# A Variational Approach to Weakly Supervised Document-Level Multi-Aspect
Sentiment Classification

Ziqian Zeng1, Wenxuan Zhou2, Xin Liu1, and Yangqiu Song1
1Department of CSE, Hong Kong University of Science and Technology, HK
2Department of CS, University of Southern California, CA, USA
1{zzengae, xliucr<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

In this paper, we propose a variational approach to weakly supervised
document-level multi-aspect sentiment classification. Instead of using user-
generated ratings or annotations provided by domain experts, we use target-
opinion word pairs as “supervision.” These word pairs can be extracted by
using dependency parsers and simple rules. Our objective is to predict an
opinion word given a target word while our ultimate goal is to learn a
sentiment polarity classifier to predict the sentiment polarity of each aspect
given a document. By introducing a latent variable, i.e., the sentiment
polarity, to the objective function, we can inject the sentiment polarity
classifier to the objective via the variational lower bound. We can learn a
sentiment polarity classifier by optimizing the lower bound. We show that our
method can outperform weakly supervised baselines on TripAdvisor and
BeerAdvocate datasets and can be comparable to the state-of-the-art supervised
method with hundreds of labels per aspect.

##  1 Introduction

Document-level multi-aspect sentiment classification (DMSC) aims to predict
the sentiment polarity of each aspect given a document which consists of
several sentences describing one or more aspects Wang et al. (2010, 2011); Yin
et al. (2017). Solving the DMSC task is useful for providing both
recommendations for users and suggestions for business owners on customer
review platforms.

Aspect based sentiment classification Tang et al. (2016a, b); Wang et al.
(2016b); Chen et al. (2017); Ma et al. (2017); Wang et al. (2018) was usually
done by supervised learning, where aspect-level annotations should be
provided. Aspect-level annotations are not easy to obtain. Even when the
platform provides the function to rate for different aspects, users are less
likely to submit all of them. For example, about 37% of the aspect ratings are
missing on TripAdvisor. If we can solve DMSC task without using aspect-level
annotations, it can save human effort to annotate data or collect user-
generated annotations on the platform.

Existing weakly supervised approaches Wang et al. (2010, 2011) use overall
polarities instead of aspect polarities as “supervision.” Compared with the
polarity of each aspect, it is relatively easy to obtain overall polarities.
Specifically, they minimize the square loss between the overall polarity and
the weighted sum of all aspect polarities. However, when users only care about
a particular rare aspect, e.g., childcare services, these approaches cannot
estimate parameters of the rare aspect incrementally. They have to re-collect
documents which mentioned this rare aspect and estimate parameters of all
aspects based on the new corpus. In addition, these approaches assume the
document is a bag-of-words, which neglects the order of the words and fails to
capture the similarity between words.

In this paper, we propose to use target-opinion word pairs as “supervision.”
Target-opinion word pairs can be helpful with our ultimate goal which is to
learn a classifier to predict the sentiment polarity of each aspect given a
document. For example, in a document “The bedroom is very spacious,” if we can
extract the target-opinion pair “bedroom-spacious,” the sentiment polarity of
the aspect room is likely to be positive. Hence, we propose to achieve the
polarity classification goal by accomplishing another relevant objective: to
predict an opinion word given a target word.

We can decompose the opinion word prediction objective into two sub-tasks. The
first sub-task is to predict the sentiment polarity based on a document. For
example, given a document “The bedroom is very spacious,” it predicts the
sentiment polarity of the aspect room to be positive. The second sub-task is
to predict the opinion word given a target word and a sentiment polarity
predicted by the first sub-task. For example, knowing the fact that the
sentiment polarity of the aspect room is positive, it predicts the opinion
word associated with the target word “room” to be “spacious.” By introducing a
latent variable, i.e., the sentiment polarity of an aspect, to the opinion
word prediction objective, we can inject the polarity classification goal (the
first sub-task) into the objective via the variational lower bound which also
incorporates the second sub-task. In this sense, our training objective is
only based on the target-opinion word pairs which can be extracted by using
dependency parsers and some manually designed rules. We consider our approach
as weakly supervised learning because there is no direct supervision from
polarity of each aspect.

In other words, our model includes two classifiers: a sentiment polarity
classifier and an opinion word classifier. In the sentiment polarity
classifier, it predicts the sentiment polarity given a document. In the
opinion word classifier, it predicts an opinion word based on a target word
and a sentiment polarity. Compared with previous approaches Wang et al. (2010,
2011), our approach can get rid of the assumption that the overall polarity
should be observed and it is a weighted sum of all aspect polarities.
Moreover, our approach can estimate parameters of a new aspect incrementally.
In addition, our sentiment polarity classifier can be more flexible to capture
dependencies among words beyond the bag-of-words representation if we use a
deep neural network architecture to extract features to represent a document.
We conducted experiments on two datasets, TripAdvisor Wang et al. (2010) and
BeerAdvocate McAuley et al. (2012), to illustrate the effectiveness of our
approach.

Our contributions are summarized as follows,

$\bullet$ We propose to solve DMSC task in a nearly unsupervised way.

$\bullet$ We propose to learn a classifier by injecting it into another
relevant objective via the variational lower bound. This framework is flexible
to incorporate different kinds of document representations and relevant
objectives.

$\bullet$ We show promising results on two real datasets and we can produce
comparable results to the supervised method with hundreds of labels per
aspect.

Code and data for this paper are available on https://github.com/HKUST-
KnowComp/VWS-DMSC.

Figure 1: A sentiment polarity classifier and an opinion word classifier
associated with the aspect price.

##  2 VWS-DMSC Approach

In this section, we describe our variational approach to weakly supervised
DMSC (VWS-DMSC). In the next section, we present how we obtain target-opinion
word pairs by using a rule-based extraction approach.

###  2.1 Overview

Our model consists of a sentiment polarity classifier and an opinion word
classifier. Our task is document-level multi-aspect sentiment classification.
For each aspect, we train a sentiment polarity classifier and an opinion word
classifier. The input of the sentiment polarity classifier of each aspect is
the same, i.e., a representation of a document. The target-opinion word pairs
used in opinion word classifiers are different for different aspects.

Figure 1 shows the relation between two classifiers (on the aspect price). The
input $\mathbf{x}$ of the sentiment polarity classifier is a representation of
a document, e.g., bag-of-words or a representation learned by recurrent neural
networks. The sentiment polarity classifier takes $\mathbf{x}$ as input and
produces a distribution of sentiment polarity $R_{a}$ of an aspect $a$,
denoted as $q(R_{a}|\mathbf{x})$. If $R_{a}$ only has two possible values,
i.e., positive and negative, then outputs of the classifier are
$q(positive|\mathbf{x})$ and $q(negative|\mathbf{x})$ respectively. The
opinion word classifier takes a target word (“price”) and a possible value of
the sentiment polarity $r_{a}$ as input, and estimates
$p(\text{``good''}|r_{a},\text{``price''})$. Our training objective is to
maximize the log-likelihood of an opinion word given a target word, e.g.,
$p(\text{``good''}|\text{``price''})$. The likelihood is estimated based on
the sentiment polarity classifier and the opinion word classifier.

###  2.2 Sentiment Polarity Classifier

The sentiment polarity classifier aims to estimate a distribution of sentiment
polarity $q(R_{a}|\mathbf{x})$, where $R_{a}$ is a discrete random variable
representing the sentiment polarity and $\mathbf{x}$ is a feature
representation of a document. We use a simple Softmax classifier here. We
denote $r_{a}$ as a possible value of the random variable $R_{a}$,
representing a possible sentiment polarity. The model estimates the
probability of class $r_{a}$ as

$q(R_{a}=r_{a}|\mathbf{x})=\frac{\exp\big{(}\mathbf{w}_{r_{a}}^{T}\mathbf{x}\big{)}}{\sum_{r_{a}^{\prime}}{\exp\big{(}\mathbf{w}_{r_{a}^{\prime}}^{T}\mathbf{x}\big{)}}}\;,$
(1)

where $\mathbf{w}_{r_{a}}$ is a vector associated with sentiment class $r_{a}$
for aspect $a$.

##### Document Representation

The representation of a document $\mathbf{x}$ can be different using different
feature extraction approaches. Traditional document representations of
sentiment classification would be bag-of-words, n-gram, or averaged word
embeddings. Recently, end-to-end recurrent neural network based models
demonstrate a powerful capacity to extract features of a document. The state-
of-the-art model in DMSC task is Yin et al. (2017). We use it as the document
representation in our model.

###  2.3 Opinion Word Classifier

The opinion word classifier aims to estimate the probability of an opinion
word $w_{o}$ given a target word $w_{t}$ and a sentiment polarity $r_{a}$:

$p(w_{o}|r_{a},w_{t})=\frac{\exp\big{(}\varphi({w_{o}},{w_{t}},r_{a})\big{)}}{\sum_{{w_{o}^{\prime}}}{\exp\big{(}\varphi({w_{o}^{\prime}},{w_{t}},r_{a})\big{)}}}\;,$
(2)

where $\varphi$ is a scoring function related to opinion word $w_{o}$, target
word $w_{t}$, and sentiment polarity $r_{a}$. Here we use the dot product as
the scoring function:

$\varphi({w_{o}},{w_{t}},r_{a})=I\big{(}(w_{t},w_{o})\in\mathcal{P},w_{t}\in\mathcal{K}_{a}\big{)}\cdot\mathbf{c}_{r_{a}}^{T}\mathbf{w}_{o}\;,$
(3)

where $\mathbf{w}_{o}$ is the word embedding of opinion word $w_{o}$,
$\mathbf{c}_{r_{a}}$ is a vector associated with $r_{a}$, $\mathcal{P}$ is the
set of pairs extracted from the document, $\mathcal{K}_{a}$ is the set of
target words associated with aspect $a$, and $I(\cdot)$ is an indicator
function where $I(true)=1$ and $I(false)=0$.

