11institutetext: Division of Medical Image Computing, German Cancer Research
Center (DKFZ), Heidelberg, Germany
22institutetext: Helmholtz Imaging, DKFZ, Heidelberg, Germany
33institutetext: Pattern Analysis and Learning Group, Department of Radiation
Oncology, Heidelberg University Hospital, Heidelberg, Germany
44institutetext: National Center for Tumor Diseases (NCT), NCT Heidelberg, A
partnership between DKFZ and University Medical Center Heidelberg
44email<EMAIL_ADDRESS>
# MultiTalent: A Multi-Dataset Approach to Medical Image Segmentation

Constantin Ulrich 1144 Fabian Isensee 1122 Tassilo Wald 1122 Maximilian Zenk
11 Michael Baumgartner 112255 Klaus H. Maier-Hein 1133

###### Abstract

The medical imaging community generates a wealth of data-sets, many of which
are openly accessible and annotated for specific diseases and tasks such as
multi-organ or lesion segmentation. Current practices continue to limit model
training and supervised pre-training to one or a few similar datasets,
neglecting the synergistic potential of other available annotated data. We
propose MultiTalent, a method that leverages multiple CT datasets with diverse
and conflicting class definitions to train a single model for a comprehensive
structure segmentation. Our results demonstrate improved segmentation
performance compared to previous related approaches, systematically, also
compared to single-dataset training using state-of-the-art methods, especially
for lesion segmentation and other challenging structures. We show that
MultiTalent also represents a powerful foundation model that offers a superior
pre-training for various segmentation tasks compared to commonly used
supervised or unsupervised pre-training baselines. Our findings offer a new
direction for the medical imaging community to effectively utilize the wealth
of available data for improved segmentation performance. The code and model
weights will be published here: [tba]

###### Keywords:

Medical image segmentation multitask learning transfer learningfoundation
model partially labeled datasets.

##  1 Introduction

Figure 1: (a) Usually only a few classes are annotated in publicly available
datasets. b) Different groundtruth label properties can generate contradicting
class predictions. For example, the heart annotation of dataset 11 differs
from the heart annotation of dataset 10, which causes the aorta of dataset 11
to overlap with the heart of dataset 10. In contrast to dataset 11, in dataset
7 the aorta is also annotated in the lower abdomen. c) Instead of training one
network for each dataset, we introduce a method to train one network with all
datasets, while retaining dataset-specific annotation protocols.

The success of deep neural networks heavily relies on the availability of
large and diverse annotated datasets across a range of computer vision tasks.
To learn a strong data representation for robust and performant medical image
segmentation, huge datasets with either many thousands of annotated data
structures or less specific self-supervised pretraining objectives with
unlabeled data are needed [33, 29]. The annotation of 3D medical images is a
difficult and laborious task. Thus, depending on the task, only a bare minimum
of images and target structures is usually annotated. This results in a
situation where a zoo of partially labeled datasets is available to the
community. Recent efforts have resulted in a large dataset of >1000 CT images
with >100 annotated classes each, thus providing more than 100.000 manual
annotations which can be used for pre-training [30]. Focusing on such a
dataset prevents leveraging the potentially precious additional information of
the above mentioned other datasets that are only partially annotated.
Integrating information across different datasets potentially yields a higher
variety in image acquisition protocols, more anatomical target structures or
details about them as well as information on different kinds of pathologies.
Consequently, recent advances in the field allowed utilizing partially labeled
datasets to train one integrated model [21]. Early approaches handled
annotations that are present in one dataset but missing in another by
considering them as background [27, 5] and penalizing overlapping predictions
by taking advantage of the fact that organs are mutually exclusive [28, 7].
Some other methods only predicted one structure of interest for each forward
pass by incorporating the class information at different stages of the network
[4, 31, 22]. Chen et al. trained one network with a shared encoder and
separate decoders for each dataset to generate a generalized encoder for
transfer learning [2]. However, most approaches are primarily geared towards
multi-organ segmentation as they do not support overlapping target structures,
like vessels or cancer classes within an organ [6, 8, 23, 12]. So far, all
previous methods do not convincingly leverage cross-dataset synergies. As Liu
et al. pointed out, one common caveat is that many methods force the resulting
model to average between distinct annotation protocol characteristics [22] by
combining labels from different datasets for the same target structure
(visualized in Figure 1 b)). Hence, they all fail to reach segmentation
performance on par with cutting-edge single dataset segmentation methods. To
this end, we introduce MultiTalent (MULTI daTAset LEarNing and pre-Training),
a new, flexible, multi-dataset training method: 1) MultiTalent can handle
classes that are absent in one dataset but annotated in another during
training. 2) It retains different annotation protocol characteristics for the
same target structure and 3) allows for overlapping target structures with
different level of detail such as liver, liver vessel and liver tumor.
Overall, MultiTalent can include all kinds of new datasets irrespective of
their annotated target structures.
MultiTalent can be used in two scenarios: First, in a combined multi-dataset
(MD) training to generate one foundation segmentation model that is able to
predict all classes that are present in any of the utilized datasets, and
second, for pre-training to leverage the learned representation of this
foundation model for a new task. In experiments with a large collection of
abdominal CT datasets, the proposed model outperformed state-of-the-art
segmentation networks that were trained on each dataset individually as well
as all previous methods that incorporated multiple datasets for training.
Interestingly, the benefits of MultiTalent are particularly notable for more
difficult classes and pathologies. In comparison to an ensemble of single
dataset solutions, MultiTalent comes with shorter training and inference
times.
Additionally, at the example of three challenging datasets, we demonstrate
that fine-tuning MultiTalent yields higher segmentation performance than
training from scratch or initializing the model parameters using unsupervised
pre-training strategies [29, 33]. It also surpasses supervised pretrained and
fine-tuned state-of-the art models on most tasks, despite requiring orders of
magnitude less annotations during pre-training.

##  2 Methods

We introduce MultiTalent, a multi dataset learning and pre-training method, to
train a foundation medical image segmentation model. It comes with a novel
dataset and class adaptive loss function. The proposed network architecture
enables the preservation of all label properties, learning overlapping classes
and the simultaneous prediction of all classes. Furthermore, we introduce a
training schedule and dataset preprocessing which balances varying dataset
size and class characteristics.

###  2.1 Problem definition

We begin with a dataset collection of $K$ datasets $D^{(k)},k\in[1,K]$, with
$N^{(k)}$ image and label pairs
$D^{(k)}=\\{(x,y)^{(k)}_{1},...,(x,y)^{(k)}_{N^{(k)}}\\}$. In these datasets,
every image voxel $x^{(k)}_{i},i\in[1,I]$, is assigned to one class $c\in
C^{(k)}$, where $C^{(k)}\subseteq C$ is the label set associated to dataset
$D^{(k)}$. Even if classes from different datasets refer to the same target
structure we consider them as unique, since the exact annotation protocols and
labeling characteristics of the annotations are unknown and can vary between
datasets: $C^{(k)}\cap C^{(j)}=\emptyset,\forall k\neq j$. This implies that
the network must be capable of predicting multiple classes for one voxel to
account for the inconsistent class definitions.

###  2.2 MultiTalent

####  2.2.1 Network modifications

We employ three different network architectures, which are further described
below, to demonstrate that our approach is applicable to any network topology.
To solve the label contradiction problem we decouple the segmentation outputs
for each class by applying a Sigmoid activation function instead of the
commonly used Softmax activation function across the dataset. The network
shares the same backbone parameters $\Theta$ but it has independent
segmentation head parameters $\Theta_{c}$ for each class. The Sigmoid
probabilities for each class are defined as
$\hat{y}_{c}=f(x,\Theta,\Theta_{c})$. This modification allows the network to
assign multiple classes to one pixel and thus enables overlapping classes and
the conservation of all label properties from each dataset. Consequently, the
segmentation of each class can be thought of as a binary segmentation task.

