Multiple Myeloma Cancer Cell Instance Segmentation

Student Name: Dikshant Sagar

Roll Number: 2017338

BTP report submitted in partial fulfillment of the requirements

for the Degree of B.Tech. in Computer Science & Social Sciences

on May 11, 2021

BTP Track: Research

BTP Advisor

Dr.Anubha Gupta

Shubham Goswami(PhD)

Indraprastha Institute of Information Technology

New Delhi

###### Abstract

Images remain the largest data source in the field of healthcare. But at the
same time, they are the most difficult to analyze. More than often, these
images are analyzed by human experts such as pathologists and physicians. But
due to considerable variation in pathology and the potential fatigue of human
experts, an automated solution is much needed. The recent advancement in Deep
learning could help us achieve an efficient and economical solution for the
same. In this research project, we focus on developing Deep Learning based
solution for detecting Multiple Myeloma cancer cells using an Object Detection
and Instance Segmentation System. We explore multiple existing solutions and
architectures for the task of Object Detection and Instance Segmentation and
try to leverage them and come up with a novel architecture to achieve
comparable and competitive performance on the required task. To train our
model to detect and segment Multiple Myeloma cancer cells, we utilise a
dataset curated by us using microscopic images of cell slides provided by
Dr.Ritu Gupta(Prof., Dept. of Oncology AIIMS).

Keywords:image analysis, machine learning, deep learning, bio-medical imaging,
computer vision, object detection, instance segmentation, multiple myeloma.

### Acknowledgments

I thank my advisor, Dr. Anubha Gupta, for giving me an opportunity to work
under her guidance. Her advice and constant support have enabled me to think
critically about the domain of research and learn to come up with concise,
effective solutions to the practical problem.

I thank Dr.Ritu Gupta(Prof., Dept. of Oncology AIIMS), for providing a large
databank of patient cell slide images, which helped us create a benchmark
dataset in the field of Multiple Myeloma Cancer Detection.

I also thank Mr.Shubham Goswami(PhD, IIIT Delhi) for his immense contribution
to the project. His vast knowledge of the various methods pertaining to Bio-
Medical Computer Vision and readiness to brainstorm solutions for the
challenges along the way have been a significant driver to this project.

Finally, I thank Mr. Bharat Giddwani (Deep Learning Solutions Architect,
NVIDIA) for providing access to DGX Workstations for us to be able to train
large scale deep learning models.

### Work Distribution

Monsoon Semester 2020 (4 credits)

  1. 1.

Dataset: Annotations and Auxiliary Dataset Exploration (Sept - Nov, 2020)

  2. 2.

Existing Deep Learning Architectures: Literature Survey (Nov - Dec, 2020)

  3. 3.

Proposed Architecture: Backbone Network (Dec, 2020)

Winter Semester 2021 (6 Credits)

  1. 1.

Proposed Architecture: Feature Network (Jan - Feb, 2021)

  2. 2.

Proposed Architecture: Heads (Feb, 2021)

  3. 3.

Experiments and Results: (March - April, 2021)

###### Contents

  1. 1 Introduction
  2. 2 Dataset
    1. 2.1 Primary Dataset
      1. 2.1.1 Overview
      2. 2.1.2 Annotation
    2. 2.2 Auxiliary Dataset
      1. 2.2.1 MS COCO
  3. 3 Existing Deep Learning Architectures
    1. 3.1 EfficientDet: Scalable and Efficient Object Detection
    2. 3.2 CSPNet: A New Backbone that can Enhance Learning Capability of CNN
    3. 3.3 SpineNet: Learning Scale-Permuted Backbone for Recognition and Localization
  4. 4 Proposed Architecture
    1. 4.1 Backbone
      1. 4.1.1 CSP + EfficientNet + SAM
    2. 4.2 Feature Network
      1. 4.2.1 NAS-FPN
    3. 4.3 Heads
      1. 4.3.1 Mask Head : Dice Loss Objective
  5. 5 Experimental Results
    1. 5.1 Backbone
    2. 5.2 Feature Network
    3. 5.3 Heads
    4. 5.4 Our Model
  6. 6 Conclusion and Future Work

##  Chapter 1 Introduction

One of the major challenges in developing nations like India is to make
medical facilities accessible to the entire population of the country. Due to
a lack of medical infrastructure throughout the country, A significant
percentage of the country’s population faces financial and physical outreach
to specialized hospitals and treatments. Cancer detection is one of the major
life-threatening health-related hazards and even a small percentage of the
population getting diagnosed with cancer can result in large numbers given
India’s population. There are specialized hospitals being developed to address
this need but in the present situation much of the population cannot afford
the expensive diagnostics like flow cytometry test, Also much of these
specialized hospitals are present or being laid out in urban areas due to
which people available in remote areas are not able to avail these facilities.

Artificial intelligence(AI) based solutions can prove themselves as no less
than a miracle in such scenarios. The advances in Deep Learning(DL a way to
achieve AI) has increased the possibilities to use DL based solutions in the
Medical domain and for diagnostics. DL-based solutions show more
generalisability on a prospective subject’s data by learning the
distinguishable characteristics from the data itself rather than relying on
handcrafted features that are used by traditional statistics. Even though DL
based solutions have shown their potential in many medical applications like
diagnostics and are becoming the state of the art solutions. There are many
fundamental challenges related to DL based solutions like explainability,
capturing heterogeneity at various levels(subject, sensor, site), label noise,
handling uncertainty, etc which need to be addressed for their usage as an
alternative to existing diagnostics.

This BTP address some of the above described fundamental challenges in DL
based solutions in the application of white blood cancer diagnostics,
Especially Multiple Myeloma(MM) cancer diagnostics. For this, We’ll be
exploring and developing DL based solutions for the detection and segmentation
of MM cancer cells. This research work is being carried out in collaboration
with the Dept. Of Oncology, AIIMS under the supervision and expert guidance of
Dr. Anubha Gupta(Prof., Dept. of ECE, IIIT-Delhi) and Dr.Ritu Gupta(Prof.,
Dept. of Oncology AIIMS). The outcomes of this research include aiming for
publications for validation of our research followed by deployment of the
proposed DL based solution in AIIMS(Delhi).

##  Chapter 2 Dataset

###  2.1 Primary Dataset

####  2.1.1 Overview

(a) Stained Cell Slide Image

(b) Annotated Masks

(c) Masks Overlayed

Figure 2.1: Our Dataset

Our primary dataset is a set of microscopic images that were captured from the
bone marrow aspirate slides of multiple subjects, as per standard
guidelines[15]. The data is compiled from the bone marrow aspirates slides of
n subjects and consists of 35,500 images with over 1,50,000 cell markers. The
slides were stained using the Jenner-Giemsa stain. The images were captured
using a Nikon Eclipse-200 microscope equipped with a digital camera at a
magnification of 1000x. These images were captured in a raw BMP format with a
size 2040 × 1536 pixels.

Figure 2.2: Challenges Posed [5]

The given dataset poses the following challenges:

  1. 1.

The task of segmentation required us to segment both the nucleus and the
cytoplasm. For some cells, the cytoplasm may be in less contrast to the
nucleus or at times, even the background. This causes difficulty to segment
out the cytoplasm.

  2. 2.

Presence of more than one type of stained and unstained cells poses another
challenge in extracting plasma cells of interest

  3. 3.

Plasma cells may be clustered together and hence, segmentation of the
overlapping/touching cells is required. Generally, this is difficult because
of different configurations:

     * •

nuclei of different cells are touching

     * •

nuclei of one and cytoplasm of another cell are touching

     * •

the cytoplasm of different cells are touching

A subset of the dataset previously released is available at the public
repository[5].

Figure 2.3: MATLAB Image Segmentor on Multiple Myeloma Images

####  2.1.2 Annotation

The Obtained data consisted of around 35,500 images of stained Multiple
Myeloma plasma cells, with each image having multiple markers to be annotated
ranging from 0-20. The data was segmented into batches of 100 images each
giving around 350 batches in total to be annotated. Students doing projects
under Dr. Anubha Gupta on Multiple Myeloma were distributed batches and were
tasked to annotate masks(See Fig. 2.1) for the markers on these images. A
proper NDA was signed to maintain data and patient confidentiality.