Given a target word $w_{t}$ and a sentiment polarity $r_{a}$, we aim to
maximize the probability of opinion words highly related to them. For example,
opinion word “good” is usually related to target word “price” for aspect value
with sentiment polarity positive, and opinion word “terrible” is usually
related to target word “traffic” for aspect location with sentiment polarity
negative.

###  2.4 Training Objective

The objective function is to maximize the log-likelihood of an opinion word
$w_{o}$ given a target word $w_{t}$. As we mentioned before, the objective
function can be decomposed into two sub-tasks. The first one corresponds to
the sentiment polarity classifier. The second one corresponds to the opinion
word classifier. After introducing a latent variable, i.e., the sentiment
polarity, to the objective function, we can derive a variational lower bound
of the log-likelihood which can incorporate two classifiers:

$\displaystyle{\mathcal{L}}$ $\displaystyle=\log p(w_{o}|w_{t})$
$\displaystyle=\log\sum_{r_{a}}p(w_{o},r_{a}|w_{t})$
$\displaystyle=\log\sum_{r_{a}}q(r_{a}|\mathbf{x})\Big{[}\frac{p(w_{o},r_{a}|w_{t})}{q(r_{a}|\mathbf{x})}\Big{]}$
$\displaystyle\geq\sum_{r_{a}}q(r_{a}|\mathbf{x})\Big{[}\log\frac{p(w_{o},r_{a}|w_{t})}{q(r_{a}|\mathbf{x})}\Big{]}$
$\displaystyle=\operatorname{\mathbb{E}}_{q(R_{a}|\mathbf{x})}\big{[}\log
p(w_{o}|r_{a},w_{t})p(r_{a}|w_{t})\big{]}+H(q(R_{a}|\mathbf{x}))$
$\displaystyle=\operatorname{\mathbb{E}}_{q(R_{a}|\mathbf{x})}\big{[}\log
p(w_{o}|r_{a},w_{t})p(r_{a})\big{]}+H(q(R_{a}|\mathbf{x}))\;,$ (4)

where $H(\cdot)$ refers to the Shannon entropy. By applying Jensen’s
inequality, the log-likelihood is lower-bounded by Eq. (4). The equality holds
if and only if the KL-divergence of two distributions, $q(R_{a}|\mathbf{x})$
and $p(R_{a}|w_{t},w_{o})$, equals to zero. Maximizing the variational lower
bound is equivalent to minimizing the KL-divergence. Hence, we can learn a
sentiment polarity classifier which can produce a similar distribution to the
true posterior $p(R_{a}|w_{t},w_{o})$. Compared with $p(R_{a}|w_{t},w_{o})$,
$q(R_{a}|\mathbf{x})$ is more flexible since it can take any kind of feature
representations as input. We assume that a target word $w_{t}$ and a sentiment
polarity $r_{a}$ are independent since the polarity assignment is not
influenced by the target word. We also assume that the sentiment polarity
$R_{a}$ follows a uniform distribution, which means $p(r_{a})$ is a constant.
We remove it in Eq. (4) to get a new objective function as follows:

$\operatorname{\mathbb{E}}_{q(R_{a}|\mathbf{x})}\left[\log
p(w_{o}|r_{a},w_{t})\right]+H(q(R_{a}|\mathbf{x}))\;.$ (5)

####  2.4.1 Approximation

The partition function of Eq. (2) requires the summation over all opinion
words in the vocabulary. However, the size of opinion word vocabulary is
large, so we use the negative sampling technique Mikolov et al. (2013) to
approximate Eq. (2). Specifically, we substitute $\log p(w_{o}|r_{a},w_{t})$
in the objective (5) with the following objective function:

$\log\sigma\big{(}\varphi({w_{o}},{w_{t}},r_{a})\big{)}+\sum_{w_{o}^{\prime}\in{\mathcal{N}}}\log\sigma\big{(}-\varphi({w_{o}^{\prime}},{w_{t}},r_{a})\big{)}\;,$
(6)

where $w_{o}^{\prime}$ is a negative sample of opinion words in the
vocabulary, ${\mathcal{N}}$ is the set of negative samples and $\sigma$ is the
sigmoid function. Then our final objective function is rewritten as:

$\begin{split}&\operatorname{\mathbb{E}}_{q(R_{a}|\mathbf{x})}\big{[}\log\sigma\big{(}\varphi({w_{o}},{w_{t}},r_{a})\big{)}\\\
&+\sum_{w_{o}^{\prime}\in{\mathcal{N}}}\log\sigma\big{(}-\varphi({w_{o}^{\prime}},{w_{t}},r_{a})\big{)}\big{]}+\alpha
H(q(R_{a}|\mathbf{x}))\;,\end{split}$ (7)

where $\alpha$ is a hyper-parameter which can adjust the expectation and
entropy terms into the same scale Marcheggiani and Titov (2016).

##  3 Target Opinion Word Pairs Extraction

Target-opinion word pairs extraction is a well studied problem Hu and Liu
(2004); Popescu and Etzioni (2005); Bloom et al. (2007); Qiu et al. (2011). We
designed five rules to extract potential target-opinion word pairs. Our method
relies on Stanford Dependency Parser Chen and Manning (2014). We describe our
rules as follows.

Rule 1: We extract pairs satisfying the grammatical relation amod (adjectival
modifier) De Marneffe and Manning (2008). For example, in phrase “very good
price,” we extract “price” and “good” as a target-opinion pair.

Rule 2: We extract pairs satisfying the grammatical relation nsubj (nominal
subject), and the head word is an adjective and the tail word is a noun. For
example, in a sentence “The room is small,” we can extract “room” and “small”
as a target-opinion pair.

Rule 3: Some verbs are also opinion words and they are informative. We extract
pairs satisfying the grammatical relation dobj (direct object) when the head
word is one of the following four words: “like”, “dislike”, “love”, and
“hate”. For example, in the sentence “I like the smell,” we can extract
“smell” and “like” as a target-opinion pair.

Rule 4: We extract pairs satisfying the grammatical relation xcomp (open
clausal complement), and the head word is one of the following word:
“seem”,“look”, “feel”, “smell”, and “taste”. For example, in the sentence
“This beer tastes spicy,” we can extract “taste” and “spicy” as a target-
opinion pair.

Rule 5: If the sentence contains some adjectives that can implicitly indicate
aspects, we manually assign them to the corresponding aspects. According to
Lakkaraju et al. (2014), some adjectives serve both as target words and
opinion words. For example, in the sentence “very tasty, and drinkable,” the
previous rules fail to extract any pair. But we know it contains a target-
opinion pair, i.e., “taste-tasty.” Most of these adjectives have the same root
form with the aspects they indicated, e.g., “clean” (cleanliness), and
“overpriced” (price). This kind of adjective can be extracted first and then
we can obtain more similar adjectives using word similarities. For example,
given “tasty,” we could get “flavorful” by retrieving similar words.

Dataset | TripAdvisor | BeerAdvocate
---|---|---
# docs | 28,543 | 27,583
# target words | 3,737 | 3,088
# opinion words | 12,406 | 9,166
# pairs from R1 | 208,676 | 249,264
# pairs from R2 | 82,944 | 28,505
# pairs from R3 | 2,241 | 1,092
# pairs from R4 | 2,699 | 6,812
# pairs from R5 | 16,537 | 55,825
Table 1: Statistics of extracted target-opinion pairs .

Table 1 shows the statistics of the rule-based extraction on our two datasets.
The first four rules can be applied to any dataset while the last one is
domain dependent which requires human effort to identify these special
adjectives. In practice, rule 5 can be removed to save human effort. The
effect of removing rule 5 is shown in experiments.

After extracting potential target-opinion word pairs, we need to assign them
to different aspects as supervision signals. We select some seed words to
describe each aspect, and then calculate similarities between the extracted
target (or opinion) word and seed words, and assign the pair to the aspect
where one of its seed words has the highest similarity. The similarity we used
is the cosine similarity between two word embeddings trained by word2vec
Mikolov et al. (2013). For example, suppose seed words $\\{$“room”, “bed”$\\}$
and $\\{$“business”, “Internet”$\\}$ are used to describe the aspect room and
business respectively, and the candidate pair “pillow - soft” will be assigned
to the aspect room if the similarity between “pillow” and “bed” is highest
among all combinations.

##  4 Experiment

In this section, we report average sentiment classification accuracies over
all aspects on binary DMSC task.

###  4.1 Datasets

We evaluate our model on TripAdvisor Wang et al. (2010) and BeerAdvocate
McAuley et al. (2012); Lei et al. (2016); Yin et al. (2017) datasets, which
contain seven aspects (value, room, location, cleanliness, check in/front
desk, service, and business) and four aspects (feel, look, smell, and taste)
respectively. We run the same preprocessing steps as Yin et al. (2017). Both
datasets are split into train/development/test sets with proportions 8:1:1.
All methods can use development set to tune their hyper-parameters. Ratings of
TripAdvisor and BeerAdvocate datasets are on scales of $1$ to $5$ and $0$ to
$5$ respectively. But in BeerAdvocate, $0$ star is rare, so we treat the scale
as $1$ to $5$. We convert original scales to binary scales as follows: 1 and 2
stars are treated as negative, 3 is ignored, and 4 and 5 stars are treated as
positive. In BeerAdvocate, most reviews have positive polarities, so to avoid
the unbalanced issue, we perform data selection according to overall
polarities. After data selection, the number of reviews with negative overall
polarities and that with positive overall polarities are equal.