####  2.2.2 Dataset and class adaptive loss function

Based on the well established combination of a Cross-entropy and Dice loss for
single dataset medical image segmentation, we employ the binary Binary Cross-
entropy loss (BCE) and a modified Dice loss for each class over all
$B,b\in[1,B]$, images in a batch:

$L_{c}=\frac{1}{I}\sum_{b,i}BCE(\hat{y}^{(k)}_{i,b,c},\;y^{(k)}_{i,b,c})-\frac{2\sum_{b,i}\hat{y}^{(k)}_{i,b,c}\;y_{{i,b,c}}^{(k)}}{\sum_{b,i}\hat{y}^{(k)}_{i,b,c}+\sum_{b,i}y_{i,b,c}^{(k)}}$
(1)

While the regular dice loss is calculated for each image within a batch, we
calculate the dice loss jointly for all images of the input batch. This
regularizes the loss if only a few voxels of one class are present in one
image and a larger area is present in another image of the same batch. Thus,
an inaccurate prediction of a few pixels in the first image has a limited
effect on the loss. In the following, we unite the sum over the image voxels
$i$ and the batch $b$ to $\sum_{z}$. We modify the loss function to be
calculated only for classes that were annotated in the corresponding partially
labeled dataset [27, 5], in the following indicated by $\mathbb{1}_{c}^{(k)}$,
where $\mathbb{1}_{c}^{(k)}=1$ if $c\in C^{(k)}$ and 0 otherwise. Instead of
averaging, we add up the loss over the classes. Hence, the loss signal for
each class prediction does not depend on the number of other classes within
the batch. This compensates for the varying number of annotated classes in
each dataset:

$L=\sum_{c}\Big{(}\mathbb{1}_{c}^{(k)}\frac{1}{I}\sum_{z}BCE(\hat{y}^{(k)}_{z,c},\;y^{(k)}_{z,c})-\frac{2\sum_{z}\mathbb{1}_{c}^{(k)}\;\hat{y}^{(k)}_{z,c}\;y_{z,c}^{(k)}}{\sum_{z}\mathbb{1}_{c}^{(k)}\;\hat{y}^{(k)}_{z,c}+\sum_{z}\mathbb{1}_{c}^{(k)}\;y_{z,c}^{(k)}}\Big{)}$
(2)

####  2.2.3 Network architectures

To demonstrate the general applicability of this approach, we applied it to
three segmentation networks. We employed a 3D U-Net [24], an extension with
additional residual blocks in the encoder (Resenc U-Net), that demonstrated
highly competitive results in previous medical image segmentation challenges
[15, 14] and a recently proposed transformer based architecture (SwinUNETR
[29]). We implemented our approach in the nnU-Net framework [13]. However, the
automatic pipeline configuration from nnU-net was not used in favor of a
manually defined configuration that aims to reflect the peculiarities of each
of the datasets, irrespective of the number of training cases they contain. We
manually selected a patch size of $[96,192,192]$ and image spacing of 1mm in
plane and 1.5mm for the axial slice thickness, which nnU-Net used to
automatically create the two CNN network topologies. For the SwinUNETR, we
adopted the default network topology.

####  2.2.4 Multi-dataset training setup

We trained MultiTalent with 13 public abdominal CT datasets with a total of
1477 3D images, including 47 classes (Multi-dataset (MD) collection) [1, 19,
9, 20, 18, 25, 3, 26, 11]. Detailed information about the datasets, can be
found in the appendix in Table 3 and Figure 3, including the corresponding
annotated classes. We increased the batch size to 4 and the number of training
epochs to 2000 to account for the high number of training images. To
compensate for the varying number of training images in each dataset, we
choose a sampling probability per case that is inversely proportional to
$\sqrt{n}$, where $n$ is the number of training cases in the corresponding
source dataset.

####  2.2.5 Transfer learning setup

We used the BTCV (small multi organ dataset [19]), AMOS (large multi organ
dataset [16]) and KiTS19 (pathology dataset [11]) datasets to evaluate the
generalizability of the MultiTalent features in a pre-training and fine tuning
setting. Naturally, the target datasets were excluded from the repective pre-
training. Fine tuning was performed with identical configuration as the source
training, except for the batch size which was set to 2. We followed the fine-
tuning schedule proposed by Kumar et al. [17]. First, the segmentation heads
were warmed up over 10 epochs with linearly increasing learning rate, followed
by a whole-network warum-up over 50 epochs. Finally, we continued with the
standard nnU-Net training schedule.

Figure 2: Comparison of Dice scores averaged over all classes, all datasets
and over classes of special interest. Difficult classes are those for which
the default nnUNet has a Dice below 75. The same color indicates the same
network architecture and the pattern implies training with multiple datasets
using MultiTalent.

Table 1: Official BTCV test set leaderboard results, Dice and 95% Hausdorff
Distance. * indicates usage of multiple datasets.

Method #models Avg. Dice Avg. HD95 nnU-Net [13] 2 models, each 88.10 17.26
5-fold ensemble UNETR [22, 10] single model 81.43 - SwinUNETR [22, 29] single
model 82.06 - Universal Model* [22] single model 86.13 - DoDNet (pretrained*)
[31] single model 86.44 15.62 PaNN* [32] single model 84.97 18.47 MultiTalent
Resenc U-Net* single model 88.82 16.35 MultiTalent Resenc U-Net* 5-fold
ensemble 88.91 14.68 Resenc U-Net (pre-trained MultiTalent*) 5-fold ensemble
89.07 15.01

Table 2: 5-fold cross validation results of different architectures and
pretraining schemes. We used the original (org.) SwinUNETR implementation and
the provided self-supervised weights as additional baseline [29].

Architecture Pretraining scheme BTCV AMOS KiTs19 KiTs19 Dice avg. Dice avg.
Kidney Dice Tumor Dice SwinUNETR from scratch 74.27 86.04 87.69 46.56 org.
implement. self-supervised [29] 74.71 86.11 87.62 43.64 SwinUNETR from scratch
81.44 87.59 95.97 76.52 supervised (~$10^{5}$ annot.) [30] 83.08 88.63 96.36
80.30 MultiTalent(~3600 annot.) 82.14 87.32 96.08 76.56 U-Net from scratch
83.76 89.40 96.56 80.69 self-supervised [33] 84.01 89.30 96.59 80.91
supervised (~$10^{5}$ annot.) [30] 84.22 89.66 96.72 82.48 MultiTalent (~3600
annot.) 84.42 89.60 96.81 83.03 Resenc U-Net from scratch 84.38 89.71 96.83
83.22 self-supervised [33] 84.27 89.70 96.82 83.53 supervised (~ $10^{5}$
annot.) [30] 84.79 89.91 96.85 83.73 MultiTalent (~3600 annot.) 84.92 89.81
96.89 84.01

###  2.3 Baselines

As a baseline for the MultiTalent, we applied the 3D U-Net generated by the
nnU-Net without manual intervention to each dataset individually. Furthermore,
we trained a 3D U-Net, a Resenc U-Net and a SwinUNETR with the same network
topology, patch and batch size as our MultiTalent for each dataset. All
baseline networks were also implemented within the nnU-Net framework and
follow the default training procedure. Additionally, we compare MultiTalent
with related work on the public BTCV leaderboard in Table 1.
Furthermore, the utility of features generated by MultiTalent is compared to
supervised and unsupervised pre-training baselines. As supervised baseline, we
used the weights resulting from training the three model architectures on the
TotalSegmentator dataset, which consists of 1204 images and 104 classes [30],
resulting in more than $10^{5}$ annotated target structures. In contrast,
MultiTalent is only trained with about 3600 annotations. We used the same
patch size, image spacing, batch size and number of epochs as for the
MultiTalent training. As unsupervised baseline for the CNNs, we pre-trained
the networks on the Multi-dataset collection based on the work of Zhou at al.
(Model Genesis [33]). Finally, for the SwinUNETR architecture, we compared the
utility of the weights from our MultiTalent with the ones provided by Tan et
al. who performed self-supervised pre-training on 5050 CT images. This
necessitated the use of the original (org.) implementation of SwinUNETR
because the recommended settings for fine tuning were used. This should serve
as additional external validation of our model. To ensure fair comparability,
we did not scale up any models. Despite using gradient checkpointing, the
SwinUNETR models requires roughly 30 GB of GPU memory, compared to less than
17 GB for the CNNs.

##  3 Results

Multi-dataset training results are presented in Figure 2. In general, the
convolutional architectures clearly outperform the transformer-inspired
SwinUNETR. MultiTalent improves the performance of the purely convolutional
architectures (U-Net and Resenc U-Net) and outperforms the corresponding
baseline models that were trained on each dataset individually. Since a simple
average over all classes would introduce a biased perception due to the highly
varying numbers of images and classes, we additionally report an average over
all datasets. For example, dataset 7 consists of only 30 training images but
has 13 classes, whereas dataset 6 has 126 training images but only 1 class.
Table 4 in the appendix provides all results for all classes. Averaged over
all datasets, the MultiTalent gains 1.26 Dice points for the Resenc U-Net
architecture and 1.05 Dice points for the U-Net architecture. Compared to the
default nnU-Net, configured without manual intervention for each dataset, the
improvements are 1.56 and 0.84 Dice points. Additionally, in Figure 2 we
analyzed two subgroups of classes. The first group includes all "difficult"
classes for which the default nnU-Net has a Dice smaller than 75 (labeled by a
"d" in Table 4 in the appendix). The second group includes all cancer classes
because of their clinical relevance. Both class groups, but especially the
cancer classes, experience notable performance improvements from MultiTalent.
For the official BTCV test set in Table 1, MultiTalent outperforms all related
work that have also incorporated multiple datasets during training, proving
that MultiTalent is substantially superior to related approaches. The
advantages of MultiTalent include not only better segmentation results, but
also considerable time savings for training and inference due to the
simultaneous prediction of all classes. The training is 6.5 times faster and
the inference is around 13 times faster than an ensemble of models trained on
13 datasets.

