and aggressiveness generally had significant effects, as did various
pairwise interactions between them. Greater equivalence in vehicle
distributions did not always correspond to more similar vehicle-level
measures. Both speed and acceleration RMSEs increased for grid4x4
(coefficients: $2.62$/$0.418$) even though the other measures were lower on
average. Also, unlike its vehicle count measures but like its time measures,
the Wagner model had significantly higher speed and acceleration RMSEs
(coefficients relative to SUMO Krauß: $2.71$/$1.26$).

###  6.2 Experiment 2: Network Scale

For total time and waiting time RMSEs, the aggressiveness and its interactions
generally had significant effects, along with the Wiedemann model and its
interactions. The best-fitting model for total time did not include a road
network-gap tolerance interaction, whereas the model for waiting time did. The
smaller ingolstadt1 network had lower but more variable RMSEs (intercepts:
$899.29$/$1170.3$), while the larger ingolstadt7 network had significantly
higher but more uniform RMSEs (coefficients: $1264.59$/$1046.07$). For
ingolstadt1, the most aggressive parameter settings led to significantly
higher RMSEs (coefficients of aggressive young relative to aggressive middle-
aged: $408.36$/$568.65$). Likewise, the Wiedemann model had significantly
higher RMSEs than other car-following models in ingolstadt1 (coefficients
relative to SUMO Krauß: $192.29$/$1917.31$), but this effect was reversed for
ingolstadt7 (coefficients: $-349.13$/$-464.13$).

For total and queued vehicle count RMSEs and KL divergences, the road network
and aggressiveness generally had significant effects, as did various pairwise
interactions between them and with the gap tolerance. Unlike Experiment 1, the
RMSE measures were lower in ingolstadt7 than in ingolstadt1 (coefficients:
$-2$/$-1.67$), but the KL divergence measures were higher (coefficients:
$0.461$/$2.48$). More aggressive parameter settings again led to significant
increases in the RMSEs and KL divergences, with a larger increase in RMSEs
(coefficients of aggressive young relative to aggressive middle-aged:
$1.35$/$2.29$) than in KL divergences (coefficients: $0.595$/$0.717$).
However, the increase in both measures was smaller in ingolstadt7
(coefficients: $0.764$/$1.507$ for RMSE, $0.283$/$0.413$ for KL divergences).
Both the Wagner and ACC models had RMSEs that significantly increased with gap
tolerance (coefficients relative to SUMO Krauß per unit of gap tolerance:
$0.151$/$0.056$ for Wagner, $0.165$/$0.092$ for ACC), but KL divergences that
significantly decreased with it (coefficients: $-0.285$/$-0.511$ for Wagner,
$-0.354$/$-0.642$ for ACC).

For vehicle speed and acceleration RMSEs, the Wagner and Wiedemann models
along with the aggressiveness had significant effects, as did various pairwise
interactions of the road network with the car-following models and
aggressiveness. Again, the road network had significant effects on both speed
and acceleration RMSEs, with these measures being higher for ingolstadt7
(coefficients: $5.51$/$0.173$). The Wagner car-following model had
significantly higher RMSEs (coefficients relative to SUMO Krauß:
$0.363$/$0.461$), whereas the Wiedemann model had significantly lower RMSEs
(coefficients: $-1.94$/$-0.957$).

###  6.3 Parameter Validity

Parameter validation was conducted by comparing the parameter settings for
car-following and lane-changing models used in the virtual experiments to
prior driving simulator and real-world studies. Overall, the dependence of
these parameters on external factors such as traffic density and speed
suggests that the settings used in traffic simulators should be calibrated to
specific road networks and conditions. However, the settings used in this work
remain largely reasonable considering the variation reported in the
literature.

For acceleration and deceleration, the settings used in the virtual
experiments were based on CapelaDias2013, with smaller values for older and
less aggressive drivers (subsection 4.1); however, they considered gender
effects to be negligible. Similar values have been reported in prior work
Nakagawa2006,Fadhloun2015,Korber2016. However, among driving simulator
studies, Korber2016 demonstrated an age effect opposite to that assumed by
Capela Dias et al. Among real-world studies, Moon2008 found a dependence of
the 95th percentile of braking decelerations on speed, and Nakagawa2006
reported a significant interaction between age and gender.

For minimum gap and headway time, the settings used in the virtual experiments
also followed CapelaDias2013, with larger values for older and less aggressive
drivers (subsection 4.1). Similar values have again been reported in prior
work Michael2000,Vogel2002. In particular, real-world studies for which
reported values are influenced by a similar age effect include
Delorme2001,TaiebMaimon2001, and Moon2008. TaiebMaimon2001 and Shangguan2019
reported larger minimum gaps and smaller minimum headways, but their
measurements were made for higher speeds (respectively over $50\
\mathrm{\nicefrac{{km}}{{h}}}$ and $100\ \mathrm{\nicefrac{{km}}{{h}}}$).

For lane-changing gap tolerance, this work used gap widths in empirical lane-
changing behavior to approximately quantify its level of variation,
specifically using the mean gaps for forced, cooperative, and free lane
changes from Sun2010’s real-world study (subsection 4.2). The settings in this
work are consistent with standard deviations in lead and lag gaps as reported
in prior studies Moridpour2008,Balal2014. Ali2020’s driving simulator study
identified significant factors that impact gap tolerance: relative to the
average male middle-aged driver, gaps are smaller for younger drivers, larger
for female drivers, and smaller as speed increases. Likewise, Hill2015’s real-
world study reported that the mean and standard deviation of lag gaps depended
on congestion. Future work could use these factors to create a taxonomy of
lane-changing behavior similar to CapelaDias2013.

##  7 Conclusion

In this work, virtual experiments compared the low-level simulation outcomes
of two traffic simulators, CityFlow and SUMO. To capture the effects of
modelling real-world heterogeneity, various parameters of driver behavior and
road network scale were varied. The results indicate a lack of distributional
equivalence between the simulators, with certain parameter settings worsening
distributional equivalence.

However, as noted in section 3, this work is insufficient to provide a
complete characterization of what the critical differences between these
simulators are. Many aspects of CityFlow and SUMO that were not controlled —
simulation control-flow, other aspects of driver behavior, the effects of
traffic signals, and randomness — could all have contributed to the observed
discrepancies. Thus, future work should perform more comprehensive, controlled
evaluations of these two simulators.

Regardless, researchers in RL for transportation must not take traffic
simulators for granted as a deus ex machina for training, and must recognize
that they may not be interchangeable. Which simulator, then, should be chosen?
This work does not aim to answer this question, but some observations can be
made:

  * •

SUMO provides a detailed simulation that models real-world heterogeneity, and
captures additional aspects of traffic management and driver behavior

  * •

CityFlow provides an efficient simulation that abstracts out and homogenizes
various details, reducing the number of parameters that need to be tuned

Therefore, as with many other problems in simulation, the core trade-off
between these two simulators (and others) involves veridicality and
efficiency. There is no one best simulator; researchers must decide whether
using a coarser abstraction of the environment is acceptable in exchange for
faster training.

But how exactly should researchers make this decision? Crucially, RL-based
ITSs may not necessarily perform better when they are trained with more
granular simulators. Prior work has shown that both introducing unnecessary
complexity Zheng2019 and removing needed complexity Genders2018 in the
observation space may harm performance. Researchers should compare training
results from different simulators and use them to design RL formulations in a
principled way. As a first step, Goel2023 report training results on both SUMO
and CityFlow, but they do not use the same road networks for both simulators.
One strategy may be to train a baseline using an efficient simulator (e.g.,
CityFlow), and then to finetune it by further training with a veridical one
(e.g., SUMO).

Ultimately, the true goal is to ensure that RL-based ITSs can perform well
under real-world traffic conditions, which requires that their training and
validation environments are close to reality. As previously noted,
distributional equivalence between simulators is a proxy measure of this goal.
Unfortunately, a chicken-and-egg problem exists in that traffic simulators are
intended to replace real-world data collection, yet cannot be validated
without it. For now, simulations should still be developed in collaboration
with stakeholders to ensure that they meet acceptable standards of fidelity.
However, the future holds promise for traffic simulations that are both
veridical and efficient: the increasing prevalence of connected vehicles
Mathew2021 means that collection of granular real-world data for validation
may be within reach.

## Acknowledgments

This work was supported by a Mobility21 research grant and the Tang Family
Endowed Innovation Fund.

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## Author Biographies

REX CHEN (email ) is a third-year PhD candidate in the Societal Computing
program of the Software and Societal Systems Department in the School of
Computer Science at Carnegie Mellon University. His research focuses on
applications of reinforcement learning and other AI techniques to socially
impactful domains such as transportation. His work aims to evaluate the impact
of real-world uncertainty and user behavior on AI algorithms designed for
these domains.

KATHLEEN M. CARLEY (email ) (H.D. University of Zurich, Ph.D. Harvard, S.B.
MIT) is a Professor of Societal Computing, in the School of Computer Science’s
Software and Societal Systems Department at Carnegie Mellon University; and
the Director of the Center for Computational Analysis of Social and
Organizational Systems (CASOS), and the Center for Informed Democracy and
Social Cybersecurity (IDeaS). She and her teams have developed infrastructure
tools for analyzing large-scale dynamic networks, bot hunters, and various
agent-based simulation systems.