Dataset | TripAdvisor | BeerAdvocate
---|---|---
| DEV | TEST | DEV | TEST
| Mean | Std | Mean | Std | Mean | Std | Mean | Std
Majority | 0.6286 | – | 0.6242 | – | 0.6739 | – | 0.6726 | –
Lexicon-R | 0.5914 | 0.0021 | 0.5973 | 0.0018 | 0.5895 | 0.0020 | 0.5881 | 0.0025
Lexicon-O | 0.7153 | 0.0012 | 0.7153 | 0.0015 | 0.6510 | 0.0023 | 0.6510 | 0.0021
Assign-O | 0.7135 | 0.0016 | 0.7043 | 0.0020 | 0.6652 | 0.0028 | 0.6570 | 0.0034
N-DMSC-O | 0.7091 | – | 0.7064 | – | 0.6386 | – | 0.6493 | –
LRR | 0.6915 | 0.0045 | 0.6947 | 0.0024 | 0.5976 | 0.0110 | 0.5941 | 0.0113
VWS-DMSC (Our) | 0.7577 | 0.0016 | 0.7561 | 0.0012 | 0.7502 | 0.0058 | 0.7538 | 0.0066
N-DMSC-50 | 0.7255 | 0.0231 | 0.7270 | 0.0204 | 0.7381 | 0.0143 | 0.7442 | 0.0157
N-DMSC-100 | 0.7482 | 0.0083 | 0.7487 | 0.0069 | 0.7443 | 0.0126 | 0.7493 | 0.0145
N-DMSC-200 | 0.7531 | 0.0040 | 0.7550 | 0.0043 | 0.7555 | 0.0096 | 0.7596 | 0.0092
N-DMSC-500 | 0.7604 | 0.0028 | 0.7616 | 0.0040 | 0.7657 | 0.0066 | 0.7713 | 0.0070
N-DMSC-1000 | 0.7631 | 0.0054 | 0.7638 | 0.0042 | 0.7708 | 0.0066 | 0.7787 | 0.0053
N-DMSC-A | 0.8281 | – | 0.8334 | – | 0.8576 | – | 0.8635 | –
BoW-DMSC-A | 0.8027 | – | 0.8029 | – | 0.8069 | – | 0.8089 | –
Table 2: Averaged accuracies on DMSC of unsupervised, weakly supervised, and
supervised methods on TripAdvisor and BeerAdvocate. Methods involve randomness
also report standard deviation.

###  4.2 Compared Methods

To demonstrate the effectiveness of our method, we compare our model with
following baselines:

Majority uses the majority of sentiment polarities in training sets as
predictions.

Lexicon means using an opinion lexicon to assign sentiment polarity to an
aspect Read and Carroll (2009); Pablos et al. (2015). We combine two popular
opinion lexicons used by Hu and Liu (2004) and Wilson et al. (2005) to get a
new one. If an opinion word from extracted pairs is in positive (negative)
lexicon, it votes for positive (negative). When the opinion word is with a
negation word, its polarity will be flipped. Then, the polarity of an aspect
is determined by using majority voting among all opinion words associated with
the aspect. When the number of positive and negative words is equal, we adopt
two different ways to resolve it. For Lexicon-R, it randomly assigns a
polarity. For Lexicon-O, it uses the overall polarity as the prediction. Since
overall polarities can also be missing, for both Lexicon-R and Lexicon-O, we
randomly assign a polarity in uncertain cases and report both mean and std
based on five trials of random assignments.

Assign-O means directly using the overall polarity of a review in the
development/test sets as the prediction for each aspect.

LRR assumes the overall polarity is a weighted sum of the polarity of each
aspect Wang et al. (2010). LRR can be regarded as the only existing weakly
supervised baseline where both algorithm and source code are available.

BoW-DMSC-A is a simple softmax classifier using all annotated training data
where the input is a bag-of-words feature vector of a document.

N-DMSC-A is the state-of-the-art neural network based model Yin et al. (2017)
(N-DMSC) in DMSC task using all annotated training data, which serves an upper
bound to our method.

N-DMSC-O is to use overall polarities as “supervision” to train an N-DMSC and
apply it to the classification task of each aspect at the test time.

N-DMSC-{50,100,200,500,1000} is the N-DMSC algorithm using partial data. In
order to see our method is comparable to supervised methods using how many
labeled data, we use $\\{50,100,200,500,1000\\}$ annotations of each aspect to
train N-DMSC and compare them to our method. In addition to annotated data for
training, there are extra $20\%$ annotated data for validation. Since the
sampled labeled data may vary for different trials, we perform five trials of
random sampling and report both mean and std of the results.

For our method, denoted as VWS-DMSC, the document representation we used is
obtained from N-DMSC Yin et al. (2017). They proposed a novel hierarchical
iterative attention model in which documents and pseudo aspect related
questions are interleaved at both word and sentence-level to learn an aspect-
aware document representation. The pseudo aspect related questions are
represented by aspect related keywords. In order to benefit from their aspect-
aware representation scheme, we train an N-DMSC to extract the document
representation using only overall polarities. In the iterative attention
module, we use the pseudo aspect related keywords of all aspects released by
Yin et al. (2017). One can also use document-to-document autoencoders Li et
al. (2015) to generate the document representation. In this way, our method
can get rid of using overall polarities to generate the document
representation. Hence, unlike LRR, it is not necessary for our method to use
overall polarities. Here, to have a fair comparison with LRR, we use the
overall polarities to generate document representation. For our method, we do
not know which state is positive and which one is negative at training time,
so the Hungarian algorithm Kuhn (1955) is used to resolve the assignment
problem at the test time.

###  4.3 Results and Analysis

We show all results in Table 2, which consists of three blocks, namely,
unsupervised, weakly supervised, and supervised methods.

For unsupervised methods, our method can outperform majority on both datasets
consistently. But other weakly supervised methods cannot outperform majority
on BeerAdvocate dataset, which shows these baselines cannot handle unbalanced
data well since BeerAdvocate is more unbalanced than TripAdvisor. Our method
outperforms Lexicon-R and Lexicon-O, which shows that predicting an opinion
word based on a target word may be a better way to use target-opinion pairs,
compared with performing a lexicon lookup using opinion words from extract
pairs. Good performance of Lexicon-O and Assign-O demonstrates the usefulness
of overall polarities in development/test sets. N-DMSC-O trained with the
overall polarities cannot outperform Assign-O since N-DMSC-O can only see
overall polarities in training set while Assign-O can see overall polarities
for both development and test sets and does not involve learning and
generalization.

For weakly supervised methods, LRR is the only open-source baseline in the
literature on weakly supervised DMSC, and our method outperforms LRR by 6% and
16% on TripAdvisor and BeerAdvocate datasets. N-DMSC-O can also be considered
as a weakly supervised method because it only uses overall polarities as
“supervision,” and we still outperform it significantly. It is interesting
that LRR is worse than N-DMSC-O. We guess that assuming that the overall
polarity is a weighted sum of all aspect polarities may not be a good strategy
to train each aspect’s polarity or the document representation learned by
N-DMSC is better than the bag-of-words representation.

For supervised block methods, BoW-DMSC-A and N-DMSC-A are both supervised
methods using all annotated data, which can be seen as the upper bound of our
algorithm. N-DMSC-A outperforms BoW-DMSC-A, which shows that the document
representation based on neural network is better than the bag-of-words
representation. Hence, we use the neural networks based document
representation as input of the sentiment polarity classifier. Our results are
comparable to N-DMSC-200 on TripAdvisor and N-DMSC-100 on BeerAdvocate.

Dataset | TripAdvisor | BeerAdvocate
---|---|---
Rule | DEV | TEST | DEV | TEST
R1 | 0.7215 | 0.7174 | 0.7220 | 0.7216
R2 | 0.7172 | 0.7180 | 0.6864 | 0.6936
R3 | 0.6263 | 0.6187 | 0.6731 | 0.6725
R4 | 0.6248 | 0.6279 | 0.6724 | 0.6717
R5 | 0.5902 | 0.5856 | 0.7095 | 0.7066
\- R1 | 0.7538 | 0.7481 | 0.7458 | 0.7474
\- R2 | 0.7342 | 0.7368 | 0.7504 | 0.7529
\- R3 | 0.7418 | 0.7397 | 0.7565 | 0.7558
\- R4 | 0.7424 | 0.7368 | 0.7518 | 0.7507
\- R5 | 0.7448 | 0.7440 | 0.7550 | 0.7548
All | 0.7577 | 0.7561 | 0.7502 | 0.7538
Table 3: Averaged accuracies on DMSC. “R1 – R5” means only using a rule while
“-R1 – -R5” means removing a rule from all the rules.

###  4.4 Ablation Study

To evaluate effects of extracted rules, we performed an ablation study. We run
our algorithm VWS-DMS with each rule kept or removed over two datasets. If no
pairs extracted for one aspect in training set, the accuracy of this aspect
will be 0.5, which is a random guess. From the Table 3 we can see that, the
rule R1 is the most effective rule for both datasets. Rules R3/R4/R5 are less
effective on their own. However, as a whole, they can still improve the
overall performance. When considering removing each of rules, we found that
our algorithm is quite robust, which indicates missing one of the rules may
not hurt the performance much. Hence, if human labor is a major concern, rule
5 can be discarded. We found that sometimes removing one rule may even result
in better accuracy (e.g., “-R3” for BeerAdvocate dataset). This means this
rule may introduce some noises into the objective function. However, “-R3” can
result in worse accuracy for TripAdvisor, which means it is still
complementary to the other rules for this dataset.