Transfer learning results are found in Table 2, which compares the fine-tuned
5-fold cross-validation results of different pre-training strategies for three
different models on three datasets. The MultiTalent pre-training is highly
beneficial for the convolutional models and outperforms all unsupervised
baselines. Although MultiTalent was trained with a substantially lower amount
of manually annotated structures (~3600 vs. ~ $10^{5}$ annotations), it also
exceeds the supervised pre-training baseline. Especially for the small multi-
organ dataset, which only has 30 training images (BTCV), and for the kidney
tumor (KiTs19), the MultiTalent pre-training boosts the segmentation results.
In general, the results show that supervised pre-training can be beneficial
for the SwinUNETR as well, but pre-training on the large TotalSegmentator
dataset works better than the MD pre-training. For the AMOS dataset, no pre-
training scheme has a substantial impact on the performance. We suspect that
it is a result of the dataset being saturated due to its large number of
training cases. The Resenc U-Net pre-trained with MultiTalent, sets a new
state-of-the-art on the BTCV leaderboard111Assuming that no additional private
data from the same data domain has been used.(Table 1).

##  4 Discussion

MultiTalent demonstrates the remarkable potential of utilizing multiple
publicly available partially labeled datasets to train a foundation medical
segmentation network, that is highly beneficial for pre-training and
finetuning various segmentation tasks. MultiTalent surpasses state-of-the-art
single-dataset models and outperforms related work for multi dataset training,
while retaining conflicting annotation protocol properties from each dataset
and allowing overlapping classes. Furthermore, MultiTalent takes less time for
training and inference, saving resources compared to training many single
dataset models. In the transfer learning setting, the feature representations
learned by MultiTalent boost segmentation performance and set a new state-of-
the-art on the BTCV leaderboard. The nature of MultiTalent imposes no
restrictions on additional datasets, which allows including any publicly
available datasets (e.g. AMOS and TotalSegmentator). This paves the way
towards holistic whole body segmentation model that is even capable of
handling pathologies.

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## Appendix

Table 3: An overview of the 13 datasets that were used for the multi-class
training. The last two datasets were only used for transfer learning
experiments.

Train Dataset Labels images Median shape Spacing [mm] 01 Decathlon Task 03 [1]
Liver, Liver tumor 131 432x512x512 (1, 0.77, 0.77) 02 Decathlon Task 06 [1]
Lung nodules 63 252x512x512 (1.24, 0.79, 0.79) 03 Decathlon Task 07 [1]
Pancreas, Pancreas tumor 281 93x512x512 (2.5, 0.80, 0.80) 04 Decathlon Task 08
[1] Hepatic vessels, Hepatic tumor 303 49x512x512 (5, 0.80, 0.80) 05 Decathlon
Task 09 [1] Spleen 41 90x512x512 (5, 0.79, 0.79) 06 Decathlon Task 10 [1]
Colon cancer 126 95x512x512 (5, 0.78, 0.78) 07 BTCV [19] 13 Abdominal Organs
30 128x512x512 (3, 0.76, 0.76) 08 Pelvis [19] Uterus, Bladder, Rectum, Small
Bowel 30 180x512x512 (2.5, 0.98, 0.98) 09 BTCV2 [9] 8 Abdominal Organs 73
111Originally 90 images, but we removed duplicated images of the BTCV test set
185x512x512 (3, 0.79, 0.79) 10 StructSeg Task 3 [20] 5 thoracic organs 50
95x512x512 (5, 1.17, 1.17) 11 SegTHOR [18] Heart, Aorta, Esophagus, Trachea 40
178x512x512 (2.5, 0.98, 0.98) 12 NIH-Pan [25, 3, 26] Pancreas 82 217x512x512
(1, 0.86, 0.86) 13 KiTS19 [11] Kidney, Kidney Tumor 210 107x512x512 (3, 0.78,
0.78) Amos [16] 15 abdominal organs 300 104x512x512 (5, 0.68, 0.68)
TotalSegmentator [30] 104 classes 1204 231x231x240 (1.5, 1.5, 1.5)

Figure 3: The circle area corresponds to each dataset size and the color
indicates the number of annotated classes. MultiTalent was trained on 1460
images with about 3600 annotations. Whereas, the TotalSegmentator dataset
consists of 1204 images with about $10^{5}$ annotations [30].

Table 4: Dice (5 fold) for all classes within all datasets. All networks but
the default nnUNet are trained with batch size 4 and patch size [96,192,192].
MD indicates the MultiTalent networks that were trained with multiple datasets
together. All cancer classes are labeled with a c and all classes, for which
the default nnUNet Dice is smaller than 75 are labeled with a d.

Classes Class groups nnU-Net SwinU U-Net Resenc MD SwinU MD U-Net MD Resenc 1
liver w/o cancer - 95.71 96.14 96.34 96.48 95.94 96.10 96.27 1 liver tumor d,
c 63.72 61.78 65.89 67.53 58.09 65.83 66.50 2 lung_nodule d, c 72.11 65.31
67.94 68.22 62.40 68.43 70.49 2 pancreas w/o cancer - 82.17 80.19 81.52 81.91
79.04 81.35 81.85 3 pancreas cancer d, c 52.74 45.53 49.73 54.18 49.12 56.67
57.71 4 hepatic vessel d 64.56 63.77 63.78 63.60 64.14 64.15 64.33 4 liver
cancer d, c 72.17 66.18 71.04 71.62 67.18 72.78 73.88 5 spleen - 96.38 97.02
97.08 97.05 96.20 96.61 96.67 6 colon cancer d, c 45.53 41.75 47.23 47.40
47.56 56.73 58.69 7 spleen - 90.83 91.14 92.72 92.34 91.04 93.75 93.61 7 right
kidney - 89.39 88.59 90.93 90.33 86.93 90.90 90.89 7 left kidney - 86.75 86.80
90.18 90.65 86.83 90.75 90.47 7 gallbladder d 66.32 66.21 69.24 69.78 67.73
69.82 72.12 7 esophagus - 78.40 77.87 78.56 78.96 77.87 79.73 79.28 7 liver -
95.57 95.36 95.74 95.95 95.14 96.12 96.23 7 stomach - 88.16 86.71 90.87 92.76
86.77 91.11 92.83 7 aorta - 92.29 92.15 92.75 92.75 89.92 91.32 91.60 7 vena
cava inferior - 86.38 85.35 86.16 86.73 84.34 86.26 87.31 7 portal and splenic
vein - 76.59 73.84 77.29 77.74 73.88 77.76 77.58 7 pancreas - 81.76 79.26
83.10 83.87 82.27 84.62 84.92 7 right adrenal gland d 71.48 67.69 70.90 72.37
66.02 72.70 72.70 7 left adrenal gland d 72.38 67.78 72.03 72.77 64.65 72.17
71.62 8 bladder - 88.93 88.26 89.26 89.41 84.02 90.73 91.59 8 uterus - 80.72
78.94 80.34 80.57 76.06 79.64 78.89 8 rectum d 73.77 71.07 73.62 75.24 67.06
73.25 74.17 8 small bowel d 56.52 54.16 56.12 57.27 52.77 57.68 56.67 9 spleen
- 94.59 93.44 95.47 95.86 93.82 95.53 95.49 9 left kidney - 93.29 92.60 93.81
94.38 91.69 93.71 93.72 9 gallbladder - 78.44 78.17 81.30 81.07 77.21 80.63
81.76 9 esophagus d 74.46 74.10 74.92 75.79 72.71 75.03 75.01 9 liver - 96.03
95.90 96.32 96.50 95.41 96.09 96.19 9 stomach - 91.77 90.90 93.32 93.09 88.75
92.17 92.84 9 pancreas - 84.04 83.20 84.95 85.48 84.05 85.74 86.16 9 duodenum
- 75.62 72.47 77.35 78.70 68.96 75.92 77.60 10 left lung - 95.99 95.89 95.99
95.97 95.48 95.95 96.03 10 right lung - 96.74 96.65 96.77 96.74 96.25 96.73
96.78 10 heart - 94.17 93.87 93.91 94.05 93.02 93.66 94.01 10 esophagus -
80.19 79.97 81.30 81.39 78.91 80.87 81.06 10 bronchies - 84.16 83.44 83.88
83.51 83.13 83.23 83.64 10 spina cord nerve - 90.14 90.17 90.08 90.14 89.51
89.69 89.94 11 esophagus - 84.95 83.18 84.83 85.56 80.53 83.21 84.61 11 heart
- 95.27 94.67 95.11 95.29 93.36 94.40 94.83 11 trachea - 90.55 90.61 90.61
90.59 89.50 90.50 90.96 11 aorta - 94.26 93.70 94.11 94.29 92.07 93.00 93.82
12 pancreas - 86.59 85.47 86.10 86.27 85.74 86.80 87.21 13 both kidneys w/o
tumor - 97.02 96.19 96.72 96.92 95.37 96.11 96.34 13 kidney tumor c 83.67
78.90 81.70 82.64 74.54 81.68 83.79 Class average - 82.62 81.11 82.96 83.44
80.28 83.35 83.84


# Space-charge effect of the time-varying electron injection in a diode:
classical and relativistic regimes111Drafted November 2016, revised 18 April
2017, POP51880 submitted 20 April, review received 1 May, revised 28 June and
16 July, POP51880A resubmitted 16 July, minor rev received 26 July, POP51880B
re-resubmitted 29 July, accepted 3 Aug 2017.