FEI FANG (email ) is the Leonardo Assistant Professor in the Software and
Societal Systems Department in the School of Computer Science at Carnegie
Mellon University. Her research interests lie in the area of artificial
intelligence and multi-agent systems, focusing on the integration of
computational game theory and machine learning to address real-world
challenges in critical domains such as security, sustainability, and mobility.

NORMAN SADEH (email ) is a professor in the School of Computer Science at CMU,
where he is affiliated with the Software and Societal Systems Department, the
Human-Computer Interaction Institute, and the CyLab Security and Privacy
Institute. His research interests span mobile computing, the IoT,
cybersecurity, privacy, machine learning, AI, and related public policy
issues. His past work includes deployed planning and scheduling technologies
for commercial systems.


# Radiation Reaction in Non-commutative Electrodynamics

V. Veera Reddy  BITS Pilani Hyderabad Campus,Hyderabad 500078, India
Sashideep Gutti  BITS Pilani Hyderabad Campus,Hyderabad 500078, India  Asrarul
Haque  BITS Pilani Hyderabad Campus,Hyderabad 500078, India
<EMAIL_ADDRESS>
###### Abstract

We study the radiation reaction acting on an accelerating charge moving in
noncommutative spacetime and obtain an expression for it. Radiation reaction,
due to a nonrelativistic point charge, is found to receive a small
noncommutative correction term. The Abraham-Lorentz equation for a point
charge in noncommutative spacetime suffers from the preacceleration and the
runaway problems. We explore as an application the radiation reaction
experienced by a charge which undergoes harmonic oscillations in a
noncommutative plane.

##  I Introduction

An accelerated charge emits electromagnetic radiation which carries away
certain amounts of the energy and momentum of the chargejac ; swi ; bar . The
loss of energy leads to a deceleration of the charge which implies that there
must be a force acting on the charge due to the electromagnetic fields it
produces. This retarding force is known as the radiation reaction. Thus, the
effect caused by the radiation on the dynamics of the charge is to modify the
equation of motion of the charge with the radiation reaction.
An accelerated charge produces electromagnetic field, say $A^{\mu}(x,y,z,t),$
whose spacetime coordinates commutate among themselves, i.e.,

$[x_{i},x_{j}]=0,~{}~{}~{}~{}[x_{i},t]=0~{};~{}~{}i,j=1,2,3.$

However, there are physical instances where the notion of spatial coordinates
may not commute

$[x_{i},x_{j}]\neq~{}0~{}~{}~{}~{}~{};~{}~{}i,j=1,2,3.$

could be realizedjack .
It is natural to ask: “How the radiation reaction and hence the dynamics of an
accelerated charge will change in a spacetime manifold equipped with
noncommuting spacetime coordinates?”
Physics at high energy, might alter the continuum nature of spacetime, serves
as one of the main motivations behind the study of the spacetime coordinates
that do not commute. One of the variants of the granular structure of
spacetime could be thought in terms of the uncertainties in the measurements
of the spacetime coordinates which give rise to the idea of the noncommutative
spacetime.
It was Snyder sny  who first introduced Noncommutative space, as a possible
cure of ultraviolet divergences which stem from the ill-defined product of the
fields at the same space-time point in quantum field theories. Moreover,
noncommutative spaces are found to arise in several different contexts. At
short distance, the interplay between quantum theory and gravitation suggests
a non-trivial structure of space-time and a noncommutative structure of space-
time is a possibility. In fact, the concept of space-time as a $c^{\infty}$
manifold may break down to the distance-scales of the order of Plank length
scale dop . Noncommutative structure of spacetime naturally appears in string
theorywit . We consider fields defined on noncommutative spacetime manifold
which obeys

$[\hat{x}^{\mu},\hat{x}^{\nu}]=i\theta^{\mu\nu}$ (1)

where $\theta^{\mu\nu}$ is a constant real antisymmetric matrix of length
dimension two. Elctromagntic field theory in commutaive space time could be
generalized to a noncommutative space-time via replacing ordinary local
product by a Moyal star product of the two functions sza

$(f*g)(x):=e^{\frac{i}{2}\theta^{\mu\nu}\partial^{x}_{\mu}\partial^{y}_{\nu}}f(x)g(y)|_{x=y}$
(2)

The question arises: can such spacetime noncommutativity induce any
modification to the radiation reaction associated with an accelerated charge
in commutative spacetime?
The goal of the present work is to compute the radiation reaction force
experienced by an accelerated charge moving in the noncommutative spacetime.
We perform the calculation of radiation reaction on an accelerating charge to
capture any effect whatsoever arising due to noncommutativity in spacetime
with a hope to find the resolutions to the pathologies and ambiguities
associated with the Abraham-Lorentz equation. We find that radiation reaction
receives modification in the noncommutative Maxwell theory. The Abraham-
Lorentz equation in noncommutative spacetime is found to be plagued with the
preacceleration and the runaway solutions. This work serves as a preliminary
framework to continue further study of the radiation reaction in the quantum
version of noncommutative electrodynamics.
In section 2, we define and formulate the radiation reaction force in
noncommutative spacetime. In section 3, we derive explicitly the expression
for radiation reaction in noncommutative electrodynamics. In section 4, we
study radiation reaction experienced by a charge which undergoes harmonic
oscillations in a noncommutative plane.

##  II Lorentz force in noncommutative spacetime

A charged particle moving in an electromagnetic field experiences Lorentz
force. Lorentz four force bar ; jac  on a charged particle in electrodynamics
in commutative spacetime is defined as

$G^{\nu}_{C}=\int k^{\nu}d^{3}{x}$ (3)

with Lorentz four force density $k^{\nu}$ given by

$k^{\nu}=\frac{1}{c}j_{\mu}F^{\mu\nu}$ (4)

where $j_{\mu}$ is four current density vector and $F^{\mu\nu}$ is
electromagnetic field tensor constructed using the electromagnetic four vector
potential $A^{\nu}$ and is given by
$F^{\mu\nu}=(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})$.
For a point charge the four current density reads

$j^{\mu}(x)=ec\int_{-\infty}^{+\infty}ds\hskip
2.0pt\dot{z}^{\mu}\delta\left[(x-z(s))\right]$ (5)

where $z(s)$ is the world line of the charged particle parametrized by $s$. In
order to calculate self force due to the point charge, electromagnetic field
tensor $F^{\mu\nu}$ is replaced by retarded field tensor $F^{\mu\nu}_{ret}$
and four potential $A^{\mu}$ is replaced by retarded potential $A^{\mu}_{ret}$
bar ; roh .

$\displaystyle
F^{\mu\nu}_{ret}=\frac{1}{2}F^{\mu\nu}_{rad}+\frac{1}{2}(F^{\mu\nu}_{ret}+F^{\mu\nu}_{adv})$
(6)

where the radiation field defined by

$\displaystyle F^{\mu\nu}_{rad}=F^{\mu\nu}_{ret}-F^{\mu\nu}_{adv}$ (7)

Radiation field $\frac{1}{2}F^{\mu\nu}_{rad}$ contributes to radiation
reaction where as $\frac{1}{2}(F^{\mu\nu}_{ret}+F^{\mu\nu}_{adv})$ contributes
to the electromagnetic mass of point charge.
Therefore the radiation reaction acting on a point charge in commutative
electrodynamics is defined by

$\displaystyle G^{\nu}_{C}$ $\displaystyle=$ $\displaystyle\frac{1}{2c}\int
j_{\mu}(x)F^{\mu\nu}_{rad}(x)d^{3}{x}.$

Radiation reaction force bar  evaluates to

$\displaystyle G_{iC}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{e^{2}\gamma^{2}}{4\pi
c^{3}}\left(\ddot{v}_{i}+\frac{3\gamma^{2}}{c^{2}}\dot{v}_{i}(\vec{v}.\dot{\vec{v}})+\frac{\gamma^{2}}{c^{2}}v_{i}(\vec{v}.\ddot{\vec{v}})\right.$
(8)
$\displaystyle\left.+\frac{\gamma^{4}}{c^{4}}(\vec{v}.\dot{\vec{v}})^{2}v_{i}\right)$

where $\textit{i}=1,2,3$ denote the spatial indices and $v_{i}$ is the three
velocity of the charged particle.
In the non-relativistic limit, radiation reaction force due to a point charge
is given by

$G^{rad}_{iC}=-\frac{2}{3}\frac{e^{2}}{4\pi c^{3}}\ddot{v}_{i}.$ (9)

Lorentz force in commutative electrodynamics could be generalized to
noncommutative electrodynamics as follows:

$\displaystyle F^{\nu}_{NC}=\int K^{\nu}\,d^{3}x$ (10)

with noncommutative Lorentz force density $K^{\nu}$ rab  given by

$\displaystyle K^{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{c}\left(j_{\mu}F^{\mu\nu}+\theta^{\alpha\beta}\partial_{\alpha}\left[A_{\beta}j_{\mu}F^{\mu\nu}\right]\right)$
(11)

where $A_{\mu}$ and $j_{\mu}$ ado  are given as:

$\displaystyle A_{\mu}$ $\displaystyle=$ $\displaystyle
A_{\mu}^{(0)}+A_{\mu}^{(1)}(\theta)+O(\theta^{2})$ (12) $\displaystyle
j_{\mu}$ $\displaystyle=$ $\displaystyle
j_{\mu}^{(0)}+j_{\mu}^{(1)}(\theta)+O(\theta^{2})$ (13)

Radiation reaction four force on a charge in noncommutative electromagnetic
theory can therefore be defined by