We used the Image Segmenter Tool from MATLAB[9] to achieve the annotated
masks.(See Fig.2.3)

We use a subset of this dataset along with previous similar dataset from
SBIlab at IIIT Delhi to create a dataset, which was also made a public
challenge in SegPC-2021. This subset contains 775 images. These are divided
into a training set of 298 images, validation set of 200 images, and a test
set of 277 images. For this BTP’s purpose we use the train set for training
and validation set for performance metric purposes.

###  2.2 Auxiliary Dataset

####  2.2.1 MS COCO

Figure 2.4: Example from COCO Dataset

The MS COCO dataset[14] is a large-scale dataset of 328k images consisting of
over 2.5 million objects. The dataset was curated by gathering images of
complex everyday scenes containing common objects in their natural context.
Objects were labeled using per-instance segmentations to aid in precise object
localization. The comparitive class distribution of the COCO dataset with
respect to the PASCAL VOC dataset is shown in Fig 2.5

Figure 2.5: Comparitive class distribution of MSCOCO w.r.t PASCAL VOC [14]

##  Chapter 3 Existing Deep Learning Architectures

We explored State of the art architectures on some of the benchmark datasets
for Object Detection and Instance Segmentation namely MS COCO and PASCAL VOC
2007 in the non-medical domain.

Following are some of the recent advancements in Object Detection and Instance
Segmentation Architectures :

###  3.1 EfficientDet: Scalable and Efficient Object Detection

Figure 3.1: EfficientDet Model Architecture [20]

Existing object detectors are mostly categorized by whether they have a
region-of interest proposal step (two-stage) or not (one stage). While two-
stage detectors tend to be more flexible and more accurate, one-stage
detectors are often considered to be simpler and more efficient by leveraging
predefined anchors. EfficientDet [20] largely follows the one-stage detectors
paradigm.It employs ImageNet-pretrained EfficientNets [19] as the backbone
network. Their proposed BiFPN[20] serves as the feature network, which takes
level 3-7 features {P3, P4, P5, P6, P7} from the backbone network and
repeatedly applies top-down and bottom-up bidirectional feature fusion. These
fused features are fed to a class and box network to produce object class and
bounding box predictions respectively (See Fig. 3.1). Similar to RetinaNet
[13], the class and box network weights are shared across all levels of
features.

###  3.2 CSPNet: A New Backbone that can Enhance Learning Capability of CNN

Figure 3.2: CSP-DenseNet Block Architecture

This Paper introduces a concept of partial skip connections in repetitive
blocks of a backbone network. They suggest that rather than passing the whole
feature map through a block and also forward on a skip connection, we pass
half the channels of the feature map through the block and half through the
skip-connection (See Fig.3.2).This partitioning of the feature map helps
achieve a richer gradient combination due to gradient flow propagating through
different network paths while reducing the amount of computation. The proposed
CSPNet[21] can be easily applied to ResNet[6], ResNeXt[24], and DenseNet[8].
After applying CSPNet on the above mentioned networks, the computation effort
can be reduced from 10% to 20%, but it outperforms in terms of accuracy, in
conducting image classification task on ImageNet[17].

Figure 3.3: Exact Fusion Model [21]

They also propose a feature aggregation network called the Exact Fusion
Model[21] inspired from YOLOv3[16] which assigns exactly one bounding-box
prior to each ground truth object.Each ground truth bounding box corresponds
to one anchor box that surpasses the threshold IoU. If the size of an anchor
box is equivalent to the field-of-view of the grid cell, then for the grid
cells of the sth scale, the corresponding bounding box will be lower bounded
by the (s - 1)th scale and upper bounded by the (s + 1)th scale. Therefore,
the EFM assembles features from the three scales. See Fig.3.3

###  3.3 SpineNet: Learning Scale-Permuted Backbone for Recognition and
Localization

(a) ResNet-50-FPN [6]

(b) SpineNet-49 [3]

Figure 3.4: The architecture comparison between a ResNet backbone (3.4(a)) and
the SpineNet backbone (3.4(b)) derived from it using NAS.

CNNs created for image tasks typically encode an input image into a sequence
of intermediate features that capture the semantics of an image (from local to
global), where each subsequent layer has a lower spatial dimension. However,
this scale-decreased model may not be able to deliver strong features for
multi-scale visual recognition tasks where recognition and localization are
both important (e.g., object detection and segmentation). Several works
including FPN[12] and DeepLabv3+[1] propose multi-scale encoder-decoder
architectures to address this issue, where a scale-decreased network (e.g., a
ResNet) is taken as the encoder (commonly referred to as a backbone model). A
decoder network is then applied to the backbone to recover the spatial
information.

While this architecture has yielded improved success for image recognition and
localization tasks, it still relies on a scale-decreased backbone that throws
away spatial information by down-sampling, which the decoder then must attempt
to recover.

They propose a meta architecture called a scale-permuted model that enables
two major improvements on backbone architecture design. First, the spatial
resolution of intermediate feature maps should be able to increase or decrease
anytime so that the model can retain spatial information as it grows deeper.
Second, the connections between feature maps should be able to go across
feature scales to facilitate multi-scale feature fusion. They then use neural
architecture search (NAS)[25] with a novel search space design that includes
these features to discover an effective scale-permuted model.

They named the learned 49-layer scale-permuted backbone architecture
SpineNet-49[3] from the ResNet-50[6] model. SpineNet-49 can be further scaled
up to SpineNet-96/143/190 by repeating blocks two, three, or four times and
increasing the feature dimension. An architecture comparison between
ResNet-50-FPN and the final SpineNet-49 is shown.(See Fig. 3.4)

##  Chapter 4 Proposed Architecture

Figure 4.1: Two Step Object Detection - Segmentation Model

We model our proposed novel architecture on a two-step Object Detection -
Segmentation Architecture(See Figure 4.1) based on the Mask-RCNN[7]. The main
reason for this choice is that though one-step models are known to be faster
in inference time, they lack at the accuracy metric when compared to two-step
models[10]. And our specific problem of detecting cancer cells require
accuracy to be the top most-priority and not the inference time.

###  4.1 Backbone

####  4.1.1 CSP + EfficientNet + SAM

One current candidate for the backbone that we have devised is a modified
EfficientNet[19] with CSP Connections[21](See Section 3.2) and a Spatial
Attention Module(SAM)[22] (See Fig.4.2) in the repetitive MBConv
Block(Inverted Residual Block)[18] of the original efficientnet. We found this
configuration and architecture to give much better performance on CIFAR10[11]
Classification Task and due to the CSP Connections the FLOPs are reduced and
the architecture remains both scalable and efficient. We call our backbone
architectures a CSP-EfficientNet Family of models with variants same as
EfficientNets[19] ranging from B0 to B7.

Figure 4.2: Spatial Attention Module

###  4.2 Feature Network

####  4.2.1 NAS-FPN

Feature networks are feature extractors and recent state of the art fetaure
networks include FPN(Feature Pyramid Network)[12].FPN composes of a bottom-up
and a top-down pathway. The bottom-up pathway is the usual convolutional
network for feature extraction. As we go up, the spatial resolution decreases.
With more high-level structures detected, the semantic value for each layer
increases. FPN also provides top-down pathway to construct higher resolution
layers from the semantic rich layers with lateral connections between
reconstructed layers and the corresponding feature maps to help the detector
to predict the location betters (See Fig. 4.3).