Yangjié Liu (刘泱杰<EMAIL_ADDRESS>Faculty of Physics and Electronic
Science, Hubei University, and Hubei Collaboration Innovation Centre for
Advanced Organic Chemical Materials – Hubei Key Laboratory of Ferro &
Piezoelectric Materials and Devices, 430062 Wuhan, Hubei Province, P. R. China
Antennas Group, School of Electrical Engineering and Computer Science, Queen
Mary University of London, E1 4NS Mile End, East London, United Kingdom  Qi
Tang  Department of Mathematical Sciences, Rensselaer Polytechnic Institute,
United States  Biyi Wu (吴比翼)  Antennas Group, School of Electrical Engineering
and Computer Science, Queen Mary University of London, E1 4NS Mile End, East
London, United Kingdom  Center for Electromagnetic Simulation, Beijing
Institute of Technology, Haidian District, Beijing, P. R. China

(compiled , submitted 29 July, accepted 3 Aug 2017 to _AIP Phys. Plasmas_)

###### Abstract

Conventionally, space-charge (SC) limited current density is defined as the
maximal current density allowed to traverse a diode under a DC voltage when a
time-invariant flow is injected from the cathode. In this work, we study the
SC limited current density under a time-varying injection for both classical
and relativistic regimes and determine the maximal amount of limited current
density under certain conditions. Our simulations show that it is unlikely
that a time-varying injection emitted from cathode exceeds the known SC
limits, in either classical or relativistic regime.

###### pacs:

52.59.Sa, 71.15.-m, 52.65.Rr, 68.43.Tj

##  I Introduction

It is of practical interest for plasma devices to seek enhanced density of
plasma sources, which typically requires consideration of space-charge (SC)
limit for a diode structure. SC limited current density describes the maximal
current density allowed to transport through a diode, for a time-invariant
injection from the cathode. For a diode with a gap spacing of $d$ and a diode
voltage of $\phi_{g}$, this is known as the well-known Child-Langmuir law
Child (1911); Langmuir (1913); Zhang  _et al._ (2017). This law extends to the
more general case of nonzero initial velocity $v_{0}$ Jaffé (1944); Liu and
Dougal (1995) and the relativistic regime Jory (1969); Howes (1966); Feng and
Verboncoeur (2008); Lindsay, Li, and Chen (2009); Liu  _et al._ (2014) when
the voltage potential becomes relativistic. In recent years the efforts to
achieve a higher electron flow have been very active, in both classical and
relativistic regimes Luginsland  _et al._ (2002); Kumar and Biswas (2008).
Note that self-magnetic effect and inductive effect can both be additional
factors to suppress SC limit Kumar and Biswas (2008).

There has been recent interest in investigating whether an injection of time-
varying current density can contribute to a larger current density that is
transmittable across the diode space Caflisch and Rosin (2012); Griswold,
Fisch, and Wurtele (2010); Zhu and Ang (2011); Griswold, Fisch, and Wurtele
(2012); Liu and Ang (2012); Pant and Ang (2013); Liu and Ang (2014); Liu
(2014); Griswold and Fisch (2016). On the other hand, time-varying current
emission through the diode anode arises naturally even when the injection
current density is _time-invariant_ , with the diode voltage being fixed. For
example, a recent work based on 1D charge sheet model investigated how the
temporal interval becomes distorted due to space-charge effect Liu  _et al._
(2015). Another recent work Caflisch and Rosin (2012) used a characteristic
method to solve the 1D pressureless Euler-Poisson equations and achieved an
over-classical limit by varying the anode boundary voltage with time, although
the relevant characteristics are not solvable in a closed form for arbitrary
temporal profiles of injection. Therefore, a general treatment for time-
varying flow of electrons is demanded. In this work, we address the practical
issue of time-varying injection current by solving the same 1D pressureless
Euler-Poisson equations, in order to unfold the physics of time-varying
injection flow of electrons. According to our simulation, it seems _unlikely_
that one can transmit a time-varying injection $j(t)$ across the diode that is
time-averagely higher than the conventional SC limited current density, in
both classical Jaffé (1944) and relativistic regimes Feng and Verboncoeur
(2008); Zhang  _et al._ (2009). In our paper, the electrons are injected into
the diode gap following the prescribed time profiles. It is assumed that SC
limit is reached when reflection occurs Chen  _et al._ (2011). Then the
average current densities flowing through the anode are computed and compared
to the previous known SC limits. This approach is also extended to the
relativistic regime Feng and Verboncoeur (2008) when the diode voltage or the
initial velocity become relativistic, and the same negative result prevails.

##  II Problem and method

In this section, we describe the problem of electron flow in a 1D diode
structure depicted in Fig. 1 in the classical regime, and the numerical
approach to determine the SC limit. They are also extended to the relativistic
regime when the initial velocity becomes comparable to the speed of light, or
when the relativistic diode voltage becomes comparable with the rest energy of
electrons.

###  II.1 Classical regime

Consider a diode with a gap of spacing $d$ and a fixed boundary potential
difference $\phi_{g}$ between the cathode and anode. The electrons are
injected from the cathode ($x=0$) with a forced time-varying current density
of $j_{0}(t)$222In 1D case, the current density $j$ are simply related to the
current $J$: $j=J/A$ where $A$ is the area.  over a pulse length $\tau_{p}$.
We take the pulse length $\tau_{p}$ to be longer than the transmit time. The
1D pressureless Euler-Poisson equations for the electron flow in a diode can
be written as

$\displaystyle\partial_{t}\rho+\partial_{x}(\rho v)$ $\displaystyle=0,$ (1)
$\displaystyle\partial_{t}v+v\partial_{x}v$ $\displaystyle=\partial_{x}\phi,$
(2) $\displaystyle\partial_{x}^{2}\phi$ $\displaystyle=\rho,$ (3)

in which scaled quantities are used for position $x$, time $t$, velocity $v$,
potential $\phi$ and charge density $\rho$ similar to the previous work
Caflisch and Rosin (2012). Note that all primed ones variables in SI units are
scaled into unprimed ones to simplify the equations:

$\displaystyle(x,t,v,\phi,\rho)=\Big{(}\frac{x^{\prime}}{L},\frac{t^{\prime}}{L},\frac{v^{\prime}}{L/T},\frac{\phi^{\prime}}{\Phi},\frac{\rho^{\prime}}{R}\Big{)},$
(4)

where $L,T$ are the characteristic length and time, and the characteristic
potential and density are $\Phi=mL^{2}/(eT^{2}),R=\epsilon_{0}\Phi/eL^{2}$ for
simplicity Caflisch and Rosin (2012). An inflow boundary condition is used at
$x=0$,

$\displaystyle\phi=0,\quad v=v_{0},\quad\rho(t)=\frac{J_{0}(t)}{v_{0}},\qquad
0\leq t\leq\tau_{p},$ (5)

to take into account the injection current density $J_{0}(t)$ at the cathode
$x=0$ (see Fig. 1). Note that for a special case when $v_{0}=0$, our algorithm
can deal with it with a different inflow boundary condition. More details will
be elaborated in Subsec. III.4. The outflow boundary condition used at $x=d$
is a fixed potential $\phi=\phi_{g}$.

$x=0$$x=d$$\phi(t,x=d)=\phi_{g}$$v(t\leq\tau_{p},x=0)=v_{0}$$j(t\leq\tau_{p},x=0)=J_{0}(t)$$\phi(t,x=0)=0$Electron
beamsCathodeAnode

Figure 1:  Schematic: 1D diode model of voltage difference $\phi_{\rm g}$ and
spacing $d$. Time-dependent injection current density $J_{0}(t)$ up to finite
time $0\leq t\leq\tau_{\rm p}$ emits from the cathode, and is dragged towards
the anode with initial velocity $v_{0}$. The blue color indicates the electron
beams in the diode spacing.