###  4.5 Parameter Sensitivity

We also conduct parameter sensitivity analysis of our approach. The parameter
$\alpha$ in Equation (7) adjusts the expectation and entropy terms on the same
scale. We test $\alpha=\\{0,0.01,0.1,1\\}$ for both of the datasets. As we can
see from Figure 2, $\alpha=0.1$ is a good choice for both datasets.

Figure 2: Parameter sensitivity analysis.

###  4.6 Implementation Details

We implemented our models using TensorFlow Abadi et al. (2016). For N-DMSC and
LRR, we used code released by Yin et al. (2017) and Wang et al. (2010)
respectively and followed their preprocessing steps and optimal settings.

Parameters are updated by using ADADELTA Zeiler (2012), an adaptive learning
rate method. To avoid overfitting, we impose weight decay and drop out on both
classifiers. The regularization coefficient and drop out rate are set to
$10^{-3}$ and $0.3$ respectively. The number of negative samples and $\alpha$
in our model are set to $10$ and $0.1$ respectively. For each document and
each aspect, multiple target-opinion pairs are extracted. The opinion word
classifier associated with an aspect will predict five target-opinion pairs at
a time. These five target-opinion pairs are selected with bias. The
probability of a pair being selected is proportional to the frequency of the
opinion word to the power of $-0.25$. In this way, opinion words with low
frequency are more likely to be selected compared to the uniform sampling. In
order to initialize both classifiers better, the word embeddings are
retrofitted Faruqui et al. (2015) using PPDB Ganitkevitch et al. (2013)
semantic lexicons.

##  5 Related Work

In this section, we review the related work on document-level multi-aspect
sentiment classification, target-opinion word pairs extraction, and
variational methods.

##### Document-level Multi-Aspect Sentiment Classification.

Wang et al. (2010) proposed a LRR model to solve this problem. LRR assumes the
overall polarity is a weighted sum of all aspect polarities which are
represented by word frequency features. LRR needs to use aspect keywords to
perform sentence segmentation to generate the representation of each aspect.
To address the limitation of using aspect keywords, LARAM Wang et al. (2011)
assumes that the text content describing a particular aspect is generated by
sampling words from a topic model corresponding to the latent aspect. Both LRR
and LARAM can only access to overall polarities in the training data, but not
gold standards of aspect polarities. Meng et al. (2018) proposed a weakly
supervised text classification method which can take label surface names,
class-related keywords, or a few labeled documents as supervision. Ramesh et
al. (2015) developed a weakly supervised joint model to identify aspects and
the corresponding sentiment polarities in online courses. They treat aspect
(sentiment) related seed words as weak supervision. In the DMSC task which is
a fine-grained text classification task, the label surface names or keywords
for some aspects would be very similar. Given that the inputs are the same and
the supervisions are similar, weakly supervised models cannot distinguish
them. So we do not consider them as our baselines. Yin et al. (2017) modeled
this problem as a machine comprehension problem under a multi-task learning
framework. It also needs aspect keywords to generate aspect-aware document
representations. Moreover, it can access gold standards of aspect polarities
and achieved state-of-the-art performance on this task. Hence, it can serve as
an upper bound. Some sentence-level aspect based sentiment classification
methods Wang et al. (2016b, 2018) can be directly applied to the DMSC task,
because they can solve aspect category sentiment classification task. For
example, given a sentence “the restaurant is expensive,” the aspect category
sentiment classification task aims to classify the polarity of the aspect
category “price” to be negative. The aspect categories are predefined which
are the same as the DMSC task. Some of them Tang et al. (2016a, b); Chen et
al. (2017); Ma et al. (2017) cannot because they are originally designed for
aspect term sentiment classification task. For example, given a sentence “I
loved their fajitas,” the aspect term sentiment classification task aims to
classify the polarity of the aspect term “fajitas” to be positive. The aspect
terms appearing in the sentence should be provided as inputs.

##### Target Opinion Word Pairs Extraction.

There are two kinds of methods, namely, rule based methods and learning based
methods to solve this task. Rule based methods extract target-opinion word
pairs by mining the dependency tree paths between target words and opinion
words. Learning based methods treat this task as a sequence labeling problem,
mapping each word to one of the following categories: target, opinion, and
other.

Hu and Liu (2004) is one of earliest rule based methods to extract target-
opinion pairs. An opinion word is restricted to be an adjective. Target words
are extracted first, and then an opinion word is linked to its nearest target
word to form a pair. Popescu and Etzioni (2005) and Bloom et al. (2007)
manually designed dependency tree path templates to extract target-opinion
pairs. If the path between a target word candidate and an opinion word
candidate belongs to the set of path templates, the pair will be extracted.
Qiu et al. (2011) identified dependency paths that link opinion words and
targets via a bootstrapping process. This method only needs an initial opinion
lexicon to start the bootstrapping process. Zhuang et al. (2006) adopted a
supervised learning algorithm to learn valid dependency tree path templates,
but it requires target-opinion pairs annotations.

Learning based methods require lots of target-opinion pairs annotations. They
trained conditional random fields (CRF) Lafferty et al. (2001) based models
Jakob and Gurevych (2010); Yang and Cardie (2012); Wang et al. (2016a) or deep
neural networks Liu et al. (2015); Wang et al. (2017); Li and Lam (2017) to
predict the label (target, opinion or other) of each word. Jakob and Gurevych
(2010) and Li et al. (2012) extracted target-opinion pairs without using using
any labeled data in the domain of interest, but it needs lots of labeled data
in another related domain.

In this paper, we only use very simple rules to extract target-opinion pairs
to validate the effectiveness of our approach. If better pairs can be
extracted, we can further improve our results.

##### Variational Methods.

Variational autoencoders Kingma and Welling (2014); Rezende et al. (2014)
(VAEs) use a neural network to parameterize a probability distribution. VAEs
consists of an encoder which parameterizes posterior probabilities and a
decoder which parameterizes the reconstruction likelihood given a latent
variable. VAEs inspire many interesting works Titov and Khoddam (2015);
Marcheggiani and Titov (2016); Šuster et al. (2016); Zhang et al. (2018); Chen
et al. (2018) which are slightly different from VAEs. Their encoders produce a
discrete distribution while the encoder in VAEs yields a continuous latent
variable. Titov and Khoddam (2015) aimed to solve semantic role labeling
problem. The encoder is essentially a semantic role labeling model which
predicts roles given a rich set of syntactic and lexical features. The decoder
reconstructs argument fillers given predicted roles. Marcheggiani and Titov
(2016) aimed to solve unsupervised open domain relation discovery. The encoder
is a feature-rich relation extractor, which predicts a semantic relation
between two entities. The decoder reconstructs entities relying on the
predicted relation. Šuster et al. (2016) tried to learn multi-sense word
embeddings. The encoder uses bilingual context to choose a sense for a given
word. The decoder predicts context words based on the chosen sense and the
given word. Zhang et al. (2018) aimed to solve knowledge graph powered
question answering. Three neural networks are used to parameterize
probabilities of a topic entity given a query and an answer, an answer based
on a query and a predicted topic, and the topic given the query. Chen et al.
(2018) aimed to infer missing links in a knowledge graph. Three neural
networks are used to parameterize probabilities of a latent path given two
entities and a relation, a relation based on two entities and the chosen
latent path, and the relation given the latent path. Our method also uses
neural networks to parameterize two discrete distributions but aims to solve
the DMSC task.

##  6 Conclusion

In this paper, we propose a variational approach to weakly supervised DMSC. We
extract many target-opinion word pairs from dependency parsers using simple
rules. These pairs can be “supervision” signals to predict sentiment polarity.
Our objective function is to predict an opinion word given a target word.
After introducing the sentiment polarity as a latent variable, we can learn a
sentiment polarity classifier by optimizing the variational lower bound. We
show that we can outperform weakly supervised baselines by a large margin and
achieve comparable results to the supervised method with hundreds of labels
per aspect, which can reduce a lot of labor work in practice. In the future,
we plan to explore better target-opinion word extraction approaches to find
better “supervision” signals.

## Acknowledgments

This paper was supported by the Early Career Scheme (ECS, No. 26206717) from
Research Grants Council in Hong Kong. Ziqian Zeng has been supported by the
Hong Kong Ph.D. Fellowship. We thank Intel Corporation for supporting our deep
learning related research. We also thank the anonymous reviewers for their
valuable comments and suggestions that help improve the quality of this
manuscript.

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# Warping effects in strongly perturbed metrics

Marco Frasca  Riccardo Maria Liberati  Massimiliano Rossi  MBDA Italia S.p.A.,
Via Monte Flavio, 45,
00131, Rome (Italy)

###### Abstract

A technique devised some years ago permits to study a theory in a regime of
strong perturbations. This translates into a gradient expansion that, at the
leading order, can recover the BKL solution in general relativity. We solve
exactly the leading order Einstein equations in a spherical symmetric case,
assuming a Schwarzschild metric under the effect of a time-dependent
perturbation, and we show that the 4-velocity in such a case is multiplied by
an exponential warp factor when the perturbation is properly applied. This
factor is always greater than one. We will give a closed form solution of this
factor for a simple case. Some numerical examples are also given.

##  I Introduction

The study of Einstein equations in certain regimes is often reduced to solve
them numerically teul99 . The reason is that they form a set of nonlinear PDEs
that are generally difficult to handle with analytical tools for most
interesting situations. Often, the reason relies on the fact that no small
parameter can be found to apply standard perturbation techniques while
analytical solutions are very rare and difficult to find. Some years ago, one
of us (M.F.) proposed an approach based on earlier works in strongly perturbed
systems fra06 . It was shown that, under a strong perturbation in the formal
limit running to infinity, the leading order is obtained by neglecting the
gradient terms in the Einstein equations. The leading order of this
perturbation series was firstly proposed by Belinsky, Kalathnikov, Lifshitz
for their famous BKL conjecture kl70 ; bkl70 ; bkl82 , as is known today.