$G^{\nu}_{NC}=\int K^{\nu}_{rad}d^{3}x$ (14)

where,

$K_{rad}^{\nu}=\frac{1}{2c}j_{\mu}F^{\mu\nu}_{rad}+\frac{1}{4c}\theta^{\alpha\beta}\partial_{\alpha}\left[j_{\mu}A_{\beta}^{rad}F^{\mu\nu}_{rad}\right].$
(15)

##  III Radiation reaction in noncommutative spacetime

We shall now turn up to the calculation of the radiation reaction in non-
commutative space-time. The radiation reaction in noncommutative spacetime
reads:

$\displaystyle G^{\nu}_{NC}$ $\displaystyle=$ $\displaystyle\frac{1}{2c}\int
J_{\mu}F^{\mu\nu}_{rad}\,d^{3}x$ (16) $\displaystyle+$
$\displaystyle\frac{\theta^{0i}}{4c}\int\partial_{0}(A_{i}^{rad}J_{\mu}F^{\mu\nu}_{rad})\,d^{3}x$
$\displaystyle+$
$\displaystyle\frac{\theta^{ij}}{4c}\int\partial_{i}(A_{j}^{rad}J_{\mu}F^{\mu\nu}_{rad})\,d^{3}x$

where i, j=1, 2, 3 are spatial indices and index $0$ stands for the temporal
index. All terms containing $\theta^{ij}$ involve total spatial derivative and
hence must vanish since the vector potential as well as the electromagnetic
field tensor vanish at the large spatial boundary. The non-trivial surviving
terms are given by

$\displaystyle G^{\nu}_{NC}=\frac{1}{2c}\int
J_{\mu}F^{\mu\nu}_{rad}\,d^{3}x+\frac{\theta^{0i}}{4c}\int\partial_{0}(A_{i}^{rad}J_{\mu}F^{\mu\nu}_{rad})\,d^{3}x$
(17)

The term $\frac{1}{2c}\int J_{\mu}F^{\mu\nu}_{rad}\,d^{3}x$ turns out bar :

$\displaystyle\frac{1}{2c}\int J_{\mu}F^{\mu\nu}_{rad}d^{3}x$ $\displaystyle=$
$\displaystyle\frac{e}{2}\int
F^{\mu\nu}_{rad}d^{3}x\int\dot{z}_{\mu}(s)\delta(x(t)-z(s))ds$ (18)
$\displaystyle=$
$\displaystyle\frac{e}{2}\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}F^{\mu\nu}_{rad}(z(s))$
$\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{e^{2}}{4\pi}\frac{1}{\dot{z}^{0}}\left(\dddot{z}^{\nu}+\dot{z}^{\nu}\ddot{z}^{2}\right)$

The second term to the RHS of the equation (17) could be simplified
decomposing into two terms (please see Appendix A) as follows:

$\displaystyle\int\frac{\partial}{\partial
x^{0}}(A_{i}^{rad}F^{\mu\nu}_{rad})J_{\mu}\,d^{3}x$
$\displaystyle=\frac{2}{3}\frac{e^{3}c}{4\pi^{2}}\left(\dddot{z}^{\nu}(s)+\dot{z}^{\nu}(s){\ddot{z}^{2}(s)}\right)\times$
$\displaystyle\frac{1}{\dot{z}^{0}(s)}\left(\frac{1}{3}\dot{z}_{i}(s)\dddot{z}_{0}(s)+\ddot{z}_{i}(s)\ddot{z}_{0}(s)+\dddot{z}_{i}(s)\dot{z}_{0}(s)\right)$
$\displaystyle+\frac{e^{3}c}{4\pi^{2}}\frac{\ddot{z}_{i}(s)}{\dot{z}^{0}(s)}\left[\frac{\dddot{z}^{0}(s)\ddot{z}^{\nu}(s)}{6}+\frac{2}{3}\ddot{z}^{0}(s)\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)\right.$
$\displaystyle\left.+\ddot{z}^{2}(s)\ddot{z}^{\nu}(s)\left(\frac{1}{2}\dot{z}^{0}(s)+\frac{2}{3}\frac{1}{\dot{z}^{0}(s)}\right)\right]$
(19)

and

$\displaystyle\int\frac{\partial J_{\mu}}{\partial
x^{0}}(A_{i}^{rad}F^{\mu\nu}_{rad})\,d^{3}x$
$\displaystyle=\frac{2}{3}\frac{e^{3}c}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)}{(\dot{z}^{0}(s))^{2}}\dot{z}^{\nu}(s)\ddot{z}_{\mu}(s)\dddot{z}^{\mu}(s)$
$\displaystyle-\frac{2}{3}\frac{e^{3}c}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)\ddot{z}^{0}(s)}{(\dot{z}^{0}(s))^{3}}\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)$
(20)

So that the radiation reaction becomes

$\displaystyle
G^{\nu}_{NC}=-\frac{2}{3}\frac{e^{2}}{4\pi}\frac{1}{\dot{z}^{0}(s)}\left(\dddot{z}^{\nu}(s)+\dot{z}^{\nu}(s)\ddot{z}^{2}(s)\right)$
$\displaystyle+{\theta}^{0i}\frac{e^{3}}{16{\pi}^{2}}\times$
$\displaystyle\left\\{\frac{\ddot{z}^{i}(s)}{\dot{z}^{0}(s)}\left[\frac{2}{3}\ddot{z}^{0}(s)\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)\left(\frac{1}{(\dot{z}^{0}(s))^{2}}-2\right)\right.\right.$
$\displaystyle\left.\left.-\frac{1}{6}\dddot{z}^{0}(s)\ddot{z}^{\nu}(s)-\ddot{z}^{2}(s)\ddot{z}^{\nu}(s)\left(\frac{1}{2}\dot{z}^{0}(s)+\frac{2}{3}\frac{1}{\dot{z}^{0}(s)}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{\dot{z}^{\nu}(s)\ddot{z}_{\mu}(s)\dddot{z}^{\mu}(s)}{\dot{z}^{0}(s)}\right]-\frac{2}{3}\left(\dddot{z}^{\nu}(s)+\dot{z}^{\nu}(s){\ddot{z}^{2}(s)}\right)\times\right.$
$\displaystyle\left.\frac{1}{\dot{z}^{0}(s)}\left(\frac{1}{3}\dot{z}^{i}(s)\dddot{z}_{0}(s)+\dddot{z}^{i}(s)\dot{z}_{0}(s)\right)\right\\}$

Moreover, in terms of 3-velocity, radiation reaction takes the following form

$\displaystyle G_{j}^{NC}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{e^{2}\gamma^{2}}{4\pi
c^{3}}\left(\ddot{v}_{j}+\frac{3\gamma^{2}}{c^{2}}\dot{v}_{j}\vec{v}.\dot{\vec{v}}+\frac{\gamma^{2}}{c^{2}}v_{j}\vec{v}.\ddot{\vec{v}}+\frac{3\gamma^{4}}{c^{4}}(\vec{v}.\dot{\vec{v}})^{2}v_{j}\right)$
(22) $\displaystyle-$
$\displaystyle\theta^{0i}\frac{e^{3}\gamma^{6}}{16\pi^{2}c^{6}}\left(\frac{2}{3}\ddot{v}_{i}\ddot{v}_{j}-\dot{v}_{i}v_{j}\dot{\vec{v}}.\ddot{\vec{v}}\right)$

where $\textit{i,~{}j}=1,2,3$.
Thus the radiation reaction force is modified by $\theta-$dependent terms and
it, in fact, depends on both space-space $\theta^{ij}$ and space-time
$\theta^{0i}$ non commutative parameters due to $\theta^{\mu\nu}$ dependence
of velocity $v_{j}$ and its various order time derivatives.
In the non-relativistic limit $v/c\to 0$, radiation reaction force turns out
to be

$\displaystyle G_{j}^{NC}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{e^{2}}{4\pi
c^{3}}\ddot{v}_{j}\left(1+\theta^{0i}\frac{e}{4\pi c^{3}}\ddot{v}_{i}\right).$
(23)

The non-relativistic expression for radiation reaction in noncommutative
electrodynamics, unlike its commutative counterpart, receives
$\theta$-correction term. Even in the case of zero space-time noncommutativity
$\theta^{0i}=0$, space-space noncommutativity $\theta^{ij}\neq 0$ is there
through $\theta^{\mu\nu}$ dependence of $\ddot{v}_{j}$ which could be seen in
the example of radiation reaction experienced by charged harmonic oscillator
in noncommutative plane which is discussed in the Section 4. It is worthy to
note at this juncture that, the non relativistic limit $\frac{v}{c}\rightarrow
0$ doesn’t imply the commutative limit $\theta^{\mu\nu}\rightarrow 0$ and vice
versa. The Abraham-Lorentz equation in noncommutative spacetime now reads:

$ma_{j}-m\tau\dot{a_{j}}-\frac{3m^{2}\tau^{2}}{2e}\theta^{0i}\dot{a_{i}}\dot{a_{j}}=F_{j}$
(24)

where $\dot{v}_{j}=a_{j},$ the characteristic time
$\tau=\frac{2}{3}\frac{e^{2}}{4\pi mc^{3}}$ and $F_{j}$ is the external force.
Let us consider one space - one time dimensions and assume
$\theta^{01}=\theta$ then the radiation reaction force becomes