Figure 4.3: FPN[12]

For the feature network in our model we incorporate a novel architecture known
as NAS-FPN[4] that is discovered using neural architecture search. The search
is a novel scalable search space covering all cross-scale connections possible
in a vanilla FPN[12]. The discovered architecture consists of a combination of
top-down and bottom-up connections to fuse features across scales. The optimal
intra-scale connections found through the Neural architecture search are
visualised in Fig.4.4

Figure 4.4: Architecture of the discovered 7-merging-cell pyramid network in
NAS-FPN with 5 input layers (yellow) and 5 output feature layers (blue). GP
and R-C-B are stands for Global Pooling and ReLU-Conv-BatchNorm,
respectively.[4]

###  4.3 Heads

####  4.3.1 Mask Head : Dice Loss Objective

Instead of the normal Sigmoid Crossentropy loss on the mask head of the
MaskRCNN type model, we incorporate a new loss using the Dice Coefficient[2]
which in theory will tend to maximise the mean IOU of the predicted masks and
the ground truth masks of the samples by minimising the negative of the Dice
Coefficient. The following is the expression for the dice coefficient :

$Dice=\frac{2|X\cap Y|}{|X|+|Y|}$ (4.1)

inspired from [2], where X is the predicted mask and Y is the ground truth
mask. Then our dice loss is given by:

$DiceLoss=-1*Dice$ (4.2)

##  Chapter 5 Experimental Results

###  5.1 Backbone

For Backbone experimentation purposes we took EfficientNetB0[19] as the base
architecture and CIFAR10[11] as the metric dataset to compare performances
which is in accordance to the norm in computer vision classification task
before training on a larger dataset like Imagenet[17]. We fixed training
hyperparamters to be same for every architecture and epochs for the
architectures to 50, as we observed complex architectures like these converged
relatively faster on a small dataset like CIFAR10[11] which can be seen in
Fig.5.1 on Train and Validation data. The accuracy results can be seen in
Table 5.1

(a) EfficientNetB0

(b) EfficientNetB0 + SAM

(c) EfficientNetB0 + SAM + CSP

Figure 5.1: Classification accuracy plots for experimented architectures. Table 5.1: Val. accuracy of experimented backbone architectures after 50 epochs on CIFAR10[11] classification task. Architecture | Val. Accuracy
---|---
EfficientNetB0 | 0.71
EfficientNetB0 + Spatial Attention | 0.75
EfficientNetB0 + Spatial Attention + CSP | 0.82

###  5.2 Feature Network

To understand the contribution of the NAS-FPN[4] feature network, we perform
ablation study on the performance of the model on the COCO dataset[14].

Table 5.2: Performace of our Model with NAS-FPN[4] and other feature networks on COCO minival[14]. Architecture | BoxAP | MaskAP
---|---|---
CSP-EfficientnetB0 + C4 | 40.3 | 35.3
CSP-EfficientnetB0 + FPN | 42.3 | 37.2
CSP-EfficientnetB0 + NAS-FPN | 43.8 | 38.3

###  5.3 Heads

For our model we train our mask head with a loss function based on IoU score
called the dice loss , unlike the conventional Mask-RCNN’s Mask head which
trains on just sigmoid binary crossentropy of the predicted binary masks from
the GT masks. So we perform ablation study to understand the contribution of
the modified loss objective of the mask head to the performance of the whole
model.

Table 5.3: MaskAP of our model(CSP-EfficientNetB0 + NAS-FPN) with and without the modified loss objective on the COCO minival[14]. Mask Head Loss | MaskAP
---|---
Sigmoid Binary Crossentropy Loss | 38.3
Dice Loss | 39.1

###  5.4 Our Model

In this section we show the performance of our model in comparison with other
existing comparable architectures on the same task.

Table 5.4: Performance of our model in comparison with comparable existing MaskRCNN[7] model architectures on the COCO minival[14]. Work | Model | BoxAP | MaskAP
---|---|---|---
| ResNet50-FPN[23] | 38.6 | 35.2
Prior | SpineNet-49 | 41.2 | 36.3
| EfficientNetB0-FPN | 41.4 | 36.7
Ours | CSP-EfficientNetB0 + NAS-FPN + Dice Loss | 43.8 | 39.1
Figure 5.2: Comparing training loss plots of vanilla MaskRCNN[7] with EfficientNetB0+FPN vs our CSP-EfficientNetB0+NAS-FPN. Table 5.5: Performance of Our model in comparison with comparable existing MaskRCNN[7] model architectures on our primary dataset. Work | Model | BoxAP | MaskAP | mIOU
---|---|---|---|---
| ResNet50-FPN | 62.4 | 63.3 | 0.82
Prior | SpineNet-49 | 63.3 | 64.1 | 0.84
| EfficientNetB0-FPN | 63.4 | 64.3 | 0.84
Our Model | CSP-EfficientNetB0 + NAS-FPN + Dice Loss | 64.7 | 65.6 | 0.86

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.3: Sample Detections on COCO minival [14] by our model.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.4: Sample Detections on our primary dataset by our model.

##  Chapter 6 Conclusion and Future Work

We find that our CSP-EfficientNet family of backbone architectures provide
increased performance from the base vanilla EfficientNet[19] Family of Models.
We also find that NAS-FPN[4] also provides an increment in performance in
comparision to other features networks. And our Modified loss objective for
the mask head using dice coefficient[2] helps us achieve better IoU of
predicted and ground truth masks and in result a better MaskAP.

Our model even on the base backbone architecture(CSP-EfficientNetB0) provides
us with competitive performance on both the benchmark datasets for Object
detection and instance segmentation. Also, it gives a competitive performance
on our primary dataset.

We plan to further improve the performance of our model by finding more novel
techniques and architectures to help us deploy a much accurate Multiple
Myeloma cancer detection system at AIIMS, Delhi to help doctors promptly
detect and diagnose Multiple Myeloma cancer patients.

## Bibliography

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# Gravitational deflection of light in Rindler-type potential as a possible
resolution to the observations of Bullet Cluster 1E0657-558

Xin Li<EMAIL_ADDRESS>Zhe Chang<EMAIL_ADDRESS>Institute of High Energy
Physics
Theoretical Physics Center for Science Facilities
Chinese Academy of Sciences, 100049 Beijing, China

###### Abstract

The surface density $\Sigma$-map and the convergence $\kappa$-map of Bullet
Cluster 1E0657-558 show that the center of baryonic matters separates from the
center of gravitational force, and the distribution of gravitational force do
not possess spherical symmetry. This hints that a modified gravity with
difference to Newtonian inverse-square law at large scale, and less symmetry
is worth investigating. In this paper, we study the dynamics in Randers-
Finsler spacetime. The Newtonian limit and gravitational deflection of light
in a Rindler-type potential is focused in particular. It is shown that the
convergence in Finsler spacetime could account for the observations of Bullet
Cluster.

###### pacs:

04.50.Kd,04.25.-g,95.35.+d

##  I Introduction

In 1933, ZwickyZwicky  analysed the velocity dispersion for the Coma cluster.
His analysis implied that the Coma cluster is full of invisible matters. There
are a great variety of observations which show that the rotational velocity
curves of all spiral galaxies tend to constant valuesTrimble . These include
the OortOort  discrepancy in the disk of the Milky WayBahcall , the velocity
dispersions of dwarf Spheroidal galaxiesVogt , and the flat rotation curves of
spiral galaxiesRubin . These facts violate sharply the prediction of Newton’s
inverse square law of gravitation.

The most widely adopted way to resolve these difficulties is the dark matter
hypothesis. It is assumed that all visible stars are surrounded by massive
nonluminous matters. Though it explains the flat rotation curves of spiral
galaxies, the hypothesis has its own weakness. No theory predicts these
matters, and they behave in such ad hoc way. There are a lot of possible
candidates of dark matter (such as axion, neutrino et al), but none of them
satisfactory. Up to now, all of them either undetected or excluded by
observation.

Because of these troubles induced by dark matter, some models have been built
for alternative of the dark matter hypothesis. Their main ideas are to assume
that the Newtonian gravity or Newton’s dynamics is invalid in galactic scale.
The famous one is Milgrom’s Modified Newtonian Dynamics (MOND)Milgrom . MOND
assumes that the Newtonian dynamics does not hold in galactic scale. As a
phenomenological model, MOND explains well the flat rotation curves with a
simple formula and a new parameter. In particular, it deduces naturally a
well-known global scaling relation for spiral galaxies, the Tully-Fisher
relationTF . By introducing several scalar, vector and tensor fields,
BekensteinBekenstein  rewrote the MOND into a covariant formulation. He showed
that the MOND satisfies all four classical tests on Einstein’s general
relativity in solar system.

However, MOND still face challenges. The strong and weak gravitational lensing
observations of Bullet Cluster 1E0657-558Bullet  could not be explained well
by MOND and Bekenstein’s relativistic version of MONDAngus . Takahashi et al.
Takahashi  have investigated weak gravitational lensing of three galaxy
clusters ((A1689, CL0024+1654, CL1358+6245) in terms of the MOND. They found
that MOND could not explain the data of three galaxy clusters unless a dark
matter halo is added. Problems still remain in fitting X-ray temperature of
galaxy clusters with MOND. In the framework of MOND, a plausible resolution of
these issues is the “marriage” of MOND with neutrino of mass $m_{\nu}\geq
2$eVAngus ; Takahashi . However, the WMAP7 data give an upper limit of 1.2eV
for the sum of neutrino mass Hannestad .