The space-charge limited current density for time-invariant injection in
scaled quantity can be written as Jaffé (1944); Caflisch and Rosin (2012)

$J_{\rm
m}=\frac{2}{9d^{2}}\Big{(}v_{0}+\sqrt{v_{0}^{2}+2\phi_{g}}\Big{)}^{3}.$ (6)

For arbitrary $J_{0}(t)$ at cathode, a Lax-Friedrichs scheme coupled with a
second-order Poisson solver provides insight into solving the system
numerically in MATLAB. In Sec. III the numerical solutions are used to observe
how the charge density and diode potential along diode space $d$ vary with
time. The magnitude of the injection density is increased from small to large,
and we determine that SC limit is reached when negative velocity _just_
appears near cathode under injection of a certain emission profile. Here we
assume that emission profile $J_{0}(t)$ can be expressed as

$J_{0}(t)=\beta_{m}J_{m}\cdot j_{m}(t),$ (7)

where $\beta_{m}$ is the dimensionless magnitude and $j_{m}$ is a particular
time-dependent profile assumed as follows,

$\displaystyle j_{0}$ $\displaystyle=1,$ (8) $\displaystyle j_{1}$
$\displaystyle=2\bar{t},$ (9) $\displaystyle j_{2}$
$\displaystyle=3\bar{t}^{2},$ (10) $\displaystyle j_{3}$
$\displaystyle=2\big{(}1-\bar{t}\big{)},$ (11) $\displaystyle j_{4}$
$\displaystyle=3-3\bar{t}^{2},$ (12) $\displaystyle j_{5}$
$\displaystyle=-6\bar{t}^{2}+6\bar{t},$ (13) $\displaystyle j_{6}$
$\displaystyle=12\left(\bar{t}-\frac{1}{2}\right)^{2},$ (14)

where the time is normalised by $\bar{t}=t/\tau_{\rm p}$. Compared with other
temporal profiles, these polynomial emission profiles are chosen based on the
understanding that a smooth temporal function should be expanded as a series
of polynomials according to Taylor series. We first choose a steady function
(8), then monotonic increasing functions (9, 10), monotonic decreasing
functions (11, 12), and lastly a valley function (13) and a mountain function
(14) 333The increasing profiles (8-10) are normalised so that within the
emission pulses they give the same account of charge $\int_{0}^{\tau_{\rm
p}}j_{m}(t){\rm d}t=\int_{0}^{\tau_{\rm p}}1{\rm d}t=\tau_{\rm p}$ and the
other profiles are chosen to be as smoothly-varying as possible. . We sweep
magnitudes $\beta_{m}$ for each $m$ in profile $j_{m}(t)$ and decide the SC
limit is reached when reflection _just_ occurs, i.e. the velocity becomes
negative, within the pulse length $0\leq t\leq\tau_{\rm p}$. As a reference
when the injection flow is time-invariant $m=0$, it is understood that
$\beta_{0}$ should be one which corresponds to the convectional SC limit
$J_{m}$ of Eq. (6), while other $\beta_{m}$’s generally differ from one.

In our simulations, we choose the electron beam length to be longer than the
typical transmit time of an electron across the spacing (cf. Figs. 2-3 below).
This is for the following reason. In empirical situations, one can only inject
an electron beam of finite length. We thus do so in order to understand the
sole role of time-dependent injection. This long pulse condition is important
as when the temporal length of the electron beam becomes shorter than Child-
Langmuir transmit time $3d/\sqrt{2\phi_{g}}$, the short-pulse limited current
density can transcend Child-Langmuir limit already Valfells  _et al._ (2002).
Therefore we intensionally implement this long pulse condition in order to
remove complication resultant from the short-pulse effect Valfells  _et al._
(2002).

###  II.2 Relativistic regime

In the relativistic regime, when $v_{0}$ becomes comparable to light velocity
$c$ or the relativistic diode voltage $U:=q\phi_{g}/{mc^{2}}\succsim 0.01$,
the charge density $\rho$ become

$\rho=\gamma\rho_{0},\quad\gamma[v]:=\left(1-\frac{v^{2}}{\hat{c}^{2}}\right)^{-1/2}.$
(15)

The 1D pressureless Euler-Poisson equations for relativistic electron flow are
changed into

$\displaystyle\partial_{t}\rho+\partial_{x}(\rho v)$ $\displaystyle=0,$ (16)
$\displaystyle\gamma^{3}(\partial_{t}v+v\partial_{x}v)$
$\displaystyle=\partial_{x}\phi,$ (17) $\displaystyle\partial_{x}^{2}\phi$
$\displaystyle=\rho.$ (18)

Note that the relativistic equations are under the same scaling of Eq. (4).
Therefore the light speed $\hat{c}={cT}/{L}$. For computational purpose, let
$L/T=10^{7}{\rm m/s}$, so the light speed $\hat{c}=30$ in this scale. The
details of the derivation are given in Appendix A. In the relativistic regime,
the space-charge limit for time-invariant injection is Feng and Verboncoeur
(2008)

$J_{\rm rm}=\frac{c^{3}}{2d^{2}}\,\Big{[}G(U,\gamma_{0})\Big{]}^{2},$ (19)

where
$G(U,\gamma_{0}):=\int_{0}^{U}\left(\sqrt{(u+\gamma_{0})^{2}-1}-\sqrt{\gamma_{0}^{2}-1}\right)^{-1/2}{\rm
d}u$, $U:=\phi_{g}/c^{2}$ and $\gamma_{0}=\gamma[v_{0}]$. The same approach
will be used to find the SC limited current density for various time profiles
as in the classical case II.1.

##  III Results

In this section, the numerical results are used to pinpoint the space-charge
limit when the injection is time-varying. To assess the space-charge effect
with respect to the conventional Child-Langmuir limit (6), we define two
characteristic ratios: the scaled charge

$\displaystyle\bar{q}:=\frac{\int_{0}^{\tau_{\rm p}}j_{A}(t)\,{\rm
d}t}{\int_{0}^{\tau_{\rm p}}J_{0}(t)\,{\rm d}t},$ (20)

and the scaled current density

$\displaystyle\bar{j}:=\frac{\int_{\tau_{1}}^{\tau_{2}}j_{A}(t)\,{\rm
d}t}{J_{m}(\tau_{2}-\tau_{1})},$ (21)

in which $j_{\rm A}$ represents the current density at the anode, and
$\tau_{1}$ and $\tau_{2}$ are the starting and finishing time for $j_{\rm A}$
being positive. In the classical case $\bar{q}=1$ indicates the critical
situation that all injected charge has transmitted through the anode at the
finishing time $\tau_{\rm p}$. Due to the leftover charge inside the diode,
this should generally be unreachable as confirmed in Fig. 3. In the
relativistic case (see (a, c, e) of Fig. 4), $\bar{q}$ becomes larger than
unity due to Lorentz length contraction Schutz (2011). Note $\bar{j}\geq 1$
means that SC limited current density with time-varying pulse for each
$j_{m}(t)$ profile is reached or even transcended.

###  III.1 Classical case

We first demonstrate that the output current density $j_{\rm A}(t)$ at the
anode is typically time-dependent, even when the injection current density is
constant over the pulse. The numerical results for the case $m=0$ are
presented in Fig. 2. In Fig. 2 (a), when $\beta_{0}$ varies from 0.90 to 1.37,
the transmitted charge at the anode (dashed line) is almost identical to the
emitted charge at cathode. A careful check indicates that the smallest amount
is 0.98. On the other hand in Fig. 2 (b), as $\beta_{m}$ increases, the
transmitted current density (red dashed) increases to the optimal value 89% of
the SC limited one (red solid line) Eq. (6) and then goes down until 86% (blue
dashed line). These observations are based on the ratios $\bar{q}$ and
$\bar{j}$ defined in Eqs. (20) and (21). When magnitude $\beta_{0}$ increases
more than 1.37, the whole beam starts to reflect and the space-charge limit is
reached. We further study three typical cases of $\beta_{0}$ found from Fig. 2
(a), which are denoted in colored circles. In Fig. 2 (b) the emitted currents
at the cathode (dash lines) and the transmitted currents at the anode (solid
lines) of those three cases are presented in matching colors. The pulse length
is chosen to be $\tau_{\rm p}=4$ in order to be longer than the time
$\tau_{1}=1.15$ when the anode current becomes positive. It is concluded that
even when the injection flow is time-invariant, the transmitted flow is time-
dependent at the anode. Moreover, when the injection is smaller than space-
charge limit (denoted as the green asterisk in Fig. 2), the transmitted
current is relatively uniform in time (the black curves). When the injection
is larger, the transmittance becomes damped around the central region and
electrons are shaped to high values at the pulse end (see red and blue curves
therein).

When the pulse length is set longer as $\tau_{\rm p}=6$, a larger current
density can transmit across the anode, seen in Fig. 2 (c). Similar saturation
effect due to space-charge limit is observed for the longer pulse length. The
transmitted current density is extended in the central part of the pulse, as
the anode current density $j_{\rm A}$ in Fig. 2 (b, d) show. To make a fair
comparison, we choose $\tau_{\rm p}=4$ in the rest classical simulations.