Some decades ago, Alcubierre proposed a solution of the Einstein equations
alcu94  that describes an observer moving with an unbounded velocity provided
the condition of positivity of the energy is violated. A recent paper Santos-
Pereira:2020puq  (see also Refs. therein) yields a short recount about
Alcubierre metric and its interaction with dust. Indeed, any kind of pathology
has emerged about it and the difficulties arise from the fact that this is an
engineered metric that is imposed on the Einstein equations. It would be
desirable to have a metric like this one emerging as a solution of the
Einstein equations and conserving the positivity of the energy. A recent
proposal goes in such a direction Lentz:2020euv . This is possible by
introducing a hyperbolic shift vector potential and the author shows how this
can emerge from a plasma.

In this paper we will show how a warp factor for the velocity can emerge when
a strong perturbation is applied to a spherical symmetric metric. So, any
Eulerian observer will get its velocity expanded when such perturbation is
acting. This extends and complete our preceding work fra06 . We will get the
exact solution of the leading perturbation equations and we will show how an
exponential factor can emerge that is systematically greater than one. We
emphasize that we are doing perturbation theory in a limit where the a
perturbation, applied to a given gravitational field, is taken much greater of
the unperturbed situation. This is the opposite limit to standard small
perturbation theory and is based on the technique devised in fra06 .

The paper is so structured. In Sec.II, we will introduce the technique to
treat strongly perturbed systems. In Sec.III, we apply this to the Einstein
equations for a spherical symmetry metric with a time-dependent perturbation.
In Sec.IV, we solve the leading order perturbation equations. In Sec.V, we
yield the geodesic equations. In Sec.VI, we show how the expansion factor
enters into the velocity providing some examples and an analytic solution. In
Sec.VII, conclusions are presented.

##  II Strong perturbations and gradient expansion

For our computations in general relativity, we need to study the case of a
strong perturbation on a given metric that we will choose to be the
Schwarzschild one. In order to prove that a gradient expansion indeed
represents a strong perturbation theory, we will study the following non-
linear equation as a toy model for the Einstein equations111We just point out
that this model can represent Einstein equations in 1+1 dimensions teitel ;
jack ; DHoker:1982wmk . (here and in the following we assume $c=1$).

$-\Box\phi+\lambda V^{\prime}(\phi)=0$ (1)

being $\Box=\nabla^{2}-\partial^{2}_{t}$ the wave operator (here and in the
following $c=1$), $\phi$ a scalar field and $V(\phi)$ its self-interaction
with a coupling $\lambda$. For 2D Einstein equations, this would be a
Liouville equation teitel ; jack ; DHoker:1982wmk . We would like to do
perturbation theory in the formal limit of $\lambda\rightarrow\infty$. This
ends up to obtain a non-trivial series in $1/\lambda$. We can accomplish our
aim by rescaling the time variable Frasca:1998ch . We take
$t\rightarrow\sqrt{\lambda}t$ and the equation above becomes

$-\nabla^{2}\phi+\lambda\partial^{2}_{t}\phi+\lambda V^{\prime}(\phi)=0.$ (2)

Then, we take

$\phi=\phi_{0}+\frac{1}{\lambda}\phi_{1}+\frac{1}{\lambda^{2}}\phi_{2}+\ldots$
(3)

and substitute this into eq.(2). This gives the set of perturbative equations

$\displaystyle\partial^{2}_{t}\phi_{0}$ $\displaystyle=$
$\displaystyle-V^{\prime}(\phi_{0})$ $\displaystyle\partial^{2}_{t}\phi_{1}$
$\displaystyle=$
$\displaystyle-V^{\prime\prime}(\phi_{0})\phi_{1}+\nabla^{2}\phi_{0}$
$\displaystyle\partial_{t}^{2}\phi_{2}$ $\displaystyle=$
$\displaystyle-V^{\prime\prime}(\phi_{0})\phi_{2}-\frac{1}{2}V^{\prime\prime\prime}(\phi_{0})\phi_{1}^{2}+\nabla^{2}\phi_{1}$
(4) $\displaystyle\vdots$ .

We see that we have obtained a set of non-trivial equations that define the
perturbation series in the formal limit $\lambda\rightarrow\infty$. This
approach can be applied, exactly in this way, to the Einstein equations. This
also shows how consistent was the original BKL approach in kl70 ; bkl70 ;
bkl82 . Indeed, we have obtained a gradient expansion.

In order to see how this technique applies to Einstein equations, we write
them down in the Arnowitt-Deser-Misner (ADM) formalism as teul99

$\displaystyle\partial_{t}\gamma_{ij}-\beta^{l}\partial_{l}\gamma_{ij}$
$\displaystyle=$
$\displaystyle\gamma_{lj}\partial_{i}\beta^{l}+\gamma_{il}\partial_{j}\beta^{l}-2\alpha
K_{ij}$ (5) $\displaystyle\partial_{t}K_{ij}-\beta^{l}\partial_{l}K_{ij}$
$\displaystyle=$ $\displaystyle
K_{il}\partial_{j}\beta^{l}+K_{jl}\partial_{i}\beta^{l}-2\alpha
K_{il}K_{j}^{l}+\alpha KK_{ij}$ (6)
$\displaystyle-\frac{1}{2}\alpha\gamma^{lm}\left\\{\partial_{l}\partial_{m}\gamma_{ij}+\partial_{i}\partial_{j}\gamma_{lm}-\partial_{i}\partial_{l}\gamma_{mj}-\partial_{j}\partial_{l}\gamma_{mi}\right.$
$\displaystyle+\gamma^{np}\left[(\partial_{i}\gamma_{jn}+\partial_{j}\gamma_{in}-\partial_{n}\gamma_{ij})\partial_{l}\gamma_{mp}\right.$
$\displaystyle+\partial_{l}\gamma_{in}\partial_{p}\gamma_{jm}-\partial_{l}\gamma_{in}\partial_{m}\gamma_{jp}\left.\right]$
$\displaystyle-\frac{1}{2}\gamma^{np}\left[(\partial_{i}\gamma_{jn}+\partial_{j}\gamma_{in}-\partial_{n}\gamma_{ij})\partial_{p}\gamma_{lm}+\partial_{i}\gamma_{ln}\partial_{j}\gamma_{mp}\left.\right]\right\\}$
$\displaystyle-\partial_{i}\partial_{j}\alpha+\frac{1}{2}\gamma^{lm}(\partial_{i}\gamma_{jm}+\partial_{j}\gamma_{im}-\partial_{m}\gamma_{ij})\partial_{l}\alpha$
$\displaystyle+\alpha\left[-8\pi GT_{ij}+4\pi G\gamma_{ij}(T-\rho)\right]$

where the energy-matter tensor $T_{\mu\nu}$ is given with the density $\rho$,
for a metric

$ds^{2}=(-\alpha^{2}+\gamma_{ij}\beta^{i}\beta^{j})dt^{2}+2\beta^{i}dx_{i}dt+\gamma_{ij}dx^{i}dx^{j}$
(7)

being $\alpha$ the lapse function, $\beta^{i}$ the shift vector, $\gamma_{ij}$
the spatial part of the metric and $K_{ij}$ the extrinsic curvature. For our
aims, we do not care about constraint equations that are just essential for
numerical computations. This set is amenable to the same treatment we applied
to the preceding example. The procedure is identical, we introduce an ordering
parameter $\lambda$ that we will set to 1 to the end of computation. Then, we
consider the perturbation series defined by

$\displaystyle\tau$ $\displaystyle=$ $\displaystyle\sqrt{\lambda}t,$ (8)
$\displaystyle K_{ij}$ $\displaystyle=$
$\displaystyle\sqrt{\lambda}\left(K_{ij}^{(0)}+\frac{1}{\lambda}K_{ij}^{(1)}+\frac{1}{\lambda^{2}}K_{ij}^{(2)}+\ldots\right),$
$\displaystyle\gamma_{ij}$ $\displaystyle=$
$\displaystyle\gamma_{ij}^{(0)}+\frac{1}{\lambda}\gamma_{ij}^{(1)}+\frac{1}{\lambda^{2}}\gamma_{ij}^{(2)}+\ldots,$
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\alpha_{0}+\frac{1}{\lambda}\alpha_{1}+\frac{1}{\lambda^{2}}\alpha_{2}+\ldots.$

Our gauge choice is to set the shift vector $\beta_{i}=0$. This approach is
completely dual to the standard computation for weak gravitational fields but
implies that the perturbation is taken formally to go to infinity. This
represents a situation where the perturbation overcomes the intensity of the
gravitational field where is applied. Typical situations where this technique
could apply are black hole collisions where, currently, only numerical
computations or analytical techniques, working given certain approximations,
are available Soffel:2019aoq . Therefore, we get the non-trivial set of
equations (we have set $\lambda=1$)