The main feature of the Bullet Cluster is that the magnitude of the
gravitational force approximately equals five times of what produced by
baryonic matters. Beside the main feature, there are two particular features
too. The surface density $\Sigma$-map reconstructed from X-ray imaging
observations gives the center of baryonic matters. The convergence
$\kappa$-map reconstructed from strong and weak gravitational lensing
observations gives the center of gravitational force. The first particular
feature is that the center of baryonic matters separates from the center of
gravitational force. The convergence $\kappa$-map manifests that the
distribution of gravitational force do not possess spherical symmetry. This is
the other particular feature. These two particular features are the reasons
why MOND could not explain well the observations of Bullet Cluster.

Due to the reasons given above, the observations of Bullet Cluster have been
regarded as a direct evidence of dark matter. The dominance of dark matter has
lasted almost six decades. However, up to now, no direct evidence claim that
dark matter is detected. It is very interesting to search an alternative to
dark matter. One famous and successful model is modified gravity (MOG)Moffat .
MOG assumes that some non geometrical fields couple to the gravitational field
of general relativity. MOG explains well the flat rotation curves of
galaxiesBrownstein . The MOG’s prediction for the $\kappa$-map results in two
baryonic components distributed across the Bullet Cluster 1E0657-558 with
averaged mass-fraction of 83% intracluster medium (ICM) gas and 17%
galaxiesBrownstein1 . One should notice that the locally measured value of
Newton’s constant $G$ varies spatially in MOG. In dark matter hypothesis, the
galaxy’s mass involve baryonic mass and the mass of dark matter. As for MOG,
the galaxy’s mass only involve baryonic mass, and the effective acceleration
depends on the running gravitational coupling $G(r)$.

In this paper, we try to introduce a possible alternative to dark matter,
which could account for the observations of Bullet Cluster. In 1912, A.
Einstein proposed his famous general relativity which gives the connection
between Riemann geometry and gravitation. In general relativity, the effects
of gravitation are ascribed to spacetime curvature instead of a force. By
mimicking Einstein, we may investigate the gravity in Finsler spacetimeBook by
Bao . Finsler geometry as a natural generalization of Riemann geometry could
provide new sight on modern physics.

The Finsler gravity modifies the Newtonian inverse-square law at large scale.
It reduces to the Newtonian inverse-square law at small distance. The center
of gravitational force may separate from the center of baryonic matters at
large scale. In general, Finsler spacetime admits less Killing vectors than
Riemann spacetime. In Finsler spacetime, the spherical symmetry may be broken
at large scale. Finsler spacetime possesses the two particular features of
Bullet Cluster. It could account for the observations of Bullet Cluster.

Recently, Grumiller Grumiller  constructed an effective model for gravity of a
central object at large scales. In Grumiller’s model, to leading order in the
large radius expansion, all terms are expected from general relativity, except
for the Rindler one. The Rindler term leads to an anomalous acceleration Wald
, which could account for the Pioneer anomaly Anderson  in solar system and
the rotational curve of galaxies. In this paper, we will show that the
Rindler-type potential could account for the observations of Bullet Cluster
within the framework of Finsler gravity.

This paper is organized as follows. In Sec.2, we present the vacuum field
equation in Finsler spacetime. In Sec.3, by making use of the post-Newtonian
approximation and the viewpoints of Zermelo navigation problem, we investigate
the dynamics in Randers-Finsler spacetime. The emphasis is focused on the
Newtonian limit and gravitational deflection of light in a Rindler-type
potential. The concluding remarks are given is Sec.4.

##  II Vacuum field equation in Finsler spacetime

Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is based on the so called Finsler structure
$F$ with the property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$,
where $x$ represents position and $y\equiv\frac{dx}{d\tau}$ represents
velocity. The Finsler metric is given asBook by Bao

$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (1)

Finsler geometry has its genesis in integrals of the form

$\int^{r}_{s}F(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau})d\tau~{}.$
(2)

The Finsler structure represents the length element of Finsler space.

The parallel transport has been studied in the framework of Cartan
connectionMatsumoto ; Antonelli ; Szabo . The notation of parallel transport
in Finsler manifold means that the length $F\left(\frac{dx}{d\tau}\right)$ is
constant. The geodesic equation for Finsler manifold is given asBook by Bao

$\frac{d^{2}x^{\mu}}{d\tau^{2}}+G^{\mu}=0,$ (3)

where

$G^{\mu}=\frac{1}{2}g^{\mu\nu}\left(\frac{\partial^{2}F^{2}}{\partial
x^{\lambda}\partial y^{\nu}}y^{\lambda}-\frac{\partial F^{2}}{\partial
x^{\nu}}\right)$ (4)

is called geodesic spray coefficient. Obviously, if $F$ is Riemannian metric,
then

$G^{\mu}=\tilde{\gamma}^{\mu}_{\nu\lambda}y^{\nu}y^{\lambda},$ (5)

where $\tilde{\gamma}^{\mu}_{\nu\lambda}$ is the Riemannian Christoffel
symbol. Since the geodesic equation (3) is directly derived from the integral
length of $\sigma$

$L=\int F\left(\frac{dx}{d\tau}\right)d\tau,$ (6)

the inner product
$\left(\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}=F\left(\frac{dx}{d\tau}\right)\right)$
of two parallel transported vectors is preserved.

In Finsler manifold, there exists a linear connection - the Chern
connectionChern . It is torsion freeness and almost metric-compatibility,

$\Gamma^{\alpha}_{\mu\nu}=\gamma^{\alpha}_{\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{\mu}}{F}\right),$
(7)

where $\gamma^{\alpha}_{\mu\nu}$ is the formal Christoffel symbols of the
second kind with the same form of Riemannian connection, $N^{\mu}_{\nu}$ is
defined as
$N^{\mu}_{\nu}\equiv\gamma^{\mu}_{\nu\alpha}y^{\alpha}-A^{\mu}_{\nu\lambda}\gamma^{\lambda}_{\alpha\beta}y^{\alpha}y^{\beta}$
and $A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})$ is the Cartan tensor (regarded as a measurement of deviation
from the Riemannian Manifold). In terms of Chern connection, the curvature of
Finsler space is given as

$R^{~{}\lambda}_{\kappa~{}\mu\nu}=\frac{\delta\Gamma^{\lambda}_{\kappa\nu}}{\delta
x^{\mu}}-\frac{\delta\Gamma^{\lambda}_{\kappa\mu}}{\delta
x^{\nu}}+\Gamma^{\lambda}_{\alpha\mu}\Gamma^{\alpha}_{\kappa\nu}-\Gamma^{\lambda}_{\alpha\nu}\Gamma^{\alpha}_{\kappa\mu},$
(8)

where $\frac{\delta}{\delta x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}_{\mu}\frac{\partial}{\partial y^{\nu}}$.

The gravity in Finsler spacetime has been investigated for a long timeTakano ;
Ikeda ; Tavakol1 ; Bogoslovsky1 . In this paper, we introduce vacuum field
equation from the extension of an analogy, which was discussed first by
PiraniPirani ; Rutz . In Newton’s theory of gravity, the equation of motion of
a test particle is given as

$\frac{d^{2}x^{i}}{dt^{2}}=-\eta^{ij}\frac{\partial\phi}{\partial x^{i}},$ (9)

where $\phi=\phi(x)$ is the gravitational potential and $\eta^{ij}$ is
Euclidean metric. For an infinitesimal transformation $x^{i}\rightarrow
x^{i}+\epsilon\xi^{i}$($|\epsilon|\ll 1$), the equation (9) becomes, up to
first order in $\epsilon$,

$\frac{d^{2}x^{i}}{dt^{2}}+\epsilon\frac{d^{2}\xi^{i}}{dt^{2}}=-\eta^{ij}\frac{\partial\phi}{\partial
x^{i}}-\epsilon\eta^{ij}\xi^{k}\frac{\partial^{2}\phi}{\partial x^{j}\partial
x^{k}}.$ (10)

Combining the above equations(9) and (10), we obtain

$\frac{d^{2}\xi^{i}}{dt^{2}}=\eta^{ij}\xi^{k}\frac{\partial^{2}\phi}{\partial
x^{j}\partial x^{k}}\equiv\xi^{k}H^{i}_{k}.$ (11)

In Newton’s theory of gravity, the vacuum field equation is given as
$H^{i}_{i}=\bigtriangledown^{2}\phi=0$. It means that the tensor $H^{i}_{k}$
is traceless in Newton’s vacuum.