$\displaystyle G^{NC}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{e^{2}}{4\pi
c^{3}}\left(\ddot{v}+\theta\frac{e}{4\pi c^{3}}\ddot{v}^{2}\right).$ (25)

Now, the equation of motion of a charge particle under a constant external
force $F$ is given by

$\frac{3}{2}\frac{m\tau^{2}\theta}{e}\dot{a}^{2}+\tau\dot{a}-a+\frac{F}{m}=0$
(26)

Equation (26) is quadratic in $\dot{a}$. The acceptable root which is
consistent in the commutative limit $\theta\to 0$ is

$\dot{a}=\frac{e}{3m\tau\theta}\left(-1+\sqrt{1+\frac{6m\theta}{e}(a-F/m)}\right)$
(27)

Suppose the constant external force $F$ acts on the charge only during the
time interval $0\leq t\leq T$. Therefore, in the region $t\leq 0,$ $F=0$ and
equation (27) leads to

$\dot{a}=\frac{e}{3m\tau\theta}\left(-1+\sqrt{1+\frac{6m\theta}{e}}a\right)$
(28)

which, after integration, yields

$\displaystyle(-1+\sqrt{1+\frac{6m\theta
a_{1}}{e}})\exp(-1+\sqrt{1+\frac{6m\theta
a_{1}}{e}})=I_{1}\exp{\frac{t}{\tau}}$ (29)

where $I_{1}$ is the integration constant. In the region $0\leq t\leq T,$ we
have

$(-1+\sqrt{1+\frac{6m\theta(a_{2}-\frac{F}{m})}{e}})\exp(-1+\sqrt{1+\frac{6m\theta(a_{2}-\frac{F}{m})}{e}})=I_{2}\exp{\frac{t}{\tau}}$
(30)

Similarly for the region $t\geq T$, external force F is zero and the
connection between acceleration and time in this region turns out

$\displaystyle(-1+\sqrt{1+\frac{6m\theta
a_{3}}{e}})\exp(-1+\sqrt{1+\frac{6m\theta
a_{3}}{e}})=I_{3}\exp{\frac{t}{\tau}}$ (31)

$I_{2}$ and $I_{3}$ are integration constants. The acceleration versus time in
the region $t\geq T$ with $T=1\tau$ is plotted as shown in the Figure 1.

Figure 1:

We notice that the acceleration in the region $t\geq T$ is increasing
continuously with time $t$. Therefore, equation (31) will admit runaway
solutions.
Thus, Abraham-Lorentz equation in noncommutative spacetime is plagued with
preacceleration and runaway solutions.

##  IV Radiation reaction in two dimensional noncommutative harmonic
oscillator

In this section, we explore the radiation reaction force due to a charge
undergoing simple harmonic motion in a noncommutative plane
$([\hat{x},\hat{y}]=i\theta)$.
The nonrelativistic Lagrangian for 2-dimensional noncommutative harmonic
oscillator dra  in terms of commutative coordinates $q_{1}$ and $q_{2}$ to the
first order in $\theta$ is given by:

$L=\frac{m}{2}\left(\dot{q}_{1}^{2}+\dot{q}_{2}^{2}\right)+\frac{m^{2}\omega^{2}\theta}{2}\left(\dot{q}_{2}q_{1}-\dot{q}_{1}q_{2}\right)-\frac{m\omega^{2}}{2}\left({q}_{1}^{2}+{q}_{2}^{2}\right).$
(32)

The Euler-Lagrange equations in $q_{1}$ and $q_{2}$, unlike commutative
counterpart, are coupled equations as given below:

$\displaystyle\ddot{q}_{1}-m\omega^{2}\theta\dot{q}_{2}+\omega^{2}q_{1}=0$
$\displaystyle\ddot{q}_{2}+m\omega^{2}\theta\dot{q}_{1}+\omega^{2}q_{2}=0.$

The effect of noncommutativity is to couple the two independent modes and
hence the system now oscillates with two distinct normal mode frequencies, say
$\omega_{+}$ and $\omega_{-}$. The general solution of the above differential
equations are given by

$\displaystyle q_{1}(t)$ $\displaystyle=$ $\displaystyle
C_{1}\cos(\omega_{+}t)+C_{2}\sin(\omega_{+}t)$
$\displaystyle{}+C_{3}\cos(\omega_{-}t)+C_{4}\sin(\omega_{-}t)$ $\displaystyle
q_{2}(t)$ $\displaystyle=$ $\displaystyle
D_{1}\cos(\omega_{+}t)+D_{2}\sin(\omega_{+}t)$
$\displaystyle{}+D_{3}\cos(\omega_{-}t)+D_{4}\sin(\omega_{-}t)$

where,

$\displaystyle\omega_{+}$ $\displaystyle=$
$\displaystyle\omega+\frac{m\theta\omega^{2}}{2}$ $\displaystyle\omega_{-}$
$\displaystyle=$ $\displaystyle-\omega+\frac{m\theta\omega^{2}}{2}.$

$C_{i}$ and $D_{i}$ are integration constants to be fixed by the initial
conditions. Moreover, there are only four independent constants
$(C_{1}=-D_{2},C_{2}=D_{1},C_{3}=-D_{4},C_{4}=D_{3})$ out of the eight ones
due to the differential relations between $q_{1}$ and $q_{2}$. The radiation
reaction experienced by the charged harmonic oscillator in the noncommutative
plane using (23) is given by

$\displaystyle\vec{F}_{rad}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{e^{2}}{4\pi
c^{3}}[{\omega_{+}}^{3}(C_{1}\sin{\omega_{+}t}-C_{2}\cos{\omega_{+}t})$ (33)
$\displaystyle+$
$\displaystyle{\omega_{-}}^{3}(C_{3}\sin{\omega_{-}t}-C_{4}\cos{\omega_{-}t})]\hat{q}_{1}$
$\displaystyle-$ $\displaystyle\frac{2}{3}\frac{e^{2}}{4\pi
c^{3}}[{\omega_{+}}^{3}(C_{1}\cos{\omega_{+}t}+C_{2}\sin{\omega_{+}t})$
$\displaystyle+$
$\displaystyle{\omega_{-}}^{3}(C_{3}\cos{\omega_{-}t}+C_{4}\sin{\omega_{-}t})]\hat{q}_{2}$

Where $\hat{q}_{1}$ and $\hat{q}_{2}$ are unit vectors along $q_{1}$ and
$q_{2}$ directions respectively. We shall commit to the specific case when the
initial conditions are such that $C_{1}=C_{2}=C_{3}=C_{4}=1$. This choice does
not alter the generalities of the interpretation of result for radiation
reaction force due to 2D noncommutative charged harmonic oscillator. The
quantity $\frac{m\theta\omega^{2}t}{2}$ for all practical purposes is rather
small and could admit large enough time to observe radiation reaction.
Therefore, we can have for our purpose
$\sin\frac{m\theta\omega^{2}t}{2}\sim\frac{m\theta\omega^{2}t}{2}$, so that

$\displaystyle\cos\left(\pm\omega+\frac{m\theta\omega^{2}}{2}\right)t$
$\displaystyle\sim$ $\displaystyle\cos{\omega
t}\mp\left(\frac{m\theta\omega^{2}t}{2}\right)\sin{\omega t}$
$\displaystyle\sin\left(\pm\omega+\frac{m\theta\omega^{2}}{2}\right)t$
$\displaystyle\sim$ $\displaystyle\pm\sin{\omega
t}+\left(\frac{m\theta\omega^{2}t}{2}\right)\cos{\omega t}$

so that,

$\displaystyle q_{1}$ $\displaystyle=$ $\displaystyle 2\cos{\omega
t}+\left(m\theta\omega^{2}t\right)\cos{\omega t}$ $\displaystyle q_{2}$
$\displaystyle=$ $\displaystyle 2\cos{\omega
t}-\left(m\theta\omega^{2}t\right)\cos{\omega t}$ $\displaystyle\dddot{q}_{1}$
$\displaystyle=$ $\displaystyle 2\omega^{3}\sin\omega
t-m\theta\omega^{4}(3\cos\omega t-\omega t\sin\omega t)$ (34)
$\displaystyle\dddot{q}_{2}$ $\displaystyle=$ $\displaystyle
2\omega^{3}\sin\omega t+m\theta\omega^{4}(3\cos\omega t-\omega t\sin\omega t)$
(35)

Radiation reaction turns out

$\vec{F}_{Rad}=(\hat{q_{1}}+\hat{q_{2}})f(t)+(\hat{q_{1}}-\hat{q_{2}})\theta
g(t)$ (36)

where

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle-\frac{2}{3}\hskip
2.0pt\frac{e^{2}}{4\pi c^{3}}2\omega^{3}\sin\omega t$ (37) $\displaystyle
g(t)$ $\displaystyle=$ $\displaystyle\frac{2}{3}\hskip 2.0pt\frac{e^{2}}{4\pi
c^{3}}m\omega^{4}(3\cos\omega t-\omega t\sin\omega t).$ (38)

Let us define

$\displaystyle f_{s}(t)$ $\displaystyle=$
$\displaystyle\frac{f(t)}{-\frac{2}{3}\hskip 2.0pt\frac{e^{2}}{4\pi
c^{3}}2\omega^{3}}$ (39) $\displaystyle g_{s}(t)$ $\displaystyle=$
$\displaystyle\frac{g(t)}{\frac{2}{3}\hskip 2.0pt\frac{e^{2}}{4\pi
c^{3}}m\omega^{4}}.$ (40)

Figure 2 shows a plot between the scaled components $f_{s}(t)$ and $g_{s}(t)$
of the radiation reaction and time.