$\displaystyle\partial_{t}\gamma_{ij}^{(0)}$ $\displaystyle=$
$\displaystyle-2\alpha_{0}K_{ij}^{(0)}$ (9)
$\displaystyle\partial_{t}\gamma_{ij}^{(1)}$ $\displaystyle=$
$\displaystyle-2\alpha_{1}K_{ij}^{(0)}-2\alpha_{0}K_{ij}^{(1)}$
$\displaystyle\vdots$ $\displaystyle\partial_{t}K_{ij}^{(0)}$ $\displaystyle=$
$\displaystyle-2\alpha_{0}K_{il}^{(0)}K_{j}^{l(0)}+\alpha_{0}K^{(0)}K_{ij}^{(0)}$
$\displaystyle\partial_{t}K_{ij}^{(1)}$ $\displaystyle=$
$\displaystyle-2\alpha_{1}K_{il}^{(0)}K_{j}^{l(0)}-2\alpha_{0}K_{il}^{(1)}K_{j}^{l(0)}-2\alpha_{0}K_{il}^{(0)}K_{j}^{l(1)}$
(10)
$\displaystyle+\alpha_{1}K^{(0)}K_{ij}^{(0)}+\alpha_{0}K^{(1)}K_{ij}^{(0)}+\alpha_{0}K^{(0)}K_{ij}^{(1)}$
$\displaystyle-\frac{1}{2}\alpha_{0}\gamma^{lm(0)}\left\\{\partial_{l}\partial_{m}\gamma_{ij}^{(0)}+\partial_{i}\partial_{j}\gamma_{lm}^{(0)}-\partial_{i}\partial_{l}\gamma_{mj}^{(0)}-\partial_{j}\partial_{l}\gamma_{mi}^{(0)}\right.$
$\displaystyle+\gamma^{np(0)}\left[(\partial_{i}\gamma_{jn}^{(0)}+\partial_{j}\gamma_{in}^{(0)}-\partial_{n}\gamma_{ij}^{(0)})\partial_{l}\gamma_{mp}^{(0)}\right.$
$\displaystyle+\partial_{l}\gamma_{in}^{(0)}\partial_{p}\gamma_{jm}^{(0)}-\partial_{l}\gamma_{in}^{(0)}\partial_{m}\gamma_{jp}^{(0)}\left.\right]$
$\displaystyle-\frac{1}{2}\gamma^{np(0)}\left[(\partial_{i}\gamma_{jn}^{(0)}+\partial_{j}\gamma_{in}^{(0)}-\partial_{n}\gamma_{ij}^{(0)})\partial_{p}\gamma_{lm}^{(0)}+\partial_{i}\gamma_{ln}^{(0)}\partial_{j}\gamma_{mp}^{(0)}\left.\right]\right\\}$
$\displaystyle-\partial_{i}\partial_{j}\alpha_{0}+\frac{1}{2}\gamma^{lm(0)}(\partial_{i}\gamma_{jm}^{(0)}+\partial_{j}\gamma_{im}^{(0)}-\partial_{m}\gamma_{ij}^{(0)})\partial_{l}\alpha_{0}$
$\displaystyle+\alpha^{(0)}\left[-8\pi GT_{ij}+4\pi
G\gamma_{ij}^{(0)}(T-\rho)\right]$ $\displaystyle\vdots$

where one sees that the energy-matter tensor contributes to the next-to-
leading order. We realize from these equations that the gradient terms, that
is components of the metric that are varying spatially, are moved to the next-
to-leading order. We will apply them in the following in the spherical
symmetric case, assuming the Schwarzschild metric as the unperturbed solution.
Here and in the following, we avoid to show explicitly the energy-matter
tensor as our perturbation series moves its contribution to the next-to-
leading order. This implies that, at the leading order, an approximation for
the energy-matter configuration can be taken to be that in absence of the
gravitational field. This is consistent with our approach for strongly
perturbed metrics.

Anyway, in order to have an idea of the main concept underlying this
approximation scheme, let us consider the Reissner-Nördstrom metric of a
charged black hole. This will be given by

$ds^{2}=-\left(1-\frac{r_{g}}{r}+\frac{r_{Q}^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{r_{g}}{r}+\frac{r_{Q}^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.$ (11)

being $r_{g}=2GM$ the Schwarzschild radius and
$r_{q}^{2}=GQ^{2}/4\pi\epsilon_{0}$ the scale introduced by the black hole
charge $Q$ with $1/4\pi\epsilon_{0}$ the Coulomb constant. This is an exact
solution of the Einstein-Maxwell equations. In our case, we assume that the
electric field overcomes largely the gravitational contribution, that is
$r_{Q}\gg r_{g}$. This appears formally as a large perturbation on a
Schwarzschild black hole and the approximate metric will be

$ds^{2}=-\left(1+\frac{r_{Q}^{2}}{r^{2}}\right)dt^{2}+\left(1+\frac{r_{Q}^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}+O(r_{g}/r).$ (12)

This should be compared with the opposite dual limit $r_{Q}/r_{g}\ll 1$ that
yields

$ds^{2}=-\left(1-\frac{r_{g}}{r}\right)dt^{2}+\left(1-\frac{r_{g}}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}+O(r_{Q}^{2}/r^{2}).$ (13)

##  III Strongly perturbed spherical symmetry metric

We are assuming a spherical symmetry metric in ADM formalism given by

$ds^{2}=-\alpha^{2}dt^{2}+\gamma_{rr}dr^{2}+\gamma_{\theta\theta}d\theta^{2}+\gamma_{\phi\phi}d\phi^{2}.$
(14)

This implies a specific choice of the gauge where all the components of the
shift vector, normally named $\beta_{i}$, are taken to be zero. Then, the
perturbation $\alpha_{1}$ is just applied to the lapse function as follows
teul99

$\alpha^{2}=\alpha_{0}^{2}+\alpha_{1}.$ (15)

Then, we specialize the set of eq.(9) to this case. Assuming as unperturbed
solution the Schwarzschild one, the exterior solution is given by (again,
$r_{g}=2GM$ is the Schwarzschild radius)

$\displaystyle\alpha_{0}^{2}$ $\displaystyle=$
$\displaystyle\left(1-\frac{r_{g}}{r}\right),$ (16)
$\displaystyle\gamma_{11}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\frac{r_{g}}{r}},$
$\displaystyle\gamma_{22}^{(0)}=r^{2},$
$\displaystyle\gamma_{33}^{(0)}=r^{2}\sin^{2}\theta,$

and the interior solution is

$\displaystyle\alpha_{0}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(3\sqrt{1-\frac{r_{g}}{r_{s}}}-\sqrt{1-\frac{r^{2}r_{g}}{r_{s}^{3}}}\right)^{2},$
(17) $\displaystyle\gamma_{11}^{(0)}$ $\displaystyle=$
$\displaystyle\left(1-\frac{r^{2}r_{g}}{r_{s}^{3}}\right)^{-1},$
$\displaystyle\gamma_{22}^{(0)}=r^{2}$
$\displaystyle\gamma_{33}^{(0)}=r^{2}\sin^{2}\theta,$

being $r_{s}$ is the value of the $r$-coordinate at the body’s surface. It
easy to see that both metrics are the same at the sphere surface for $r=r_{s}$
granting continuity. We also have, with our gauge’s choice $\beta_{i}=0$, the
general formula

$K_{ij}=-\frac{1}{2\alpha}\partial_{t}\gamma_{ij}.$ (18)

In our case is

$\alpha^{2}=\alpha_{0}^{2}+\alpha_{1}=\alpha_{0}^{2}+Af(r,t),$ (19)

being $A$ the amplitude of the perturbation. We emphasize that the
perturbations we are going to consider are time-dependent. This yields

$\partial_{t}\gamma_{ij}^{(1)}=\alpha_{1}\frac{1}{\alpha_{0}}\partial_{t}\gamma_{ij}^{(0)}-2\alpha_{0}K_{ij}^{(1)},$
(20)

that reduces to

$\partial_{t}\gamma_{ij}^{(1)}=-2\alpha_{0}K_{ij}^{(1)},$ (21)

as $\gamma_{ij}^{(0)}$ does not depend on time variable. Now, one has

$\displaystyle\partial_{t}K_{ij}^{(1)}$ $\displaystyle=$
$\displaystyle-2\alpha_{1}K_{il}^{(0)}K_{j}^{l(0)}-2\alpha_{0}K_{il}^{(1)}K_{j}^{l(0)}-2\alpha_{0}K_{il}^{(0)}K_{j}^{l(1)}$
$\displaystyle+\alpha_{1}K^{(0)}K_{ij}^{(0)}+\alpha_{0}K^{(1)}K_{ij}^{(0)}+\alpha_{0}K^{(0)}K_{ij}^{(1)}$
$\displaystyle+\alpha_{1}K^{(0)}K_{ij}^{(0)}+\alpha_{0}K^{(1)}K_{ij}^{(0)}+\alpha_{0}K^{(0)}K_{ij}^{(1)}$
$\displaystyle-\frac{1}{2}\alpha_{0}\gamma^{lm(0)}\left\\{\partial_{l}\partial_{m}\gamma_{ij}^{(0)}+\partial_{i}\partial_{j}\gamma_{lm}^{(0)}-\partial_{i}\partial_{l}\gamma_{mj}^{(0)}-\partial_{j}\partial_{l}\gamma_{mi}^{(0)}\right.$
$\displaystyle+\gamma^{np(0)}\left[(\partial_{i}\gamma_{jn}^{(0)}+\partial_{j}\gamma_{in}^{(0)}-\partial_{n}\gamma_{ij}^{(0)})\partial_{l}\gamma_{mp}^{(0)}\right.$
$\displaystyle+\partial_{l}\gamma_{in}^{(0)}\partial_{p}\gamma_{jm}^{(0)}-\partial_{l}\gamma_{in}^{(0)}\partial_{m}\gamma_{jp}^{(0)}\left.\right]$
$\displaystyle-\frac{1}{2}\gamma^{np(0)}\left[(\partial_{i}\gamma_{jn}^{(0)}+\partial_{j}\gamma_{in}^{(0)}-\partial_{n}\gamma_{ij}^{(0)})\partial_{p}\gamma_{lm}^{(0)}+\partial_{i}\gamma_{ln}^{(0)}\partial_{j}\gamma_{mp}^{(0)}\left.\right]\right\\}$
$\displaystyle-\partial_{i}\partial_{j}\alpha_{0}+\frac{1}{2}\gamma^{lm(0)}(\partial_{i}\gamma_{jm}^{(0)}+\partial_{j}\gamma_{im}^{(0)}-\partial_{m}\gamma_{ij}^{(0)})\partial_{l}\alpha_{0}$
$\displaystyle\vdots$ .