In general relativity, the geodesic deviation gives similar equation

$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\xi^{\nu}H^{\mu}_{\nu},$ (12)

where
$H^{\mu}_{\nu}=\tilde{R}^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$.
Here, $\tilde{R}^{~{}\mu}_{\lambda~{}\nu\rho}$ is Riemannian curvature tensor,
$D$ denotes the covariant derivative alone the curve $x^{\mu}(\tau)$. The
vacuum field equation in general relativity gives
$\tilde{R}_{\mu\nu}=\tilde{R}^{~{}\mu}_{\lambda~{}\lambda\nu}=0$Weinberg . It
implies that the tensor $H^{\mu}_{\nu}$ is also traceless,
$H=H^{\mu}_{\mu}=0$.

In Finsler spacetime, the geodesic deviation givesBook by Bao

$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\xi^{\nu}H^{\mu}_{\nu},$ (13)

where
$H^{\mu}_{\nu}=R^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$.
Here, $R^{~{}\mu}_{\lambda~{}\nu\rho}$ is Finsler curvature tensor defined in
(8), $D$ denotes covariant derivative
$\frac{D\xi^{\mu}}{D\tau}=\frac{d\xi^{\mu}}{d\tau}+\xi^{\nu}\frac{dx^{\lambda}}{d\tau}\Gamma^{\mu}_{\nu\lambda}(x,\frac{dx}{d\tau})$.
Since the vacuum field equations of Newton’s gravity and general relativity
have similar form, we may assume that vacuum field equation in Finsler
spacetime holds similar requirement as the case of Netwon’s gravity and
general relativity. It implies that the tensor $H^{\mu}_{\nu}$ in Finsler
geodesic deviation equation should be traceless, $H=0$.

The notion of Ricci tensor in Finsler geometry was first introduced by Akbar-
ZadehAkbar

$Ric_{\mu\nu}=\frac{\partial^{2}\left(\frac{1}{2}F^{2}R\right)}{\partial
y^{\mu}\partial y^{\nu}},$ (14)

where $R=\frac{y^{\mu}}{F}R^{~{}\kappa}_{\mu~{}\kappa\nu}\frac{y^{\nu}}{F}$.
And the scalar curvature in Finsler geometry is given as
$S=g^{\mu\nu}Ric_{\mu\nu}$. Constructing a physical Finslerian theory of
gravity in arbitrary Finsler spacetime is a difficult task. However, it has
been pointed out that constructing a Finslerian theory of gravity in Finlser
spacetime of Berwald type is viable Tavakol . A Finsler spacetime is said to
be of Berwald type if the Chern connection (7) has no $y$ dependenceBook by
Bao . In light of the research of Tavakol et al. Tavakol , the gravitational
field equation in Berwald-Finsler space has been studied in Ref. Finsler DM ;
Lixin . In Berwald-Finsler space, the Ricci tensor reduces to

$Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu}).$
(15)

It is manifestly symmetric and covariant. Apparently the Ricci tensor will
reduce to the Riemann-Ricci tensor if the metric tensor $g_{\mu\nu}$ does not
depend on $y$. We starts from the second Bianchi identities on Berwald-Finsler
spaceBook by Bao

$R^{~{}\alpha}_{\mu~{}\lambda\nu|\beta}+R^{~{}\alpha}_{\mu~{}\nu\beta|\lambda}+R^{~{}\alpha}_{\mu~{}\beta\lambda|\nu}=0,$
(16)

where the $|$ means the covariant derivative. The metric-compatibility

$g_{\mu\nu|\alpha}=0~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}g^{\mu\nu}_{~{}~{}|\alpha}=0,$
(17)

and contraction of (16) with $g^{\mu\beta}$ gives that

$R^{\mu\alpha}_{~{}~{}\lambda\nu|\mu}+R^{\mu\alpha}_{~{}\nu\mu|\lambda}+R^{\mu\alpha}_{~{}\mu\lambda|\nu}=0.$
(18)

Lowering the index $\alpha$ and contracting with $g^{\alpha\lambda}$, we
obtain

$\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]_{|\mu}+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}_{|\mu}=0,$
(19)

where

$B_{\mu\nu\alpha\beta}=-A_{\mu\nu\lambda}R^{~{}\lambda}_{\theta~{}\alpha\beta}y^{\theta}/F.$
(20)

Thus, the counterpart of the Einstein’s field equation on Berwald - Finsler
space takes the form

$\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}=8\pi
GT_{\mu\nu}.$ (21)

In Eq. (21), the term in “[ ]” is symmetrical tensor, and the term in “{}” is
asymmetrical tensor. By making use of the equation (21), the vacuum field
equation in Finsler spacetime of Berwald type implies

$Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu})=0.$
(22)

It means that the tensor
$H^{\mu}_{\nu}=R^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$
is traceless in Finsler spacetime of Berwald type. Therefore, the analogy from
the geodesic deviation equation is valid at least in Finsler spacetime of
Berwald type. For this reason, we may suppose that this analogy is valid in
general Finsler spacetime. For the rest of the paper, we adopt $H=0$ as a
requirement of vacuum field equation in Finsler spacetime.

##  III The dynamics in Randers-Finsler spacetime

###  III.1 The post-Newtonian approximation in Randers-Finsler spacetime

The Randers space is a special kind of Finsler geometry with Finsler structure
$F$ on the slit tangent bundle $TM\backslash 0$ of a manifold $M$,

$F(x,y)\equiv\alpha(x,y)+\beta(x,y),$ (23)

where

$\displaystyle\alpha(x,y)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\tilde{a}_{\mu\nu}(x)y^{\mu}y^{\nu}},$ (24)
$\displaystyle\beta(x,y)$ $\displaystyle\equiv$
$\displaystyle\tilde{b}_{\mu}(x)y^{\mu},$ (25)

and $\tilde{a}_{ij}$ is Riemannian metric.

The geodesic spray coefficient $G^{\mu}$ (4) in Randers-Finsler spacetime
readsBook by Bao

$\displaystyle
G^{\mu}=(\tilde{\gamma}^{\mu}_{\nu\lambda}+l^{\mu}\tilde{b}_{\nu|\lambda})y^{\nu}y^{\lambda}+(\tilde{a}^{\mu\nu}-l^{\mu}\tilde{b}^{\nu})(\tilde{b}_{\nu|\lambda}-\tilde{b}_{\lambda|\nu})\alpha\left(\frac{dx}{d\tau}\right)y^{\lambda},$
(26)

where $l^{\mu}\equiv y^{\mu}/F$, $\tilde{\gamma}^{\mu}_{\nu\lambda}$ is the
Christoffel symbol of Riemannian metric $\tilde{a}$ and
$\tilde{b}_{\nu|\lambda}$ denotes the covariant derivative with respect to the
Riemannian metric $\tilde{a}$

$\tilde{b}_{\nu|\lambda}=\frac{\partial\tilde{b}_{\nu}}{\partial
x^{\lambda}}-\tilde{\gamma}^{\mu}_{\nu\lambda}\tilde{b}_{\mu}.$ (27)

In the rest of the paper, we just consider the case that $\beta$ is a closed
1-form. Thus, the geodesic equation of such Randers spacetime is given as

$\frac{d^{2}x^{\mu}}{d\tau^{2}}+(\tilde{\gamma}^{\mu}_{\nu\lambda}+l^{\mu}\tilde{b}_{\nu|\lambda})y^{\nu}y^{\lambda}=0.$
(28)

In post-Newtonian approximation, to first order ($GM/r$), the non vanish
Christoffel symbols of Riemannian metric $\tilde{a}$ are
$\tilde{\gamma}^{i}_{00},~{}\tilde{\gamma}^{0}_{0i},~{}\tilde{\gamma}^{i}_{jk}$.
Here, $|GM/r|$ denotes the typical value of Newtonian potential. Deducing from
the definition of $l^{\mu}$, we find that $l^{0}$ is the same order with $1$
and $l^{i}$ is the same order with $GM/r$. Therefore, we obtain the
approximation formula of $G^{\mu}$

$\displaystyle G^{0}$ $\displaystyle=$ $\displaystyle
2\tilde{\gamma}^{0}_{0i}y^{0}y^{i}-l^{0}\tilde{b}_{i}(\tilde{\gamma}^{i}_{00}y^{0}y^{0}+\tilde{\gamma}^{i}_{jk}y^{j}y^{k}),$
(29) $\displaystyle G^{i}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}^{i}_{00}y^{0}y^{0}+\tilde{\gamma}^{i}_{jk}y^{j}y^{k}.$
(30)