Figure 2:

Radiation reaction in the commutative space vanishes at all times
$t=n\pi/\omega,$ where $n=0,1,2,3...$. However, radiation reaction in the non-
commutative space does not vanish at any point in time.

##  V Conclusion

The present paper explores whether Maxwell’s electromagnetic theory in the
noncommutative spacetime might affect the radiation reaction and could
possibly resolve the pathologies and ambiguities associated with the Abraham-
Lorentz equation. To this end, we have derived the radiation reaction force
acting on an accelerating charge due to the electromagnetic fields that it
produces while moving in the noncommutative spacetime. We find that an
accelerating point charge in noncommutative Maxwell theory experiences
radiation reaction which receives a small $\theta-$dependent correction term.
Moreover, it turns out that the Araham-Lorentz equation in noncommutative
spacetime suffers from the same long-standing problems of the preacceleration
and runaway solutions.

## Appendix A : Calculation of radiation reaction in noncommutative
electrodynamics

We have

$\displaystyle\int
d^{3}x\,\partial_{0}\left(A_{i}^{rad}(x)J_{\mu}(x)F^{\mu\nu}_{rad}(x)\right)$
$\displaystyle=\int d^{3}x\,\frac{\partial}{\partial
x^{0}}\left(A_{i}^{rad}(x)F^{\mu\nu}_{rad}(x)\right)J_{\mu}(x)$
$\displaystyle+\int d^{3}x\,\frac{\partial J_{\mu}(x)}{\partial
x^{0}}A_{i}^{rad}(x)F^{\mu\nu}_{rad}(x)$ (41)

Four vector current density due to a point charge is given by

$\displaystyle J_{\mu}(x)$ $\displaystyle=$ $\displaystyle ec\int
ds\,\dot{z}_{\mu}(s)\delta^{4}(x-z(s))$ (42) $\displaystyle=$ $\displaystyle
ec\frac{\dot{z}_{\mu}(s)\,\delta^{3}(\vec{x}-\vec{z}(s))}{\dot{z}^{0}(s)}\rvert_{x^{0}=z^{0}(s)}$

The first term in the above equation after inserting (42) gets simplified as

$\displaystyle\int d^{3}x\,\frac{\partial}{\partial
x^{0}}(A_{i}^{rad}(x)F^{\mu\nu}_{rad}(x))J_{\mu}(x)$ (43) $\displaystyle=$
$\displaystyle
ec\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}F^{\mu\nu}_{rad}(z)\left(\frac{\partial
A_{i}^{rad}}{\partial x^{0}}\right)_{x=z(s)}$ $\displaystyle+$ $\displaystyle
ec\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}A_{i}^{rad}(z)\left(\frac{\partial
F^{\mu\nu}_{rad}}{\partial x^{0}}\right)_{x=z(s)}$

The second term in the equation (41) becomes

$\displaystyle{}\int d^{3}x\frac{\partial J_{\mu}(x)}{\partial
x^{0}}(A_{i}^{rad}(x)F^{\mu\nu}_{rad}(x))$
$\displaystyle=ec\frac{A^{rad}_{i}(z)}{\dot{z}^{0}(s)}{F^{\mu\nu}_{rad}(z)}\frac{\partial}{\partial
s}\left(\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}\right)$ (44)

Four vector potential is defined as

$\displaystyle A_{\mu}^{rad}(x)$ $\displaystyle=$ $\displaystyle e\int
ds\,D(x-z(s))\dot{z}_{\mu}(s)$

At $x=z(s^{\prime})$, we have

$\displaystyle A_{\mu}^{rad}z(s^{\prime})$ $\displaystyle=$ $\displaystyle
e\int D(z(s^{\prime})-z(s))\dot{z}_{\mu}(s)ds$ (45)

Let $s=s^{\prime}+u$ where u is a small parameter.

$\displaystyle z_{\mu}(s)$ $\displaystyle=$ $\displaystyle
z_{\mu}(s^{\prime})+u\dot{z}_{\mu}(s^{\prime})+\frac{u^{2}}{2}\ddot{z}_{\mu}(s^{\prime})+\frac{u^{3}}{6}\dddot{z}_{\mu}(s^{\prime})+...$
$\displaystyle\dot{z}_{\mu}(s)$ $\displaystyle=$
$\displaystyle\dot{z}_{\mu}(s^{\prime})+u\ddot{z}_{\mu}(s^{\prime})+\frac{u^{2}}{2}\dddot{z}_{\mu}(s^{\prime})+...$

We know bar

$\displaystyle D(z(s^{\prime})-z(s))$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{d\delta(u)}{du}$

Therefore

$\displaystyle
A^{rad}_{\mu}(z(s^{\prime}))=-\frac{e}{2\pi}\int\delta(u)du\,(\ddot{z}_{\mu}(s^{\prime})+u\dddot{z}_{\mu}(s^{\prime})+..)$

At $u=0,s=s^{\prime}$. Hence

$\displaystyle A^{rad}_{\mu}(z(s))$ $\displaystyle=$
$\displaystyle-\frac{e}{2\pi}\ddot{z}_{\mu}(s)$ (46)

##  A1: Calculation of $\mathbf{(\partial_{\nu}A_{\mu}^{rad})_{x=z(s)}:}$

$\displaystyle\partial_{\nu}A_{\mu}^{rad}(x)=e\frac{\partial}{\partial
x^{\nu}}\int ds\,D(x-z(s))\dot{z}_{\mu}(s)$ $\displaystyle=e\int
ds\,D(x-z(s))\frac{d}{ds}\left[\frac{\dot{z}_{\mu}(s)(x-z(s))_{\nu}}{(x-z(s))^{\sigma}\dot{z}_{\sigma}(s)}\right]$

At $x=z(s^{\prime})$

$\displaystyle\partial_{\nu}A_{\mu}^{rad}(z(s^{\prime}))=$ $\displaystyle
e\int
ds\,D((z(s^{\prime})-z(s))\frac{d}{ds}\left(\frac{\dot{z}_{\mu}(s)(z(s^{\prime})-z(s))_{\nu}}{(z(s^{\prime})-z(s))^{\sigma}\dot{z}_{\sigma}}\right)$

Now:

$(z(s^{\prime})-z(s))^{\sigma}\dot{z}_{\sigma}=-u+o(u^{3})$ (47)
$\displaystyle\dot{z}_{\mu}(s)(z(s^{\prime})-z(s))_{\nu}$
$\displaystyle=-u\left\\{\dot{z}_{\mu}(s^{\prime})\dot{z}_{\nu}(s^{\prime})+\frac{u}{2}\dot{z}_{\mu}(s^{\prime})\ddot{z}_{\nu}(s^{\prime})+u\ddot{z}_{\mu}(s^{\prime})\dot{z}_{\nu}(s^{\prime})\right.$
$\displaystyle\left.+\frac{u^{2}}{6}\dot{z}_{\mu}(s^{\prime})\dddot{z}_{\nu}(s^{\prime})+\frac{u^{2}}{2}\ddot{z}_{\mu}(s^{\prime})\ddot{z}_{\nu}(s^{\prime})\right.$
$\displaystyle\left.+\frac{u^{2}}{2}\dddot{z}_{\mu}(s^{\prime})\dot{z}_{\nu}(s^{\prime})+...\right\\}$
(48) $\displaystyle(\partial_{\nu}A_{\mu}^{rad})_{x=z(s)}=$
$\displaystyle-\frac{e}{2\pi}\left(\frac{1}{3}\dot{z}_{\mu}(s)\dddot{z}_{\nu}(s)+\ddot{z}_{\mu}(s)\ddot{z}_{\nu}(s)+\dddot{z}_{\mu}(s)\dot{z}_{\nu}(s)\right)$
(49)

Therefore:

$\displaystyle F^{\mu\nu}_{rad}(z(s))$ $\displaystyle=$
$\displaystyle\left(\partial^{\mu}A^{\nu}_{rad}-\partial^{\nu}A^{\mu}_{rad}\right)_{x=z(s)}$
(50) $\displaystyle=$
$\displaystyle\frac{e}{3\pi}\left(\dot{z}^{\nu}(s)\dddot{z}^{\mu}(s)-\dddot{z}^{\nu}(s)\dot{z}^{\mu}(s)\right)$

## A2: Calculation of the term

$\mathbf{ec\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}F^{\mu\nu}_{rad}(z(s))(\frac{\partial
A_{i}^{rad}}{\partial x^{0}})_{x=z(s)}:}$
From equation (49)

$\displaystyle(\partial_{0}A_{i}^{rad})_{x=z(s)}=$
$\displaystyle-\frac{e}{2\pi}\left(\frac{1}{3}\dot{z}_{i}(s)\dddot{z}_{0}(s)+\ddot{z}_{i}(s)\ddot{z}_{0}(s)+\dddot{z}_{i}(s)\dot{z}_{0}(s)\right)$