Figure 2:  (a, c) Scaled transmitted charge [cf. Eq. (20)] and scaled current
density [cf. Eq. (21)]. (b, d) Transmittance profiles [solid curves, color
marked according to circle marks in (a, c)] at the anode for time-_invariant_
injection ones in classical regime. The injection profiles at the cathode are
plotted in dashed curves and the SC limit in green asterisks. Parameters:
$m=0,v_{0}=\hat{c}/60$; (a-b) $\tau_{\rm p}=4$, (c-d) $\tau_{\rm p}=6$.

Next, we study the time-varying injection of the cases $m=1\sim 6$. All
results for six time-varying injection profiles are plotted in Fig. 3(a-l).
The output density for time-varying injection is again time-varying, similar
to the constant injection case before. For each $m$ there exists an optimal
magnitude $\beta_{m}$ that the transmittance researches the maximum, closest
to SC limit in Eq. 6. For example when $m=4$ in Fig. 3 (g-h), the maximal
transmittance reached at $\beta_{4}=0.54$ (in red circle) is about $90\%$ of
the space charge limit. If $\beta_{4}$ is larger than $0.54$, the
transmittance decreases. When $\beta_{4}>0.62$ (see the blue circle in Fig. 3
(h)), the electron beam starts to reflect towards the cathode and it is
therefore determined that space charge limit has reached. The three cases in
Fig. 3 (h) indicates that the transmittance starts to damp in the central
region due to the clamping SC effect (the most pronounced for the blue curve).
It shows that the SC effect dominates the time-dependent electron flow before
there would be a larger current density across the diode. For all other
profiles, similar clamping effects due to space charge repulsion are observed
from the results. Based on these simulations, we conclude that _time-varying
injection seem unlikely to contribute more than the space-charge limit ( 6)_.
This is the main result of this paper.

Figure 3:  (a, c, e, g, i, k) Scaled transmitted charge [cf. Eq. (20)], scaled
current density [cf. Eq. (21)]. (b, d, f, h, j, l) Transmittance profiles at
the anode for time-varying injection ones. Other marks are the same as Fig. 2.
Parameters: $v_{0}=\hat{c}/60,\tau_{\rm p}=4$, (a-b) $m=1$; (c-d) $m=2$; (e-f)
$m=3$; (g-h) $m=4$; (i-j) $m=5$; (k-l) $m=6$.

###  III.2 Relativistic case

In the relativistic regime, we demonstrate that time-varying injection cannot
result in a transmittance higher than its SC limit Feng and Verboncoeur
(2008). Let the anode voltage $\phi_{g}=900$, and $U=\phi_{g}/\hat{c}^{2}=1$
so that the relativistic effect becomes noticeable.  When the injection is
time-invariant with $v_{0}=\hat{c}/60$, the transmittance goes below the known
SC limit (see Fig. 4 (a-b)). Note again that they are scaled quantities
following Ref. Caflisch and Rosin (2012), and all velocities are scaled with
respect to $L/T$ so light speed $\hat{c}=30$. When the injection becomes time-
varying ($m=4$ and $v_{0}=\hat{c}/60$ for example), the SC limit similarly
clamps the transmittance profiles as Fig. 4 (c-d) shows. The SC limit also
holds when initial velocity is changed to $v_{0}=\hat{c}/10$ (see Fig. 4
(e-f)). Fig. 4 (b, d, f) shows the transmittance profiles become damped at the
pulse trail when $\beta_{m}$ increases.

Figure 4:  (a, c, e) Scaled transmitted charge [cf. Eq. (20)] and scaled
current density [cf. Eq. (21)]. (b, d, f) Transmittance profiles at anode for
injection ones in relativistic regime ($m=0,4$). Other marks are the same as
Fig. 2. Parameters: $\tau_{\rm p}=0.1$, $\phi_{g}=900$. (a-b)
$m=0,v_{0}=\hat{c}/60$; (c-d) $m=4,v_{0}=\hat{c}/60$; (e-f)
$m=4,v_{0}=\hat{c}/10$.

Note that $\bar{q}$ becomes larger than unity because the charge density
$\rho$ becomes denser due to the relativistic factor (cf.
$j_{A}=\rho_{A}v_{A}$ in Eq. (20)), and this increasing effect becomes less
significant as the injection magnitude $\beta_{m}$ increases (see Fig. 4 (a,
c, e)).

###  III.3 Time-dependent quantities

More details can be observed from time-dependent plots of voltage potential
$\phi(x,t)$ and charge density $\rho(x,t)$. Fig. 5 presents $\phi(x,t)$ and
$\rho(x,t)$ at three instantaneous time points for a classical case and a
relativistic one in Fig. 5, when SC limit is reached respectively. We choose
$t=2.63$ and $0.053$ to be the time when the injection current density become
positive (cf. Figs. 3(g-h) and 4(c-d)) respectively. Fig. 5 shows that as the
charge traverses the diode with time, the electric field at the cathode can
become positive ($E_{\rm C}=-\partial_{x}\phi$) and space charge accumulates
near cathode to repel more electrons to emit. This is when SC limit dominates
the electron flow process and the virtual diode forms. It is also speculated
that the zero cathode field $E_{\rm C}=0$ is _not_ a critical condition for
space charge limit for time-varying injection cases, as previous work Jory
(1969); Liu  _et al._ (2015) consider so. We speculate that time-dependent
space-charge limit may involve more dynamic process of virtual cathode than a
static zero cathode field. In our simulation, it is observed that virtual
diode oscillation occurs when the scaled $\bar{j}$ in Eq. (21) transcends
unity. However, we do not include them in Sec. III because the SC limit is
always reached when $\bar{j}$ remains under unity.

Figure 5:  Electric potential $\phi(x,t)$ and charge density $\rho(x,t)$ for
classical and relativistic cases. Parameters: $m=4$. Classical: (a-b)
$\beta_{4}=0.62$, $v_{0}=\hat{c}/60$, $\phi_{g}=1$; relativistic: (c-d)
$\beta_{4}=0.97$, $v_{0}=0.1\hat{c}$, $\phi_{g}=900$.

###  III.4 Caveat: null emission velocity $v_{0}=0$

In the last example, we present the result of the special case when the
emission velocity $v_{0}$ is zero. This example is compared with the time
varying cases and serves as a validation case since this case is well
understood theoretically. The inflow boundary condition Eq. (5) at $x=0$
becomes

$\displaystyle\phi=0,\quad v=v_{0},\quad\rho(t)=\beta_{m}\sigma_{m}(t),\qquad
0\leq t\leq\tau_{p},$ (22)

where $\beta_{m}$ is also the dimensionless amplitude and $\sigma_{m}(t)$ the
particular time-dependent profiles assumed for the inflow charge density.
Note that we apply a slightly different inflow boundary condition at $x=0$. It
is difficult to apply a boundary condition of a finite injection current
density with a null emission velocity at $x=0$ in the current model. For such
a case, a kinetic model may be necessary to understand its limited current
density. The same outflow boundary condition $\phi=\phi_{g}$ applies as in the
nonzero injection velocity case. The particular profiles $\sigma_{m}(t)$ are
assumed to be polynomial for the same reason as the injection current
densities $j_{m}(t)$ in Eqs. (8-14). This is the slightly different inflow
boundary condition we apply in this subsection. Similar to Figs. 4, we present
a result in Fig. 6 for $v_{0}=0,\rho_{5}(t)=\beta_{5}\sigma_{5}(t)$ as the
inflow boundary condition, because this represents a reasonable injection
current density in pragmatic situations. First in Fig. 6(a), as $\beta_{5}$
increases, the scaled current density increases until 89% of the SC limit (cf.
blue circle) and most charge remains in the diode spacing at the time of
$\tau_{\rm p}$ (cf. $\bar{q}<<1$ on the left axis). Second at
$\beta_{5}=10.6$, the transmittance profile is distorted to the most extent
(see Fig. 6(b)) due to SC effect. Then we show that our simulation also treats
the null-velocity case for the time-dependent charge in-flow.

Figure 6:  (a) Scaled transmitted charge [cf. Eq. (20)] and scaled current
density [cf. Eq. (21)]. (b) Injection charge density $\rho(t)$ (dashed curves)
and transmittance profiles at anode $j_{A}(t)$ (solid curves). Note the curves
in (a-b) are plotted in different scales labeled on the left and right sides
of the line graph. Other marks are the same as Fig. 2. Parameters:
$m=5,\tau_{\rm p}=4,v_{0}=0,\phi_{g}=1$. [Associated dataset available at
http://dx.doi.org/10.5281/zenodo.836163] Liu (2017)

##  IV Conclusion

In summary, this paper addresses the question whether time-varying injection
from cathode can push a transmitted current density through anode more than
the conventional SC limit. Note that in our paper the injection beam is
assumed to be cold and possesses no thermal spread at any instant in time for
simplicity Lau (2001); Zhang  _et al._ (2017). From our simulations based on
the 1D pressureless Euler-Poisson equations, we speculate that the answer is
_negative_ under the condition of a fixed voltage difference across the diode.
We also extend the classical diode to relativistic regime when initial
velocity of electrons is close to the speed of light or when the diode voltage
contributes to relativistic acceleration.