In Finsler geometry, one can get the Chern connection from the geodesic spray
coefficient $G^{\mu}$Book by Bao

$N^{\mu}_{\nu}=\frac{1}{2}\frac{\partial G^{\mu}}{\partial
y^{\nu}},~{}~{}\Gamma^{\mu}_{\nu\lambda}=\frac{\partial
N^{\mu}_{\nu}}{\partial
y^{\lambda}}+\frac{1}{2}g^{\mu\rho}y_{\kappa}\frac{\partial^{2}N^{\kappa}_{\rho}}{\partial
y^{\nu}\partial y^{\lambda}}.$ (31)

By making use of the approximation formula of $G^{\mu}$ (29) and (30), we
obtain the approximation formula of the non vanish Chern connection

$\Gamma^{i}_{00}=\tilde{\gamma}^{i}_{00},~{}\Gamma^{0}_{0i}=\tilde{\gamma}^{0}_{0i},~{}\Gamma^{i}_{jk}=\tilde{\gamma}^{i}_{jk}.$
(32)

Therefore, to first order ($GM/r$), the Chern connection is none other than
the Christoffel symbols of Riemannian metric $\tilde{a}$. It implies that
curvature tensors approximately reduce to Riemannian, to first order ($GM/r$).
Following the deduction in Weinberg , we obtain the result

$\tilde{a}_{00}=-1+\delta a_{00},~{}~{}\tilde{a}_{ij}=\delta_{ij}(1+\delta
a_{00}),$ (33)

where $\delta a_{00}$ is a perturbation term with order of $GM/r$.

###  III.2 The Zermelo navigation problem

Zermelo aimed to find minimum time trajectories in a Riemannian manifold
($M,h$), which under the influence of a wind represented by a vector field
$W$Zermelo . Shen Shen  proved that the minimum time trajectories are exactly
the geodesic of a particular Finsler geometry-Randers spaceRanders , if the
wind is time independent. As a particular case of Finsler geometry, the
geometrical properties of Randers space have been studied Bao . Recently,
Randers space has drawn physicists’s attention. Gibbons et al. described
connection between the Randers spacetime and the Zermelo viewpoints by casting
the former in a Painlevé-Gullstrand formGibbons . It is interesting to study a
particle moving in Randers-Finsler spacetime.

In the following, we study the Zermelo navigation problem which is regarded as
an alternative of Randers-Finsler spacetime. For a given Riemannian metric $h$
and a wind $W$, one can set up a connection between Riemannnian data $(h,W)$
and Randers-Finsler structure $F$,

$\tilde{a}_{ij}=\frac{\lambda
h_{ij}+W_{i}W_{j}}{\lambda^{2}},~{}\tilde{b}_{i}=-\frac{W_{i}}{\lambda},~{}\lambda=1-h_{ij}W^{i}W^{j},$
(34)

where $W^{i}=h^{ij}W_{j}$ and $\tilde{a}^{ij}=\lambda(h^{ij}-W^{i}W^{j})$.

Schwarzschild metric as an exact solution of Einstein’s vacuum field equation
has been used to study four classical tests of general relativity. However,
the problems we mentioned in the beginning of the paper can not be solved in
the framework of Schwarzschild spacetime, if the gravitational source only
involves baryonic matters. Here, we investigate the equation of motion in
Finsler spacetime, which may be regarded as a candidate to solve the problems.

First, we start with the space part of Schwarzschild metric

$h_{ij}dx^{i}dx^{j}=\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\varphi^{2},$
(35)

and consider the influence of a radial wind $W(r)\equiv W_{r}dr$. Then, the
solution of Zermelo navigation problem gives the Randers metric, whose Finsler
structure is given as

$\displaystyle Fd\tau$ $\displaystyle=$
$\displaystyle\sqrt{\lambda^{-1}\left(\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\varphi^{2}\right)+\lambda^{-2}W_{r}^{2}dr^{2}}-\lambda^{-1}W_{r}dr$
(36) $\displaystyle=$
$\displaystyle\sqrt{\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+\lambda^{-1}(r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\varphi^{2})}-\lambda^{-1}W_{r}dr,$

where we have used the formula of $\lambda$ (34) to get the second equation of
(36).

Next, we extend the structure (36) into relativistic form. By making use of
the approximate solution (33) of Finslerian vacuum field equation, we find
that the Randers-Finsler structure is of this form

$Fd\tau=\sqrt{-\lambda^{2}\left(1-\frac{2GM}{r}\right)dt^{2}+\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+\lambda^{-1}(r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\varphi^{2})}-\lambda^{-1}W_{r}dr,$
(37)

while $|GM/r|\ll 1$.

The Schwarzschild metric is time independent and spatial isotropy, it means
that there are Killing vectors which correspond to the conserve quantities
$p_{t}=\left(1-\frac{2GM}{r}\right)\frac{dt}{d\tau}$ and
$p_{\varphi}=r^{2}\sin^{2}\theta\frac{d\phi}{d\tau}$. The Killing equations of
Randers space are given as Finsler PF

$\displaystyle K_{V}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha}(V_{\mu|\nu}+V_{\nu|\mu})y^{\mu}y^{\nu},$ (38)
$\displaystyle K_{V}(\beta)$ $\displaystyle=$
$\displaystyle\left(V^{\mu}\frac{\partial\tilde{b}_{\nu}}{\partial
x^{\mu}}+\tilde{b}_{\mu}\frac{\partial V^{\mu}}{\partial
x^{\nu}}\right)y^{\nu},$ (39)

where $``|"$ denotes the covariant derivative with respect to the Riemannian
metric $\alpha$. It is obvious that the vectors $V^{0}=C^{0}$ and
$V^{\varphi}=C^{\varphi}$ are solutions of (38) in Rander-Finsler spacetime
with structure (37), where $C^{0}$ and $C^{\varphi}$ are constant. Such
solutions $V^{0}=C^{0}$ and $V^{3}=C^{\varphi}$ are also satisfy the equation
(39). Thus, $V^{0}=C^{0}$ and $V^{3}=C^{\varphi}$ are Killing vectors in
Rander-Finsler spacetime with structure (37). This symmetry is the same with
Schwarzschild spacetime, it implies that conserve quantities exist in Rander-
Finsler spacetime with structure (37), like $p_{t}$ and $p_{\varphi}$ in
Schwarzschild spacetime.

###  III.3 The equation of motion

One should notice that the Rander-Finsler structure (37) got in above
subsection depends only on $r,dx^{\mu}$. Therefore, $\tilde{b}_{\mu}dx^{\mu}$
is a closed 1-form, the geodesic equation in such spacetime (37) is of the
form

$\frac{d^{2}x^{\mu}}{d\tau^{2}}+(\tilde{\gamma}^{\mu}_{\nu\lambda}+l^{\mu}\tilde{b}_{\nu|\lambda})\frac{dx^{\nu}}{d\tau}\frac{dx^{\nu}}{d\tau}=0.$
(40)

It is convenient to denote $B(r)\equiv\lambda^{2}\left(1-\frac{2GM}{r}\right)$
and $A(r)\equiv\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}$. By making use
of the nonvanishing components of the connection
$\tilde{\gamma}^{\mu}_{\nu\lambda}$, we find from (40) that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}t}{d\tau^{2}}+\frac{B^{\prime}}{B}\frac{dt}{d\tau}\frac{dr}{d\tau}+\frac{dt}{d\tau}f\left(x,\frac{dx}{d\tau}\right),$
(41) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}r}{d\tau^{2}}+\frac{A^{\prime}}{2A}\left(\frac{dr}{d\tau}\right)^{2}+\frac{B^{\prime}}{2A}\left(\frac{dt}{d\tau}\right)^{2}-\frac{1}{2A}\left(\frac{d\theta}{d\tau}\right)^{2}\frac{d}{dr}\left(\frac{r^{2}}{\lambda}\right)-\frac{\sin^{2}\theta}{2A}\left(\frac{d\varphi}{d\tau}\right)^{2}\frac{d}{dr}\left(\frac{r^{2}}{\lambda}\right)+\frac{dr}{d\tau}f\left(x,\frac{dx}{d\tau}\right),$
(42) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}\theta}{d\tau^{2}}+\frac{\lambda}{r^{2}}\frac{d}{dr}\left(\frac{r^{2}}{\lambda}\right)\frac{d\theta}{d\tau}\frac{dr}{d\tau}-\sin\theta\cos\theta\left(\frac{d\varphi}{d\tau}\right)^{2}+\frac{d\theta}{d\tau}f\left(x,\frac{dx}{d\tau}\right),$
(43) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}\varphi}{d\tau^{2}}+\frac{\lambda}{r^{2}}\frac{d}{dr}\left(\frac{r^{2}}{\lambda}\right)\frac{d\varphi}{d\tau}\frac{dr}{d\tau}+2\cot\theta\frac{d\varphi}{d\tau}\frac{d\theta}{d\tau}+\frac{d\varphi}{d\tau}f\left(x,\frac{dx}{d\tau}\right),$
(44)

where
$f\left(x,\frac{dx}{d\tau}\right)\equiv\tilde{b}_{\nu|\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\nu}}{d\tau}/F$
and a prime denotes $d/dr$. Furthermore, we know that structure (37) only
depends on $r,dx^{\mu}$. Thus, the field is isotropic. It is convenient to
consider the orbit of particle confined to the equatorial plane
$\theta=\pi/2$. Then, the equation (43) is satisfied. Solving the equations
(41) and (44), we get