Therefore

$\displaystyle
ec\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}F^{\mu\nu}_{rad}(z(s))(\partial_{0}A_{i}^{rad})_{x=z(s)}=$
$\displaystyle-\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}\frac{2}{3}\frac{e^{2}c}{2\pi}\left(\dot{z}^{\nu}(s)\dddot{z}^{\mu}(s)-\dddot{z}^{\nu}(s)\dot{z}^{\mu}(s)\right)\times$
$\displaystyle{}\frac{e}{2\pi}\left(\frac{1}{3}\dot{z}_{i}(s)\dddot{z}_{0}(s)+\ddot{z}_{i}(s)\ddot{z}_{0}(s)+\dddot{z}_{i}(s)\dot{z}_{0}(s)\right)$

We know that :

$\displaystyle\dot{z}^{\mu}(s)\dot{z}_{\mu}(s)=1,\dot{z}_{\mu}(s)\ddot{z}^{\mu}(s)=0$
$\displaystyle\dot{z}_{\mu}(s)\dddot{z}^{\mu}(s)+\ddot{z}_{\mu}(s)\ddot{z}^{\mu}(s)=0$
$\displaystyle\dot{z}_{\mu}(s)\dddot{z}^{\mu}(s)=-{\ddot{z}}^{2}(s)$

Now,

$\displaystyle
ec\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}F^{\mu\nu}_{rad}(z(s))(\partial_{0}A_{i}^{rad})_{x=z(s)}$
$\displaystyle=\frac{2}{3}\frac{e^{3}c}{4\pi^{2}}\left(\dddot{z}^{\nu}(s)+\dot{z}^{\nu}(s){\ddot{z}^{2}(s)}\right)\times$
$\displaystyle\frac{1}{\dot{z}^{0}(s)}\left(\frac{1}{3}\dot{z}_{i}(s)\dddot{z}_{0}(s)+\ddot{z}_{i}(s)\ddot{z}_{0}(s)+\dddot{z}_{i}(s)\dot{z}_{0}(s)\right)$
(51)

## A3: Calculation of the term

$\mathbf{ec\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}A_{i}^{rad}(z(s))(\partial_{0}F^{\mu\nu}_{rad}(x))_{x=z(s)}:}$
We have,

$\partial_{0}F^{\mu\nu}_{rad}(x)=\frac{\partial}{\partial
x^{0}}(\partial^{\mu}A^{\nu}(x)-\partial^{\nu}A^{\mu}(x))$ (52)

Where,

$\displaystyle\frac{\partial}{\partial x^{0}}(\partial^{\nu}A^{\mu}(x))$
$\displaystyle=e\int ds\,\frac{\partial D(x-z(s))}{\partial
x^{0}}\frac{d}{ds}\left[\frac{\dot{z}^{\mu}(s)(x-z(s))^{\nu}}{(x-z(s))^{\sigma}\dot{z}_{\sigma}(s)}\right]+$
$\displaystyle e\int ds\,D(x-z(s))\frac{d}{ds}\frac{\partial}{\partial
x^{0}}\left[\frac{\dot{z}^{\mu}(s)(x-z(s))^{\nu}}{(x-z(s))^{\sigma}\dot{z}_{\sigma}(s)}\right]$
(53)

The first term of the above equation (53) is

$\displaystyle e\int ds\,\frac{\partial D(x-z(s))}{\partial
x^{0}}\frac{d}{ds}\left[\frac{\dot{z}^{\mu}(s)(x-z(s))^{\nu}}{(x-z(s))^{\sigma}\dot{z}_{\sigma}}\right]$
$\displaystyle=e\int
ds\,D(x-z(s))\frac{d}{ds}\left\\{\frac{(x-z(s))^{0}}{(x-z(s))^{\rho}\dot{z}_{\rho}(s)}\times\right.$
$\displaystyle\left.\frac{d}{ds}\left[\frac{\dot{z}^{\mu}(s)(x-z(s))^{\nu}}{(x-z(s))^{\sigma}\dot{z}_{\sigma}(s)}\right]\right\\}$

At $x=z(s^{\prime})$ the above equation becomes:

$\displaystyle=$ $\displaystyle e\int
ds\,D(z(s^{\prime})-z(s))\left\\{\frac{d}{ds}\left(\frac{(z(s^{\prime})-z(s))^{0}}{(z(s^{\prime})-z(s))^{\rho}\dot{z}_{\rho}(s)}\right)\right.$
(54)
$\displaystyle\left.\times\frac{d}{ds}\left[\frac{\dot{z}^{\mu}(s)(z(s^{\prime})-z(s))^{\nu}}{(z(s^{\prime})-z(s))^{\sigma}\dot{z}_{\sigma}(s)}\right]\right\\}$
$\displaystyle+$ $\displaystyle e\int
ds\,D(z(s^{\prime})-z(s))\left\\{\frac{(z(s^{\prime})-z(s))^{0}}{(z(s^{\prime})-z(s))^{\rho}\dot{z}_{\rho}(s)}\right.$
$\displaystyle\left.\times\frac{d^{2}}{ds^{2}}\left[\frac{\dot{z}^{\mu}(s)(z(s^{\prime})-z(s))^{\nu}}{(z(s^{\prime})-z(s))^{\sigma}\dot{z}_{\sigma}(s)}\right]\right\\}$

The first term in equation (54) can be expressed as

$\displaystyle=\frac{e}{2\pi}\int
du\,\frac{d\delta(u)}{du}\left\\{\left[\frac{d}{du}(\dot{z}^{0}(s^{\prime})\right.\right.$
$\displaystyle\left.\left.+\frac{u}{2}\ddot{z}^{0}(s^{\prime})+\frac{u^{2}}{6}\dddot{z}^{0}(s^{\prime})....)\right]\right.$
$\displaystyle\left.\times\frac{d}{du}\left(\dot{z}^{\mu}(s^{\prime})\dot{z}^{\nu}(s^{\prime})+\frac{u}{2}\dot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})\right.\right.$
$\displaystyle\left.\left.+u\ddot{z}^{\mu}(s^{\prime})\dot{z}^{\nu}(s^{\prime})+\frac{u^{2}}{6}\dot{z}^{\mu}(s^{\prime})\dddot{z}^{\nu}(s^{\prime})\right.\right.$
$\displaystyle\left.\left.+\frac{u^{2}}{2}\ddot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})+\frac{u^{2}}{2}\dddot{z}^{\mu}(s^{\prime})\dot{z}^{\nu}(s^{\prime})\right.\right.$
$\displaystyle\left.\left.+\frac{u^{3}}{6}\ddot{z}^{\mu}(s^{\prime})\dddot{z}^{\nu}(s^{\prime})+\frac{u^{3}}{4}\dddot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})...\right)\right\\}$
$\displaystyle=-\frac{e}{2\pi}\int
du\,\delta(u)\left(\frac{d^{2}P(u)}{du^{2}}\frac{dQ(u)}{du}+\frac{dP(u)}{du}\frac{d^{2}Q(u)}{du^{2}}\right)$
(55)

where

$\displaystyle
P(u)=(\dot{z}^{0}(s^{\prime})+\frac{u}{2}\ddot{z}^{0}(s^{\prime})+\frac{u^{2}}{6}\dddot{z}^{0}(s^{\prime})....)$
$\displaystyle
Q(u)=\dot{z}^{\mu}(s^{\prime})\dot{z}^{\nu}(s^{\prime})+\frac{u}{2}\dot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})$
$\displaystyle+u\ddot{z}^{\mu}(s^{\prime})\dot{z}^{\nu}(s^{\prime})+\frac{u^{2}}{6}\dot{z}^{\mu}(s^{\prime})\dddot{z}^{\nu}(s^{\prime})$
$\displaystyle+\frac{u^{2}}{2}\ddot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})+\frac{u^{2}}{2}\dddot{z}^{\mu}(s^{\prime})\dot{z}^{\nu}(s^{\prime})$
$\displaystyle+\frac{u^{3}}{6}\ddot{z}^{\mu}(s^{\prime})\dddot{z}^{\nu}(s^{\prime})+\frac{u^{3}}{4}\dddot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})...$
(56)

The second term in the equation (54) can be expressed as

$\displaystyle=-\frac{e}{2\pi}\int
du\,\delta(u)\left(\frac{dP(u)}{du}\frac{d^{2}Q(u)}{du^{2}}+P(u)\frac{d^{3}Q(u)}{du^{3}}\right)$
(57)

Now, equation (54) becomes:

$\displaystyle=-\frac{e}{2\pi}\int
du\,\delta(u)\left\\{\frac{d^{2}P(u)}{du^{2}}\frac{dQ(u)}{du}\right.$
$\displaystyle\left.+2\frac{dP(u)}{du}\frac{d^{2}Q(u)}{du^{2}}+P(u)\frac{d^{3}Q(u)}{du^{3}}\right\\}$
$\displaystyle=-\frac{e}{2\pi}\left[\frac{\dddot{z}^{0}(s)}{3}\left(\frac{\dot{z}^{\mu}(s)\ddot{z}^{\nu}(s)}{2}+\ddot{z}^{\mu}(s)\dot{z}^{\nu}(s)\right)\right.$
$\displaystyle\left.+\ddot{z}^{0}(s)\\{\dddot{z}^{\nu}(s)\frac{\dot{z}^{\mu}(s)}{3}(s)+\ddot{z}^{\mu}(s)\ddot{z}^{\nu}(s)\right.$
$\displaystyle\left.+\dddot{z}^{\mu}(s)\dot{z}^{\nu}(s)\\}+\dot{z}^{0}(s)\\{\ddot{z}^{\mu}(s)\dddot{z}^{\nu}(s)\right.$
$\displaystyle\left.+\frac{3}{2}\dddot{z}^{\mu}(s)\ddot{z}^{\nu}(s)\\}\right]$
(58)