This set of equations, written in this way, are too difficult to manage. As we
will see below, we can restate them to find an exact leading order solution.

It is correct to ask why the Birkhoff theorem does not apply in our case. The
reason is that the problems we are treating are similar to the one of the
ringdown of a Schwarzschild black-hole where a strong perturbation, due to the
collision between two black holes, modifies the metric making it varying in
time, after coalescence, until the oscillations are damped out and the
spherical symmetry is recovered Merritt:2004gc , in agreement with Birkhoff
theorem. It should be said that such problems are better managed in the Kerr
metric but we do not consider rotations to avoid too much computations
cluttering formulas.

##  IV Solving perturbation equations

In order to have more manageable equations, let us start from the following
rewriting of the ADM equations of motion in exact form. We will get (as
already said, our gauge is $\beta_{i}=0$)

$\displaystyle\partial_{t}\gamma_{ij}$ $\displaystyle=$ $\displaystyle-2\alpha
K_{ij}$ $\displaystyle\partial_{t}K_{ij}$ $\displaystyle=$
$\displaystyle\alpha\left[R_{ij}-2K_{il}K_{j}^{l}+KK_{ij}\right]-\partial_{i}\partial_{j}\alpha.$
(23)

The Ricci tensor $R_{ij}$ refers to the $\gamma_{ij}$ and all Latin indexes
run from 1 to 3. We can exploit these equations for the diagonal elements to
obtain

$\displaystyle\partial_{t}\gamma_{11}$ $\displaystyle=$ $\displaystyle-2\alpha
K_{11},$ $\displaystyle\partial_{t}\gamma_{22}$ $\displaystyle=$
$\displaystyle-2\alpha K_{22},$ $\displaystyle\partial_{t}\gamma_{33}$
$\displaystyle=$ $\displaystyle-2\alpha K_{33},$
$\displaystyle\partial_{t}K_{11}$ $\displaystyle=$
$\displaystyle\alpha\left[R_{11}-2K_{1l}K_{1}^{l}+KK_{11}\right]-\partial_{1}^{2}\alpha,$
$\displaystyle\partial_{t}K_{22}$ $\displaystyle=$
$\displaystyle\alpha\left[R_{22}-2K_{2l}K_{2}^{l}+KK_{22}\right]-\partial_{2}^{2}\alpha,$
$\displaystyle\partial_{t}K_{33}$ $\displaystyle=$
$\displaystyle\alpha\left[R_{33}-2K_{3l}K_{3}^{l}+KK_{33}\right]-\partial_{3}^{2}\alpha.$
(24)

We notice that $K_{i}^{l}=\gamma^{kl}K_{ik}$ and $K=\gamma^{kl}K_{kl}$. We
expect that off-diagonal terms should be perturbatively negligible and so, we
neglect them here in view of a gradient expansion. Indeed, for $i\neq j$, we
will have

$\displaystyle K_{ij}$ $\displaystyle=$
$\displaystyle-\frac{\partial_{t}\gamma_{ij}}{2\alpha}$
$\displaystyle\partial_{t}K_{ij}$ $\displaystyle=$
$\displaystyle\alpha\left[R_{ij}-2K_{il}\gamma^{lk}K_{jk}+\gamma^{kl}K_{kl}K_{ij}\right]-\partial_{i}\partial_{j}\alpha.$
(25)

This will give

$-\partial_{t}\left(\frac{\partial_{t}\gamma_{ij}}{2\alpha}\right)=\alpha\left[R_{ij}+\frac{1}{2\alpha^{2}}\partial_{t}\gamma_{il}\gamma^{lk}\partial_{t}\gamma_{jk}+\frac{1}{4\alpha^{2}}\gamma^{kl}\partial_{t}\gamma_{kl}\partial_{t}\gamma_{ij}\right]-\partial_{i}\partial_{j}\alpha.$
(26)

In a gradient expansion, where we neglect both $R_{ij}$ and
$\partial_{i}\partial_{j}\alpha$ as we will show below, at the leading order
the off-diagonal terms will remain 0 if they were zero initially because this
is a solution of eq.(26) for $i\neq j$. Therefore,

$\displaystyle\partial_{t}\gamma_{11}$ $\displaystyle=$ $\displaystyle-2\alpha
K_{11},$ (27) $\displaystyle\partial_{t}\gamma_{22}$ $\displaystyle=$
$\displaystyle-2\alpha K_{22},$ $\displaystyle\partial_{t}\gamma_{33}$
$\displaystyle=$ $\displaystyle-2\alpha K_{33},$
$\displaystyle\partial_{t}K_{11}$ $\displaystyle=$
$\displaystyle\alpha\left[R_{11}-\gamma^{11}K_{11}^{2}+(\gamma^{22}K_{22}+\gamma^{33}K_{33})K_{11}\right]-\partial_{1}^{2}\alpha,$
$\displaystyle\partial_{t}K_{22}$ $\displaystyle=$
$\displaystyle\alpha\left[R_{22}-\gamma^{22}K_{22}^{2}+(\gamma^{11}K_{11}+\gamma^{33}K_{33})K_{22}\right]-\partial_{2}^{2}\alpha,$
$\displaystyle\partial_{t}K_{33}$ $\displaystyle=$
$\displaystyle\alpha\left[R_{33}-\gamma^{33}K_{33}^{2}+(\gamma^{11}K_{11}+\gamma^{22}K_{22})K_{33}\right]-\partial_{3}^{2}\alpha.$

These equation can be stated in a single set of equations for the $\gamma$s as

$\displaystyle\partial_{t}^{2}\gamma_{11}$ $\displaystyle=$
$\displaystyle-2\dot{\alpha}K_{11}-2\alpha\dot{K}_{11}=$
$\displaystyle\frac{\dot{\alpha}}{\alpha}\dot{\gamma}_{11}-2\alpha^{2}\left[R_{11}-\gamma^{11}\frac{1}{4\alpha^{2}}(\dot{\gamma}_{11})^{2}+\frac{1}{4\alpha^{2}}\gamma^{22}\dot{\gamma}_{22}\dot{\gamma}_{11}+\frac{1}{4\alpha^{2}}\gamma^{33}\dot{\gamma}_{33}\dot{\gamma}_{11}\right]+2\alpha\partial_{1}^{2}\alpha$
$\displaystyle\partial_{t}^{2}\gamma_{22}$ $\displaystyle=$
$\displaystyle-2\dot{\alpha}K_{22}-2\alpha\dot{K}_{22}=$
$\displaystyle\frac{\dot{\alpha}}{\alpha}\dot{\gamma}_{22}-2\alpha^{2}\left[R_{22}-\gamma^{22}\frac{1}{4\alpha^{2}}(\dot{\gamma}_{22})^{2}+\frac{1}{4\alpha^{2}}\gamma^{11}\dot{\gamma}_{11}\dot{\gamma}_{22}+\frac{1}{4\alpha^{2}}\gamma^{33}\dot{\gamma}_{33}\dot{\gamma}_{22}\right]+2\alpha\partial_{2}^{2}\alpha$
$\displaystyle\partial_{t}^{2}\gamma_{33}$ $\displaystyle=$
$\displaystyle-2\dot{\alpha}K_{33}-2\alpha\dot{K}_{33}=$
$\displaystyle\frac{\dot{\alpha}}{\alpha}\dot{\gamma}_{33}-2\alpha^{2}\left[R_{33}-\gamma^{33}\frac{1}{4\alpha^{2}}(\dot{\gamma}_{33})^{2}+\frac{1}{4\alpha^{2}}\gamma^{11}\dot{\gamma}_{11}\dot{\gamma}_{33}+\frac{1}{4\alpha^{2}}\gamma^{22}\dot{\gamma}_{22}\dot{\gamma}_{33}\right]+2\alpha\partial_{3}^{2}\alpha.$

and so on for the other components. As said into Sec.III, this set of
equations can be solved perturbatively by the change of variable
$\tau=\sqrt{\lambda}t$ being $\lambda$ just an ordering parameter that we will
set to 1 to the end of computations. This means that we can neglect spatial
gradients at the leading order, yielding

$\displaystyle\partial_{\tau}^{2}\gamma_{11}$ $\displaystyle=$
$\displaystyle\frac{\dot{\alpha}}{\alpha}\dot{\gamma}_{11}+\frac{1}{2}\gamma^{11}(\dot{\gamma}_{11})^{2}-\frac{1}{2}\gamma^{22}\dot{\gamma}_{22}\dot{\gamma}_{11}-\frac{1}{2}\gamma^{33}\dot{\gamma}_{33}\dot{\gamma}_{11}=$
(29)
$\displaystyle\frac{\dot{\alpha}}{\alpha}\dot{\gamma}_{11}+\frac{1}{2}(\gamma_{11})^{-1}(\dot{\gamma}_{11})^{2}-\frac{1}{2}(\gamma_{22})^{-1}\dot{\gamma}_{22}\dot{\gamma}_{11}-\frac{1}{2}(\gamma_{33})^{-1}\dot{\gamma}_{33}\dot{\gamma}_{11}$

This can be rewritten as

$\partial_{\tau}^{2}\gamma_{11}=\dot{\gamma}_{11}\frac{d}{d\tau}\left[\ln\alpha+\frac{1}{2}\ln\left(\frac{\gamma_{11}}{\gamma_{22}\gamma_{33}}\right)\right]$
(30)