$\displaystyle\frac{d}{d\tau}\left(\ln\frac{dt}{d\tau}+\ln B\right)$
$\displaystyle=$ $\displaystyle\frac{d\ln J_{1}}{d\tau},$ (45)
$\displaystyle\frac{d}{d\tau}\left(\ln\frac{d\varphi}{d\tau}+\ln\frac{r^{2}}{\lambda}\right)$
$\displaystyle=$ $\displaystyle\frac{d\ln J_{1}}{d\tau},$ (46)

where we involved a new quantity $J_{1}$. It is defined as$\frac{d\ln
J_{1}}{d\tau}\equiv-f\left(x,\frac{dx}{d\tau}\right)$. Deducing from the
equations (45) and (46), we obtain two constants $(E,J)$ of motion. It
satisfies

$\displaystyle\frac{B}{J_{1}}\frac{dt}{d\tau}$ $\displaystyle=$ $\displaystyle
E,$ (47) $\displaystyle\frac{r^{2}}{\lambda J_{1}}\frac{d\varphi}{d\tau}$
$\displaystyle=$ $\displaystyle J.$ (48)

By multiplying the equation (42) with $2Adr/d\tau$ and making use of the
equations (47,48), we obtain

$B\left(\frac{dt}{d\tau}\right)^{2}-A\left(\frac{dr}{d\tau}\right)^{2}-\frac{r^{2}}{\lambda}\left(\frac{d\varphi}{d\tau}\right)^{2}=CJ_{1}^{2},$
(49)

where $C$ is a constant.

In the following, the value of $C$ and the formula of $J_{1}$ will be derived.
The derivative of the term $\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}$ gives

$\displaystyle\frac{d}{d\tau}\left(\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}\right)$
$\displaystyle=$ $\displaystyle\frac{dx^{\nu}}{d\tau}\frac{\partial}{\partial
x^{\nu}}\left(\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}\right)=\frac{dx^{\nu}}{d\tau}\left(\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}\right)_{|\nu}$
(50) $\displaystyle=$
$\displaystyle\tilde{b}_{\alpha|\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}+\tilde{b}_{\mu}\left(\frac{d^{2}x^{\mu}}{d\tau^{2}}+\tilde{\gamma}^{\mu}_{\nu\lambda}\right)\frac{dx^{\nu}}{d\tau}\frac{dx^{\nu}}{d\tau}$
$\displaystyle=$
$\displaystyle\left(1-\frac{\tilde{b}_{\mu}}{F}\frac{dx^{\mu}}{d\tau}\right)\tilde{b}_{\alpha|\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau},$

where $``|"$ denotes the covariant derivative with respect to the Riemannian
metric $\alpha$. Here, we have used the fact that the term
$\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}$ is a scaler in Riemannian spacetime
with metric $\tilde{a}_{\mu\nu}$, to get the second equation of (50). And we
have used the geodesic equation (40) to get the last equation of (50).
Noticing that $F$ is constant along the geodesic, we find from equation (50)
that

$\frac{d\ln\left(F-\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}\right)}{d\tau}=-\tilde{b}_{\nu|\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\nu}}{d\tau}/F=-f\left(x,\frac{dx}{d\tau}\right).$
(51)

It implies that

$J_{1}=F-\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau},$ (52)

with normalization of $\tau$. Combining the equations (49) and (52), we find
$C=1$ for massive particles. The equations (49) and (52) can not determine the
value of $C$ for photons. While $\tilde{b}$ vanishes, the equation of motion
in Randers-Finsler spacetime must return to the one in Riemannian spacetime.
This physical requirement implies $C=0$ for photons.

At last, we list what we got. In Rander-Finsler spacetime with structure (37),
there are two constants of motion $E,J$, which again support our discussion
about Killing vectors in subsection B. There is equation of motion (49)
corresponding to the constancy of $F$. All equations (47,48,49) involve the
term $J_{1}$, which could be regarded as deviation from the equation of motion
in Schwarzschild spacetime. The constant $C$ in equation of motion equals 1 or
0, corresponds to massive particles and photons, respectively. Start with
these results, we could find the trajectory of particles moving in Randers-
Finsler spacetime.

###  III.4 The Newtonian limit and gravitational deflection of light

In the above subsection, we derived the equation of motion for particles in
Randers-Finsler spacetime. In the following, we will study the orbit of
particles. Recently, Grumiller constructed an effective model for gravity of a
central object at large scalesGrumiller . It predicted a Rindler-type
acceleration Wald . We will show that if the Finslerian parameter $\lambda$ is
of the form

$\lambda=1+\frac{GM}{r_{s}^{2}}r,$ (53)

it will deduce a Rindler-type acceleration, where $r_{s}$ is a constant which
denotes the physical scale of gravitational system. First, we investigate the
Newtonian limit. By making use of the equations (47,48,49), we obtain the
relation between the radial distant $r$ and time $t$

$\frac{AE^{2}}{B^{2}}\left(\frac{dr}{dt}\right)^{2}+\frac{J^{2}\lambda}{r^{2}}-\frac{E^{2}}{B}=-1~{}.$
(54)

In Newtonian limit, particles move slowly in a weak field. Thus, the
quantities
$\frac{J^{2}}{r^{2}},\left(\frac{dr}{dt}\right)^{2},E^{2}-1,\frac{GM}{r}$ all
are small. And to first order in these quantities, the equation (54) reduces
to

$\frac{1}{2}\left(\frac{dr}{dt}\right)^{2}+\frac{J^{2}}{2r^{2}}-\left(1-\frac{r^{2}}{r_{s}^{2}}\right)\frac{GM}{r}=\varepsilon,$
(55)

where $\varepsilon\equiv\frac{1}{2}(E^{2}-1)$ is the energy in Newton’s
theory. The equation (55) implies the effective Newtonian potential is
modified

$\phi_{M}=-\left(1-\frac{r^{2}}{r_{s}^{2}}\right)\frac{GM}{r}~{}.$ (56)

Therefore, the effective acceleration is Rindler-type acceleration

$a_{M}=-\nabla\phi_{M}=-\left(\frac{GM}{r^{2}}+\frac{GM}{r_{s}^{2}}\right)~{}.$
(57)

Then, the velocity of galaxy rotational curve have the approximate relation

$v\approx\sqrt{\frac{GM}{r}+\frac{GM}{r_{s}^{2}}r}~{}.$ (58)

For a dwarf galaxies with mass scale $10^{8}$ solar masses, and
$r_{s}=1$kiloparsec (a typical length scale of dwarf galaxies), we find that
the term $\frac{GM}{r_{s}^{2}}$ in (58) equals $10^{-62}$ in natural units
($c=\hbar=G=1$). For a spiral galaxies with mass scale $10^{11}$ solar masses,
and $r_{s}=30$kiloparsec (a typical length scale of spiral galaxies), we find
that the term $\frac{GM}{r_{s}^{2}}$ in (58) also equals $10^{-62}$ in natural
units ($c=\hbar=G=1$). Therefore, for a reasonable parameter $r_{s}$, our
result is compatible with Grumiller’s results of galaxies rotational curve
Grumiller . It means that our result (58) could account for galaxies
rotational curve.