###### Acknowledgements.

L. Y. was supported by travel grant of Hangzhou Dianzi University
[JGB06205-2017031] and UK EPSRC [grant number EP/I034548/1]. He owes the
starting idea of time-dependent space-charge effect from L. K. Ang and thanks
Lang Jiandong for useful discussion. Q. T. is supported by Eliza Ricketts
Postdoctoral Fellowship.

##  Appendix A Detailed derivation of relativistic extension for 1D electron
flow

In this section, we start from the full relativistic formulation for the same
1D electron flow problem in Sec. II, using covariant four-vector Landau and
Lifshitz (1975) and then simplify it to a new differential equations system,
Eqs. (16), (17) and (18) therein. There can be many relativistic versions of
our problem, for instance see Ref. Chan, Wong, and Yuen (2016). We choose to
formulate in a simple way to be reducible to classical regime when $v\ll c$.

We use four-vector in SI units (instead of natural units as convention) to
represent every vectorial physical quantity in contra-variant components but
the final main equations will be scaled under the same scaling of Eq. (4).
Hence, the classical current density $j$, vector potential $\mathbf{A}$ and
static field $E_{x}$ are replaced by four current $\mathcal{J}^{\mu}$, four
potential $A^{\alpha}$, the electromagnetic tensor $F^{\alpha\beta}$
respectively with the metric tensor $g^{\mu\nu}$ and four velocity $U$ being
used. The detailed definitions are given as

$\displaystyle\mathcal{J}^{\mu}$ $\displaystyle=$
$\displaystyle(c\rho,\mathbf{j}),$ (23) $\displaystyle A^{\alpha}$
$\displaystyle=$ $\displaystyle(\frac{\phi}{c},\mathbf{A}).$ (24)
$\displaystyle\partial^{\nu}:$ $\displaystyle=$
$\displaystyle\big{(}\frac{\partial}{c\partial t},-\nabla\big{)},$ (25)
$\displaystyle\partial_{\nu}:$ $\displaystyle=$
$\displaystyle\big{(}\frac{\partial}{c\partial t},\nabla\big{)},$ (26)
$\displaystyle F^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}=\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\cr
E_{x}/c&0&-B_{z}&B_{y}\cr E_{y}/c&B_{z}&0&-B_{x}\cr
E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix},$ (27) $\displaystyle g^{\mu\nu}$
$\displaystyle=$ $\displaystyle{\rm diag}(1,-1,-1,-1).$ (28) $\displaystyle
U^{\alpha}$ $\displaystyle=$ $\displaystyle\gamma(c,\mathbf{v}),$ (29)
$\displaystyle U_{\alpha}$ $\displaystyle=$
$\displaystyle\gamma(c,-\mathbf{v}).$ (30)

Therefore, the covariant formulation for the electron flow problem in the
three equations 444Note that the proper time $\tau$ is operated in full
differential, instead of partial differential because $\tau$ is frame-
independent for a certain particle.  is written as

$\displaystyle\partial_{\mu}\mathcal{J}^{\mu}$ $\displaystyle=$ $\displaystyle
0,$ (31) $\displaystyle\frac{{\rm d}U^{\alpha}}{{\rm d}\tau}$ $\displaystyle=$
$\displaystyle\frac{q}{m}F^{\alpha\beta}U_{\beta},$ (32)
$\displaystyle\partial_{\alpha}F^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\mu_{0}\mathcal{J}^{\mu}$ (33)

We choose Coulomb gauge so that Poisson equation remains unchanged Jackson
(1999)

$\displaystyle\nabla^{2}\phi$ $\displaystyle=$
$\displaystyle\frac{\rho(x,t)}{\epsilon_{0}},$ (34)
$\displaystyle(\nabla^{2}-\frac{1}{c^{2}}\partial_{t}^{2})\mathbf{A}$
$\displaystyle=$ $\displaystyle-\mu_{0}\mathbf{j}_{t}.$ (35)

Since only 1D situation is considered, $\mathbf{j}=\hat{x}j$ and
$\nabla\times\mathbf{j}=0$ means the transverse current vanishes $j_{t}=0$
Jackson (1999). It is legitimate to choose

$\mathbf{A}=0.$ (36)

Then the magnetic field also vanishes, which is consistent with that 1D time-
dependent electric field induces no magnetic field from Faraday’s law of
induction Jackson (1999). Eq. (32) leads to

$\frac{{\rm d}(\gamma v)}{\gamma{\rm
d}\tau}=\frac{q}{mc}(-\partial_{x}\phi)c=\frac{q}{m}F^{10}c=\frac{q}{m}E_{x}.$
(37)

It is noted that the equation Eq. (37) differs from
$\gamma\partial_{t}v=qE_{x}/m$. From Eq. (37) and $\gamma{\rm d}\tau={\rm d}t$
we have the scaled relativistic dynamical equation [cf. Eq. (4)] as

$\displaystyle\frac{\rm d}{{\rm d}t}\big{[}\gamma(v)\cdot
v(x,t)\big{]}=\partial_{x}\phi,$ (38)

which is expanded as

$\displaystyle\gamma^{3}\frac{\rm d}{{\rm d}t}v(x,t)=\partial_{x}\phi.$ (39)

The sign appears reverse because of the negative sign of $q$. Eqs. (31), (39)
and (34) can be written as Eqs. (16), (17) and (18) $\square$.

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$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u(r_{1},\bar{v})}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}\,|\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}|^{2}\lesssim$
$\displaystyle\,\int_{v_{0}}^{v}d\bar{v}\int_{\mathbb{S}^{2}_{\infty,\bar{v}}}\sin\theta\,d\theta\,d\phi\,|\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}|^{2}$
$\displaystyle+\int_{v_{0}}^{v}d\bar{v}\int_{u(r_{1},\bar{v})}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}\,|\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}})|^{2}\,.$

We now exploit the estimate (164) and Proposition 9.2 to bound the right hand
side and obtain

$\int_{v_{0}}^{v}d\bar{v}\int_{u(r_{1},\bar{v})}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}\,|\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}|^{2}\lesssim\mathbb{E}_{0}\,.$
(165)

To make the estimate (165) global, one exploits the equation

$\operatorname{\not{\nabla}}_{4}(\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}})=\frac{2}{r^{2}}\,\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})$

and Proposition 9.5. ∎

######  Remark 9.23.

The estimate (164) could have alternatively been obtained by exploiting the
equation (45) and Propositions 9.6 and 9.18.

###  9.5 Top-order estimates for the linearised curvature components

In this section, we derive top-order (degenerate) integrated energy decay for
the linearised curvature components
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}$.
These estimates do not lose derivatives. Combined with Propositions 8.3 and
8.6, this section completes the top-order estimates for all linearised
curvature components.

We start with two propositions which state integrated energy decay for four
angular derivatives of the linearised outgoing and ingoing shears.

######  Proposition 9.24 (Refined integrated decay for the linearised outgoing
shear).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[2]}r^{2}\not{\mathcal{D}}_{2}^{\star}\not{\textup{div}}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(166)

###### Proof.

We take the $\mathcal{A}^{[2]}r^{2}\not{\mathcal{D}}_{2}^{\star}$-derivative
of equation (66) and exploit Propositions 9.7, 9.9, 9.10 and 9.14. ∎

######  Remark 9.25.

The integrated energy decay estimate of Proposition 9.24 gains one derivative
and loses one power of $r$ if compared to the estimate of Proposition (9.5)
for three angular derivatives of the linearised outgoing shear. Proposition
9.24 suffices to prove a top-order integrated energy decay estimate for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}$ which does not
lose in $r$-weight compared to the estimate of Proposition 9.7.

######  Proposition 9.26 (Refined integrated decay for the linearised ingoing
shear).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[2]}r^{2}\not{\mathcal{D}}_{2}^{\star}\not{\textup{div}}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(167)

###### Proof.

We take the $\mathcal{A}^{[2]}r^{2}\not{\mathcal{D}}_{2}^{\star}$-derivative
of equation (67) and exploit Propositions 9.7, 9.9, 9.10 and 9.16. ∎

The following is the main proposition of the section.