At $x=z(s^{\prime})$ the second term in the equation (53) is

$\displaystyle e\int ds\,D(z(s^{\prime})-z(s))\times$
$\displaystyle\frac{d}{ds}\frac{\partial}{\partial
z^{0}(s^{\prime})}\left[\frac{\dot{z}^{\mu}(s)(z(s^{\prime})-z(s))^{\nu}}{(z(s^{\prime})-z(s))^{\sigma}\dot{z}_{\sigma}(s)}\right]$
$\displaystyle=-\frac{e}{2\pi}\int
du\,\frac{1}{\dot{z}^{0}(s^{\prime})}\delta(u)\times$
$\displaystyle\frac{d^{2}}{du^{2}}\left(\dot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})+\ddot{z}^{\mu}(s^{\prime})\dot{z}^{\nu}(s^{\prime})+\frac{u}{2}\dot{z}^{\mu}(s^{\prime})\dddot{z}^{\nu}(s^{\prime})\right.$
$\displaystyle\left.+\frac{3u}{2}\ddot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})+u\dddot{z}^{\mu}(s^{\prime})\dot{z}^{\nu}(s^{\prime})\right.$
$\displaystyle\left.+\frac{2u^{2}}{3}\ddot{z}^{\mu}(s^{\prime})\dddot{z}^{\nu}(s^{\prime})+u^{2}\dddot{z}^{\mu}(s^{\prime})\ddot{z}^{\nu}(s^{\prime})\right)$
$\displaystyle=-\frac{e}{2\pi}\frac{1}{\dot{z}^{0}(s)}\left(\frac{4}{3}\dddot{z}^{\nu}(s)\ddot{z}^{\nu}(s)+2\dddot{z}^{\mu}(s)\ddot{z}^{\nu}(s)\right)$
(59)

By substituting equations (58) and (59) in the equation (53), we have,

$\displaystyle\frac{\partial}{\partial
x^{0}}(\partial^{\nu}A^{\mu}(x))_{x=z(s)}$
$\displaystyle=-\frac{e}{2\pi}\left[\frac{1}{3}\dddot{z}^{0}(s)\left\\{\frac{1}{2}\dot{z}^{\mu}(s)\ddot{z}^{\nu}(s)+\ddot{z}^{\mu}(s)\dot{z}^{\nu}(s)\right\\}\right.$
$\displaystyle\left.+\ddot{z}^{0}(s)\left\\{\frac{1}{3}\dddot{z}^{\nu}(s)\dot{z}^{\mu}(s)+\ddot{z}^{\mu}(s)\ddot{z}^{\nu}(s)+\dddot{z}^{\mu}(s)\dot{z}^{\nu}(s)\right\\}\right.$
$\displaystyle\left.+\dot{z}^{0}(s)\left\\{\ddot{z}^{\mu}(s)\dddot{z}^{\nu}(s)+\frac{3}{2}\dddot{z}^{\mu}(s)\ddot{z}^{\nu}(s)\right\\}\right.$
$\displaystyle\left.+\frac{1}{\dot{z}^{0}(s)}\left\\{\frac{4}{3}\dddot{z}^{\nu}(s)\ddot{z}^{\mu}(s)+2\dddot{z}^{\mu}(s)\ddot{z}^{\nu}(s)\right\\}\right]$
(60)

Now:

$\displaystyle(\partial_{0}F^{\mu\nu}(x))_{x=z(s)}$
$\displaystyle=\frac{e}{2\pi}\left[\frac{\dddot{z}^{0}(s)}{6}\left(\ddot{z}^{\mu}(s)\dot{z}^{\nu}(s)-\ddot{z}^{\nu}(s)\dot{z}^{\mu}(s)\right)\right.$
$\displaystyle\left.+\frac{2}{3}\ddot{z}^{0}(s)\left(\dddot{z}^{\mu}(s)\dot{z}^{\nu}(s)-\dddot{z}^{\nu}(s)\dot{z}^{\mu}(s)\right)\right.$
$\displaystyle\left.+\left(\dddot{z}^{\mu}(s)\ddot{z}^{\nu}(s)-\dddot{z}^{\nu}(s)\ddot{z}^{\mu}(s)\right)\left(\frac{1}{2}\dot{z}^{0}(s)+\frac{2}{3}\frac{1}{\dot{z}^{0}(s)}\right)\right]$

Therefore

$\displaystyle
ec\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}A_{i}^{rad}(z(s))(\partial_{0}F^{\mu\nu}_{rad}(x))_{x=z(s)}$
$\displaystyle=\frac{e^{3}c}{4\pi^{2}}\frac{\ddot{z}_{i}(s)}{\dot{z}^{0}(s)}\left[\frac{\dddot{z}^{0}(s)\ddot{z}^{\nu}(s)}{6}\right.$
$\displaystyle\left.+\frac{2}{3}\ddot{z}^{0}(s)\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)\right.$
$\displaystyle{}\left.+\ddot{z}^{2}(s)\ddot{z}^{\nu}(s)\left(\frac{1}{2}\dot{z}^{0}(s)+\frac{2}{3}\frac{1}{\dot{z}^{0}(s)}\right)\right]$
(61)

## A4: Calculation of the term

$\mathbf{ec\frac{A^{rad}_{i}(z(s))}{\dot{z}^{0}(s)}{F^{\mu\nu}_{rad}(z(s))}\frac{\partial}{\partial
s}\left(\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}\right):}$
We have,

$\displaystyle\frac{A^{rad}_{i}(z(s))}{\dot{z}^{0}(s)}{F^{\mu\nu}_{rad}(z(s))}\frac{\partial}{\partial
s}\left[\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)}\right]$
$\displaystyle=\frac{\ddot{z}_{\mu}(s)}{(\dot{z}^{0}(s))^{2}}F^{\mu\nu}_{rad}(z(s))A_{i}^{rad}(z(s))-\frac{\dot{z}_{\mu}\ddot{z}^{0}}{(\dot{z}^{0})^{3}}F^{\mu\nu}_{rad}(z)A_{i}^{rad}(z)$

Now,

$\displaystyle\frac{\ddot{z}_{\mu}(s)}{(\dot{z}^{0}(s))^{2}}F^{\mu\nu}_{rad}(z(s))A_{i}^{rad}(z(s))$
$\displaystyle=-\frac{2}{3}\frac{e^{2}}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)}{(\dot{z}^{0}(s))^{2}}\left(\dot{z}^{\mu}(s)\ddot{z}_{\mu}(s)\dddot{z}^{\nu}(s)-\dot{z}^{\nu}(s)\ddot{z}_{\mu}(s)\dddot{z}^{\mu}(s)\right)$
$\displaystyle{}=\frac{2}{3}\frac{e^{2}}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)}{(\dot{z}^{0}(s))^{2}}\dot{z}^{\nu}(s)\ddot{z}_{\mu}(s)\dddot{z}^{\mu}(s)$
(62)

And,

$\displaystyle\frac{\ddot{z}^{0}(s)\dot{z}_{\mu}(s)}{(\dot{z}^{0}(s))^{3}}F^{\mu\nu}_{rad}(z(s))A_{i}^{rad}(z(s))$
$\displaystyle=\frac{2}{3}\frac{e^{2}}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)\ddot{z}^{0}(s)}{(\dot{z}^{0}(s))^{3}}\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)$
(63)

Therefore

$\displaystyle
ec\frac{A^{rad}_{i}(z(s))}{\dot{z}^{0}(s)}{F^{\mu\nu}_{rad}(z(s))}\frac{\partial}{\partial
s}(\frac{\dot{z}_{\mu}(s)}{\dot{z}^{0}(s)})$
$\displaystyle=\frac{2}{3}\frac{e^{3}c}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)}{(\dot{z}^{0}(s))^{2}}\dot{z}^{\nu}(s)\ddot{z}_{\mu}(s)\dddot{z}^{\mu}(s)$
$\displaystyle-\frac{2}{3}\frac{e^{3}c}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)\ddot{z}^{0}(s)}{(\dot{z}^{0}(s))^{3}}\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)$
(64)

From equations (51) and (61)

$\displaystyle\int\frac{\partial}{\partial
x^{0}}(A_{i}^{rad}F^{\mu\nu}_{rad})J_{\mu}d^{3}x$
$\displaystyle=\frac{2}{3}\frac{e^{3}c}{4\pi^{2}}\left(\dddot{z}^{\nu}(s)+\dot{z}^{\nu}(s){\ddot{z}^{2}(s)}\right)\times$
$\displaystyle\frac{1}{\dot{z}^{0}(s)}\left(\frac{1}{3}\dot{z}_{i}(s)\dddot{z}_{0}(s)+\ddot{z}_{i}(s)\ddot{z}_{0}(s)+\dddot{z}_{i}(s)\dot{z}_{0}(s)\right)$
$\displaystyle+\frac{e^{3}c}{4\pi^{2}}\frac{\ddot{z}_{i}(s)}{\dot{z}^{0}(s)}\left[\frac{\dddot{z}^{0}(s)\ddot{z}^{\nu}(s)}{6}+\frac{2}{3}\ddot{z}^{0}(s)\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)\right.$
$\displaystyle\left.+\ddot{z}^{2}(s)\ddot{z}^{\nu}(s)\left(\frac{1}{2}\dot{z}^{0}(s)+\frac{2}{3}\frac{1}{\dot{z}^{0}(s)}\right)\right]$
(65)