Next, we investigate the gravitational deflection of light. We limit us to the
case of weak gravitational field. It implies $\frac{GM}{r}\ll 1$. By making
use of the equations (47,48,49), and noticing that $C=0$ for photons, we
obtain the relation between the radial distant $r$ and angle $\varphi$

$\displaystyle\left(\frac{1}{r^{2}}\frac{dr}{d\varphi}\right)^{2}$
$\displaystyle=$
$\displaystyle\left(\frac{E}{J}\right)^{2}\frac{1}{AB\lambda^{2}}-\frac{1}{Ar^{2}\lambda}$
(59) $\displaystyle=$
$\displaystyle\left(\frac{E}{J\lambda}\right)^{2}-\frac{\lambda}{r^{2}}\left(1-\frac{2GM}{r}\right).$

In terms of variable $u=\frac{GM}{r}$, the equation (59) changes as

$\left(\frac{du}{d\varphi}\right)^{2}=\left(\frac{EGM}{J\lambda}\right)^{2}-\lambda
u^{2}(1-2u).$ (60)

At the closest approach to the gravitational source $M$,
$\varphi=\varphi_{m},~{}u=u_{m},~{}\lambda=\lambda_{m}$ and $du/d\varphi$
vanishes, thus

$\left(\frac{EGM}{J}\right)^{2}=\lambda_{m}^{3}u_{m}^{2}(1-2u_{m}).$ (61)

Substituting the equation (61) into (60), we obtain

$\frac{d\varphi}{du}=\frac{\lambda}{\sqrt{\lambda_{m}^{3}u_{m}^{2}(1-2u_{m})-\lambda^{3}u^{2}(1-2u)}}.$
(62)

Changing variable to $x=u/u_{m}$, to first order in $u_{m}$, we obtain from
(53,62) that

$\displaystyle\varphi_{m}-\varphi_{\infty}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\frac{dx}{\sqrt{1-x^{2}-2u_{m}(1-x^{3})+3u_{s}^{2}/u_{m}-2u_{s}^{2}/u_{m}x-xu_{s}^{2}/u_{m}}}$
(63) $\displaystyle=$
$\displaystyle\int^{1}_{0}\frac{dx}{\sqrt{1-x^{2}}}\left(1+u_{m}\frac{1-x^{3}}{1-x^{2}}-\frac{3u_{s}^{2}/u_{m}-2u_{s}^{2}/u_{m}x-xu_{s}^{2}/u_{m}}{2(1-x^{2})}\right)$
$\displaystyle=$
$\displaystyle\frac{\pi}{2}+2u_{m}+\frac{u_{s}^{2}}{u_{m}}\left(\frac{3\sqrt{1-x^{2}}}{2(1+x)}-\log\left(\frac{1+\sqrt{1-x^{2}}}{x}\right)\right)\bigg{|}^{1}_{0}$

The term $\frac{\pi}{2}+2u_{m}$ is what we expect in general relativity, and
the third term in (63) tends to infinity at $x=0$. This infinity is due to
that the Rindler potential (56) is infinity at infinity distance $(u=0)$. It
is reasonable to consider that the Rindler-type potential vanishes outsider a
given cut off scale $r_{\rm out}$. Therefore, we deduce from (63) that the
deflection angle for the Rindler-type acceleration in Finsler spacetime is of
the form

$\alpha(r_{m})=\frac{GM}{r_{m}}\left(4+2\frac{r_{m}^{2}}{r_{s}^{2}}\left[\log\left(\frac{r_{\rm
out}+\sqrt{r_{\rm out}^{2}-r_{m}^{2}}}{r_{m}}\right)-\frac{3\sqrt{r_{\rm
out}^{2}-r_{m}^{2}}}{2(r_{\rm out}+r_{m})}\right]\right).$ (64)

While $r_{m}\ll r_{s}$, the deflection angle (64) reduces to the familiar one
in general relativity.

The strong and weak gravitational lensing survey of Bullet cluster
1E0657-558Bullet  obtained a convergence $\kappa$-map. The convergence
$\kappa$ is defined asPeacock

$\kappa=\frac{D_{LS}}{2D_{S}}\nabla_{\theta}\alpha(\theta_{I})=\frac{D_{LS}D_{L}}{2D_{S}}\nabla_{\vec{\xi}}~{}\alpha(\vec{\xi}),$
(65)

where $\alpha(\vec{\xi})$ is deflection angle, $D_{LS}$ is the angular
distance from the lens plane to a source galaxy, $D_{L}$ is the angular
distance to the lens plane, $D_{S}$ is the angular distance to a source
galaxy, $\theta_{I}$ specify the observed position of the source galaxy,
$\vec{\xi}$ is the two dimensional vector in lens plane and
$\nabla_{\vec{\xi}}$ is two dimensional gradient operator. The surface density
$\Sigma(\vec{\xi})$ is derived as

$\Sigma(\vec{\xi})=\int\rho(\vec{\xi},z)dz,$ (66)

where $\rho(\vec{\xi},z)$ is the density of gravitational source $M$ and $z$
is the direction perpendicular to the lens plane. In weak field , the
deflection angle can be obtained as the superposition of the deflections

$\alpha(\vec{\xi})=4G\int\bar{\Sigma}(\vec{\xi}^{\prime})\frac{\vec{\xi}-\vec{\xi}^{\prime}}{|\vec{\xi}-\vec{\xi}^{\prime}|^{2}},$
(67)

where

$\bar{\Sigma}(\vec{\xi})=\Sigma(\vec{\xi})\left(1+\frac{|\vec{\xi}|^{2}}{2r_{s}^{2}}\left[\log\left(\frac{r_{\rm
out}+\sqrt{r_{\rm
out}^{2}-|\vec{\xi}|^{2}}}{|\vec{\xi}|}\right)-\frac{3\sqrt{r_{\rm
out}^{2}-|\vec{\xi}|^{2}}}{2(r_{\rm out}+|\vec{\xi}|)}\right]\right).$ (68)

Substituting (67) into (65), we obtain

$\kappa=\frac{4\pi
GD_{LS}D_{L}}{D_{S}}\bar{\Sigma}(\vec{\xi})\equiv\frac{\bar{\Sigma}(\vec{\xi})}{\Sigma_{c}}~{}.$
(69)

It is obvious from (68,69) that the convergence $\kappa$ does not reach its
maximum at the center of $\Sigma(\xi)$. It also implies that the position
where $\kappa$ reaches its maximum value is separated from the center of
$\Sigma(\xi)$. Thus, the convergence $\kappa$ deduced in Finsler gravity
satisfies the features of Bullet Cluster. The Rindler-type potential in
Finsler spacetime could account for the observations of Bullet Cluster.

##  IV Conclusions

In this paper, we presented the vacuum field equation in Finsler spacetime. By
making use of the post-Newtonian approximation and the viewpoints of Zermelo
navigation problem, we investigated the dynamics in Randers-Finsler spacetime.
The Newtonian limit and gravitational deflection of light was obtained
explicitly. Within the framework of Finsler spacetime, the deflection angle
and the convergence $\kappa$ in Rindler-type potential were given.

The surface density $\Sigma$-map and the convergence $\kappa$-map of Bullet
Cluster 1E0657-558Bullet  show that the center of baryonic matters separate
from the center of gravitational force, and the distribution of gravitational
force do not possess spherical symmetry. The formula (57) manifests that the
gravity in Finsler spacetime modified the Newtonian inverse-square law at
large scale. It is obvious from (68,69) that the convergence $\kappa$ does not
reach its maximum at the center of $\Sigma(\xi)$. It also implies that the
position where $\kappa$ reaches its maximum value is separated from the center
of $\Sigma(\xi)$. Thus, our model satisfies the first particular feature of
Bullet Cluster. The particular feature of spherical symmetry broken implies
that all modified gravity models with central potential need improvements for
accounting the observations of Bullet Cluster. However, for simplicity, the
central potential could regarded as the zero order term of the final modified
gravity model for Bullet Cluster. The spherical symmetry broken may be deduced
by the next leading order term of the final modified gravity model.

The convergence $\kappa$ in Rindler-type potential could account for
observations of Bullet Cluster. The numerical analysis is in progress. In
future work, we will consider the non-central potential in Finsler spacetime,
and investigate the effect of spherical symmetry broken.

###### Acknowledgements.

We would like to thank Prof. C. J. Zhu, M. H. Li and S. Wang for useful
discussions. The work was supported by the NSF of China under Grant No.
10875129 and 11075166.

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