######  Proposition 9.27 (Top-order integrated decay for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}(|\mathcal{A}^{[4]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})|^{2}+|\mathcal{A}^{[4]}r\not{\mathcal{D}}_{2}^{\star}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}})|^{2})\lesssim\mathbb{E}_{0}\,.$
(168)

###### Proof.

The proof follows the same argument as the proof of Proposition 9.7, but one
now takes the $\mathcal{A}^{[4]}$-derivative of the gauge invariant linearised
quantities. We recall that the initial energy flux
$\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]$
only controls
$\mathcal{A}^{[4]}(r^{4}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})$ with
degeneration at trapping ($r=3M$), the same holding for the flux
$\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]$
relative to
$\mathcal{A}^{[4]}(r^{3}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}})$.
∎

##  10 The proof of the main theorem

The proof of Theorem 7.1 follows from Propositions 8.6, 9.27, 9.12, 9.5, 9.6,
9.14, 9.16, 9.19, 9.17, 9.8, 9.10, 9.9, 9.18, 9.20, 9.21 and 9.22 (with the
propositions ordered according to how the estimates appear in the statement of
the theorem).

##  Appendix A Well-posedness of the system of linearised gravity

We present the proof of Proposition 5.2.

###### Proof.

Given smooth seed initial data, we start by showing that all the linearised
quantities can be determined on the initial hypersurface
$\mathcal{S}_{u_{0},v_{0}}$. On $\underline{C}_{v_{0}}$, one uniquely
determines (in order)

$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})\,,\,(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})$
using the equations (42) and (43),
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}$
using the equation (39),
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}$
using the equation (45),
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$
using the equation (49),
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}$
using the equation (67) on $\mathbb{S}^{2}_{\infty,v_{0}}$ and equation (77),
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}$
using the equation (65) on $\mathbb{S}^{2}_{\infty,v_{0}}$ and equation (75),
$\displaystyle(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})$
using the equation (51),
$\displaystyle(\text{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})$
using the equation (47),
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}$
using the expression (88),
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}$
using the equation (68) on $\mathbb{S}^{2}_{\infty,v_{0}}$ and equation (74),
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}$
using the equations (41) and (64),
$\displaystyle(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})$
using the equation (55).

We note that all the linearised equations intrinsic to the ingoing null cone
$\underline{C}_{v_{0}}$ were used, with the exception of the equations (57)
and (59) on $\underline{C}_{v_{0}}$ and the equations (65), (67) and (68) on
$\underline{C}_{v_{0}}\setminus\mathbb{S}^{2}_{\infty,v_{0}}$. One can check
that the equations (57) and (59) are obtained by taking a
$\operatorname{\not{\nabla}}_{3}$-derivative of the equations (42) and (43),
the ($\not{\text{curl}}$-commuted) equation (64) is obtained by taking a
$\operatorname{\not{\nabla}}_{3}$-derivative of the equation (65) and that the
constraint equations (67) and (68) are propagated from the initial horizon
sphere $\mathbb{S}^{2}_{\infty,v_{0}}$ to the whole initial ingoing null cone
$\underline{C}_{v_{0}}$.

Using, in order, the equations (44), (48), (35), (38) and (56), one uniquely
determines

$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}},\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}\,,\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4}\,,\,(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})$
on $C_{u_{0}}$.

Using the equation (53), one uniquely determines

$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}$
on $\underline{C}_{v_{0}}$.

The equation (66) then gives
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}$ on
$S^{2}_{u_{0},v_{0}}$, which, using (in order) the equations (70), (73) and
(72), yields

$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}$
on $C_{u_{0}}$.

Using the equations (71) and (69), one uniquely determines

$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$
on $\underline{C}_{v_{0}}$.

This completes the computation of all the linearised quantities in the system
on $\underline{C}_{v_{0}}$.

One can check that all the remaining linearised quantities can be uniquely
determined on $C_{u_{0}}$. To determine all the linearised quantities in the
system on $\mathcal{R}_{u_{0},v_{0}}$, one starts from the decoupled
linearised quantities
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$
solving the spin $\pm 2$ Teukolsky equations. One can then compute all the
gauge dependent linearised quantities on $\mathcal{R}_{u_{0},v_{0}}$ and check
that the linearised constraint equations are propagated. ∎

##  Appendix B Early estimates for the linearised induced metric and frame
coefficients

As an alternative to what done in Section 9, one can exploit integrated energy
decay for the gauge invariant linearised quantities (Proposition 8.3) and the
linearised outgoing shear (Proposition 9.5) to obtain integrated energy decay
for the linearised induced metric and frame coefficients (with the exception
of Proposition 9.20) _without relying on any control over the linearised
ingoing shear_.

The following is the alternative route that one would have to take after
proving Proposition 9.5:

  * •

By repeating the proof of integrated energy decay for (angular derivatives of)
the linearised induced metric

$\displaystyle(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}$ (169)

in Propositions 9.21 and 9.22, one realises that the proof only relies on
Propositions 8.3 and 9.5 and, in the case of
$(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})$, on Proposition
9.14. We note that Proposition 9.14 can be proven independently from the rest
of the linearised system of equations, see Remark 9.15. See also the related
Remark 9.23.

######  Remark B.1.

The proof of integrated energy decay for
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}$ in Proposition
9.22 does _not_ use Proposition 9.14 for
$(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\chi})$.

######  Remark B.2.

Integrated energy decay for the linearised quantities (169) can be obtained
_before_ and independently from integrated energy decay for the linearised
frame coefficients.

  * •

By using the identities

$\displaystyle\mathcal{A}^{[i]}(r^{4}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})$
$\displaystyle=\mathcal{A}^{[i]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})-\frac{3M}{r}\,\mathcal{A}^{[i]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})\,,$
$\displaystyle\mathcal{A}^{[i]}\operatorname{\not{\nabla}}_{3}(r^{4}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})$
$\displaystyle=\mathcal{A}^{[i]}r\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}_{3}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})-\frac{3M}{r^{2}}\,\mathcal{A}^{[i]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})-\frac{3M}{r}\,\mathcal{A}^{[i]}\operatorname{\not{\nabla}}_{3}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})$

and exploiting Propositions 8.3 and 9.5, one can prove the integrated energy
decay estimate for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}$ stated in
Proposition 9.7 and the estimates

$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[2]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}))|^{2}\lesssim\mathbb{E}_{0}\,,$
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,\left(1-\frac{3M}{r}\right)^{2}|\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}))|^{2}\lesssim\mathbb{E}_{0}\,,$

where we note the degenerate (at trapping, $r=3M$) factor in the second
estimate.

  * •

By commuting once with $\operatorname{\not{\nabla}}_{3}$ (and angular
derivatives) the equation (60), one obtains the red-shifted equation

$\operatorname{\not{\nabla}}_{4}(\operatorname{\not{\nabla}}_{3}\mathcal{A}^{[i]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}))+\operatorname{\hat{\omega}}\,(\operatorname{\not{\nabla}}_{3}\mathcal{A}^{[i]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}))=-\frac{1}{r^{2}}\operatorname{\not{\nabla}}_{3}\mathcal{A}^{[i]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})-\frac{2}{r^{3}}\mathcal{A}^{[i]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})\,.$
(170)

Equation (170) can be used to prove the estimates

$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\operatorname{\not{\nabla}}_{3}\mathcal{A}^{[2]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,\left(1-\frac{3M}{r}\right)^{2}|\operatorname{\not{\nabla}}_{3}\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,.$ (171)

We note that

$\lVert\operatorname{\not{\nabla}}_{3}\mathcal{A}^{[i]}\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}\rVert_{L^{2}(\underline{C}_{v_{0}})}\lesssim\mathbb{E}_{0}$

for $i\leq 3$.

Let $r_{1}=5M/2$. As noted in the proof of Proposition 9.22, one can use an
estimate like (171) to control the following integral close to the event
horizon

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}(r_{1},\bar{v})}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(172)

The constant $r_{1}$ is chosen so that one can remove the degenerate (at
trapping, $r=3M$) factor from (171). We also note that to derive the estimate
(172), one relies on control over the horizon energy flux

$\int_{\mathcal{H}^{+}(v_{0},v)}d\bar{v}\,\sin\theta\,d\theta\,d\phi\,\Omega^{2}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})|^{2}\lesssim\mathbb{E}_{0}\,,$
(173)

which one can deduce from Proposition 9.1 (see Remark 9.3). The estimate (172)
can be made global by exploiting the equation (60), thus achieving Proposition
9.8.

We emphasise that, in contrast with what done in our scheme of Section 9 (i.e.
in the proof of Proposition 9.8), we achieve Proposition 9.8 avoiding the use
of equation (53), which involves (and requires control over) the linearised
ingoing shear.

  * •

One can now repeat the proof of Propositions 9.9 and 9.18 for the linearised
frame coefficients

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}$

respectively.

This alternative route may find interesting applications to the analysis of
the system of linearised gravity of [2] in the general $|a|<M$ case, in which
the outgoing transport equation for the linearised ingoing shear couples with
the linearised induced metric and frame coefficients on the right hand side.

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