From equation (62) and (63), we have

$\displaystyle\int\frac{\partial J_{\mu}}{\partial
x^{0}}(A_{i}^{rad}F^{\mu\nu}_{rad})d^{3}x$
$\displaystyle=\frac{2}{3}\frac{e^{3}c}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)}{(\dot{z}^{0}(s))^{2}}\dot{z}^{\nu}(s)\ddot{z}_{\mu}(s)\dddot{z}^{\mu}(s)$
$\displaystyle-\frac{2}{3}\frac{e^{3}c}{4{\pi}^{2}}\frac{\ddot{z}_{i}(s)\ddot{z}^{0}(s)}{(\dot{z}^{0}(s))^{3}}\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)$
(66)

Radiation reaction becomes:

$\displaystyle
G^{\nu}_{NC}=-\frac{2}{3}\frac{e^{2}}{4\pi}\frac{1}{\dot{z}^{0}(s)}\left(\dddot{z}^{\nu}(s)+\dot{z}^{\nu}(s)\ddot{z}^{2}(s)\right)$
$\displaystyle+{\theta}^{0i}\frac{e^{3}}{16{\pi}^{2}}$
$\displaystyle\left\\{\frac{\ddot{z}^{i}(s)}{\dot{z}^{0}(s)}\left[\frac{2}{3}\ddot{z}^{0}(s)\left(\dddot{z}^{\nu}(s)+\ddot{z}^{2}(s)\dot{z}^{\nu}(s)\right)\left(\frac{1}{(\dot{z}^{0}(s))^{2}}-2\right)\right.\right.$
$\displaystyle\left.\left.-\frac{1}{6}\dddot{z}^{0}(s)\ddot{z}^{\nu}(s)-\ddot{z}^{2}(s)\ddot{z}^{\nu}(s)\left(\frac{1}{2}\dot{z}^{0}(s)+\frac{2}{3}\frac{1}{\dot{z}^{0}(s)}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{\dot{z}^{\nu}(s)\ddot{z}_{\mu}(s)\dddot{z}^{\mu}(s)}{\dot{z}^{0}(s)}\right]-\frac{2}{3}\left(\dddot{z}^{\nu}(s)+\dot{z}^{\nu}(s){\ddot{z}^{2}(s)}\right)\times\right.$
$\displaystyle\left.\frac{1}{\dot{z}^{0}(s)}\left(\frac{1}{3}\dot{z}^{i}(s)\dddot{z}_{0}(s)+\dddot{z}^{i}(s)\dot{z}_{0}(s)\right)\right\\}$
(67)

Where,

$\displaystyle\frac{\ddot{z}^{i}(s)\ddot{z}^{0}(s)}{\dot{z}^{0}(s)}\left(\dddot{z}^{j}(s)+\dot{z}^{j}(s)\ddot{z}^{2}(s)\right)\left(\frac{1}{(\dot{z}^{0}(s))^{2}}-2\right)$
$\displaystyle=-\frac{1}{\gamma}\frac{\gamma^{2}}{c^{2}}\left(\dot{v}_{i}+\frac{\gamma^{2}}{c^{2}}v_{i}(\vec{v}.\dot{\vec{v}})\right)\frac{\gamma^{4}}{c^{3}}(\vec{v}.\dot{\vec{v}})\frac{\gamma^{3}}{c^{3}}\times$
$\displaystyle\left(\ddot{v}_{j}+\frac{3\gamma^{2}}{c^{2}}\dot{v}_{j}(\vec{v}.\dot{\vec{v}})+\frac{\gamma^{2}}{c^{2}}v_{j}(\vec{v}.\ddot{\vec{v}})+\frac{3\gamma^{4}}{c^{4}}(\vec{v}.\dot{\vec{v}})^{2}v_{j}\right)\left(1+\frac{v^{2}}{c^{2}}\right)$
$\displaystyle=-\frac{\gamma^{8}}{c^{8}}\dot{v}_{i}(\vec{v}.\dot{\vec{v}})\ddot{v}_{j}-\frac{\gamma^{8}}{c^{10}}v^{2}\dot{v}_{i}(\vec{v}.\dot{\vec{v}})\ddot{v}_{j}$
(68)

with,

$\displaystyle\frac{\ddot{z}^{i}(s)}{\dot{z}^{0}(s)}\ddot{z}^{j}\dddot{z}^{0}(s)=\frac{1}{\gamma}\frac{\gamma^{8}}{c^{8}}\left(\dot{v}_{i}+\frac{\gamma^{2}}{c^{2}}v_{i}(\vec{v}.\dot{\vec{v}})\right)\times$
$\displaystyle\left(\dot{v}_{j}+\frac{\gamma^{2}}{c^{2}}v_{j}(\vec{v}.\dot{\vec{v}})\right)\left((\vec{v}.\ddot{\vec{v}})+\dot{v}^{2}+4\frac{\gamma^{2}}{c^{2}}(\vec{v}.\dot{\vec{v}})^{2}\right)$
$\displaystyle=\frac{\gamma^{8}}{c^{8}}\dot{v}_{i}\left((\vec{v}.\ddot{\vec{v}})+\dot{v}^{2}\right)\dot{v}_{j}$
(69)

And,

$\displaystyle\frac{\ddot{z}^{i}(s)}{\dot{z}^{0}(s)}\ddot{z}^{2}(s)\ddot{z}^{j}(s)\left(\frac{z^{0}(s)}{2}+\frac{2}{3\dot{z}^{0}(s)}\right)$
$\displaystyle=\frac{\gamma^{2}}{c^{2}}\left(\dot{v}_{i}+\frac{\gamma^{2}}{c^{2}}v_{i}(\vec{v}.\dot{\vec{v}})\right)\frac{\gamma^{2}}{c^{2}}\left(\dot{v}_{j}+\frac{\gamma^{2}}{c^{2}}v_{j}(\vec{v}.\dot{\vec{v}})\right)\times$
$\displaystyle(-\frac{\gamma^{4}}{c^{4}})\left(\dot{v}^{2}+2\frac{\gamma^{2}}{c^{2}}(\vec{v}.\dot{\vec{v}})^{2}+\frac{\gamma^{4}}{c^{4}}v^{2}(\vec{v}.\dot{\vec{v}})^{2}-\frac{\gamma^{4}}{c^{2}}(\vec{v}.\dot{\vec{v}})^{2}\right)\times$
$\displaystyle\left(\frac{1}{2}+\frac{2}{3}(1-\frac{v^{2}}{c^{2}})\right)$
$\displaystyle=-\frac{7\gamma^{8}}{6c^{8}}\dot{v}_{i}\dot{v}^{2}\dot{v}_{j}+\frac{2\gamma^{8}}{3c^{10}}\dot{v}_{i}\dot{v}^{2}\dot{v}_{j}v^{2}$
(70)

Here,

$\displaystyle\frac{\ddot{z}^{i}}{(\dot{z}^{0})^{2}}\dot{z}^{j}\ddot{z}_{\mu}\dddot{z}^{\mu}$
$\displaystyle=$
$\displaystyle-\frac{\gamma^{6}}{c^{6}}\dot{v}_{i}v_{j}\dot{\vec{v}}.\ddot{\vec{v}}-3\frac{\gamma^{8}}{c^{8}}\dot{v}_{i}v_{j}\dot{v}^{2}\vec{v}.\dot{\vec{v}}$
(71) $\displaystyle-$
$\displaystyle\frac{\gamma^{8}}{c^{8}}\dot{v}_{i}v^{j}\vec{v}.\dot{\vec{v}}\left(\dot{v}^{2}+\vec{v}.\ddot{\vec{v}}\right)$
$\displaystyle-$
$\displaystyle\frac{\gamma^{8}}{c^{8}}\dot{v}^{i}v^{j}\left(\vec{v}.\dot{\vec{v}}\right)\left(\vec{v}.\ddot{\vec{v}}\right)$
$\displaystyle-$
$\displaystyle\frac{\gamma^{8}}{c^{10}}v^{i}v^{j}\left(\vec{v}.\dot{\vec{v}}\right)\left(\dot{\vec{v}}.\ddot{\vec{v}}\right)$

And,

$\displaystyle\frac{1}{\dot{z}^{0}}\left(\dddot{z}^{i}\dot{z}^{0}+\frac{1}{3}\dot{z}^{i}\dddot{z}^{0}\right)\left(\dddot{z}^{j}+\dot{z}^{j}\ddot{z}^{2}\right)$
$\displaystyle=\frac{\gamma^{6}}{c^{6}}\ddot{v}_{i}\ddot{v}_{j}+3\frac{\gamma^{8}}{c^{8}}\ddot{v}_{i}\dot{v}_{j}\left(\vec{v}.\dot{\vec{v}}\right)$
$\displaystyle+\frac{\gamma^{8}}{c^{8}}\ddot{v}_{i}v_{j}\left(\vec{v}.\ddot{\vec{v}}\right)+3\frac{\gamma^{8}}{c^{8}}\dot{v}_{i}\ddot{v}_{j}\left(\vec{v}.\dot{\vec{v}}\right)$
$\displaystyle+\frac{4}{3}\frac{\gamma^{8}}{c^{8}}v_{i}\ddot{v}_{j}\left(\dot{v}^{2}+\vec{v}.\ddot{\vec{v}}\right)$
(72)

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