Then,

$\partial_{\tau}\ln\dot{\gamma}_{11}=\frac{d}{d\tau}\left[\ln\alpha+\frac{1}{2}\ln\left(\frac{\gamma_{11}}{\gamma_{22}\gamma_{33}}\right)\right]$
(31)

and finally

$\ln\dot{\gamma}_{11}=\left[\ln\left(r_{k}\frac{\alpha}{\alpha_{0}}\right)+\frac{1}{2}\ln\left(\frac{\gamma_{11}}{\gamma_{22}\gamma_{33}}\right)\right]$
(32)

where we have properly fixed the integration constant in such a way that, in
absence of perturbation, the contribution from $\alpha$ disappears while
dimensions are kept with the constant $r_{k}=r_{g}$ for the exterior solution
and $r_{k}=r_{s}$ for the interior solution. This gives the following set of
differential equations

$\displaystyle\dot{\gamma}_{11}$ $\displaystyle=$ $\displaystyle
r_{k}\frac{\alpha}{\alpha_{0}}\sqrt{\frac{\gamma_{11}}{\gamma_{22}\gamma_{33}}}=r_{k}\frac{\alpha}{\alpha_{0}}\gamma_{11}\gamma^{-\frac{1}{2}}$
$\displaystyle\dot{\gamma}_{22}$ $\displaystyle=$ $\displaystyle
r_{k}\frac{\alpha}{\alpha_{0}}\sqrt{\frac{\gamma_{22}}{\gamma_{11}\gamma_{33}}}=r_{k}\frac{\alpha}{\alpha_{0}}\gamma_{22}\gamma^{-\frac{1}{2}}$
$\displaystyle\dot{\gamma}_{33}$ $\displaystyle=$ $\displaystyle
r_{k}\frac{\alpha}{\alpha_{0}}\sqrt{\frac{\gamma_{33}}{\gamma_{11}\gamma_{22}}}=r_{k}\frac{\alpha}{\alpha_{0}}\gamma_{33}\gamma^{-\frac{1}{2}}$
(33)

This set can be solved exactly by multiplying in the following way

$\displaystyle\dot{\gamma}_{11}\gamma_{22}\gamma_{33}$ $\displaystyle=$
$\displaystyle r_{k}\frac{\alpha}{\alpha_{0}}\gamma^{\frac{1}{2}}$
$\displaystyle\dot{\gamma}_{22}\gamma_{11}\gamma_{33}$ $\displaystyle=$
$\displaystyle r_{k}\frac{\alpha}{\alpha_{0}}\gamma^{\frac{1}{2}}$
$\displaystyle\dot{\gamma}_{33}\gamma_{11}\gamma_{22}$ $\displaystyle=$
$\displaystyle r_{k}\frac{\alpha}{\alpha_{0}}\gamma^{\frac{1}{2}}$ (34)

and summing up the three equations obtained in this way giving

$\dot{\gamma}=3r_{k}\frac{\alpha}{\alpha_{0}}\gamma^{\frac{1}{2}}$ (35)

that has as a solution

$\gamma(t)=\left[\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}\right]^{2}$
(36)

and, e.g. one has

$\gamma(0)=|\gamma_{11}^{(0)}\gamma_{22}^{(0)}\gamma_{33}^{(0)}|=\frac{r^{4}\sin^{2}\theta}{1-\frac{r_{g}}{r}}.$
(37)

for the exterior solution. This yields the set of equations

$\displaystyle\dot{\gamma}_{11}$ $\displaystyle=$
$\displaystyle\frac{r_{k}\alpha_{0}^{-1}\alpha}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\gamma_{11}$
$\displaystyle\dot{\gamma}_{22}$ $\displaystyle=$
$\displaystyle\frac{r_{k}\alpha_{0}^{-1}\alpha}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\gamma_{22}$
$\displaystyle\dot{\gamma}_{33}$ $\displaystyle=$
$\displaystyle\frac{r_{k}\alpha_{0}^{-1}\alpha}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\gamma_{33}.$
(38)

These can be solved exactly by

$\displaystyle\gamma_{11}(t)$ $\displaystyle=$
$\displaystyle\exp\left[r_{k}\alpha_{0}^{-1}\int_{0}^{t}dt^{\prime\prime}\frac{\alpha(t^{\prime\prime})}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t^{\prime\prime}}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\right]\gamma_{11}^{(0)}$
$\displaystyle\gamma_{22}(t)$ $\displaystyle=$
$\displaystyle\exp\left[r_{k}\alpha_{0}^{-1}\int_{0}^{t}dt^{\prime\prime}\frac{\alpha(t^{\prime\prime})}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t^{\prime\prime}}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\right]\gamma_{22}^{(0)}$
$\displaystyle\gamma_{33}(t)$ $\displaystyle=$
$\displaystyle\exp\left[r_{k}\alpha_{0}^{-1}\int_{0}^{t}dt^{\prime\prime}\frac{\alpha(t^{\prime\prime})}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t^{\prime\prime}}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\right]\gamma_{33}^{(0)}.$
(39)

We can derive the volume expansion from the equation alcu94

$\Theta=-\alpha{\rm
Tr}K=-\alpha\gamma^{ij}K_{ij}=\frac{1}{2}\gamma^{ij}{\dot{\gamma}_{ij}}$ (40)

and $K_{ij}$ are given by eq.(18). Then,

$\Theta=\frac{3}{2}\frac{r_{k}\alpha_{0}^{-1}\alpha}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\exp\left[r_{k}\alpha_{0}^{-1}\int_{0}^{t}dt^{\prime\prime}\frac{\alpha(t^{\prime\prime})}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t^{\prime\prime}}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\right].$
(41)

Here we can see the first appearance of the expansion (warp) factor given by

$U(r,\theta,t)=\exp\left[r_{k}\alpha_{0}^{-1}\int_{0}^{t}dt^{\prime\prime}\frac{\alpha(t^{\prime\prime})}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t^{\prime\prime}}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\right].$
(42)

As we will see, this is always greater than one..

##  V Geodesic equations

For the sake of completeness, we give below the geodesic equations in such a
perturbed metric. For this aim, we need to consider

$ds^{2}=-\alpha^{2}(r,t)dt^{2}+\gamma_{11}(r,\theta,t)dr^{2}+\gamma_{22}(r,\theta,t)d\theta^{2}+\gamma_{33}(r,\theta,t)d\phi^{2}.$
(43)

From this it is easy to derive the Lagrangian

$L=-\left(-\alpha^{2}(r,t){\dot{t}}^{2}+\gamma_{11}(r,\theta,t)(\dot{r})^{2}+\gamma_{22}(r,\theta,t){\dot{\theta}}^{2}+\gamma_{33}(r,\theta,t){\dot{\phi}}^{2}\right)^{\frac{1}{2}}$
(44)

where the dot means derivative with respect to the proper time $s$. Then,
using the Euler-Lagrange equations one has

$\displaystyle\frac{d}{ds}\left(\alpha^{2}{\dot{t}}\right)-\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
t}{\dot{t}}^{2}+\frac{1}{2}\frac{\partial\gamma_{11}(r,\theta,t)}{\partial
t}{\dot{r}}^{2}+\frac{1}{2}\frac{\partial\gamma_{22}(r,\theta,t)}{\partial
t}{\dot{\theta}}^{2}+\frac{1}{2}\frac{\partial\gamma_{33}(r,\theta,t)}{\partial
t}{\dot{\phi}}^{2}=0$
$\displaystyle\frac{d}{ds}\left(\gamma_{11}{\dot{r}}\right)+\frac{\partial\alpha^{2}}{\partial
r}{\dot{t}}^{2}-\frac{\partial\gamma_{11}(r,\theta,t)}{\partial
r}{\dot{r}}^{2}-\frac{\partial\gamma_{22}(r,\theta,t)}{\partial
r}{\dot{\theta}}^{2}-\frac{\partial\gamma_{33}(r,\theta,t)}{\partial
r}{\dot{\phi}}^{2}=0$
$\displaystyle\frac{d}{ds}\left(\gamma_{22}{\dot{\theta}}\right)-\frac{\partial\gamma_{11}(r,\theta,t)}{\partial\theta}{\dot{r}}^{2}-\frac{\partial\gamma_{22}(r,\theta,t)}{\partial\theta}{\dot{\theta}}^{2}-\frac{\partial\gamma_{33}(r,\theta,t)}{\partial\theta}{\dot{\phi}}^{2}=0$
$\displaystyle\frac{d}{ds}\left(\gamma_{33}\dot{\phi}\right)=0.$ (45)

Then, finally

$\displaystyle\frac{d}{ds}[\alpha^{2}(r,t){\dot{t}}]-\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
t}{\dot{t}}^{2}+\frac{1}{2}\frac{\partial\gamma_{11}(r,\theta,t)}{\partial
t}{\dot{r}}^{2}+\frac{1}{2}\frac{\partial\gamma_{22}(r,\theta,t)}{\partial
t}{\dot{\theta}}^{2}+\frac{1}{2}\frac{\partial\gamma_{33}(r,\theta,t)}{\partial
t}{\dot{\phi}}^{2}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{d}{ds}[\gamma_{11}(r,\theta,t){\dot{r}}]+\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
r}{\dot{t}}^{2}-\frac{1}{2}\frac{\partial\gamma_{11}(r,\theta,t)}{\partial
r}{\dot{r}}^{2}-\frac{1}{2}\frac{\partial\gamma_{22}(r,\theta,t)}{\partial
r}{\dot{\theta}}^{2}-\frac{1}{2}\frac{\partial\gamma_{33}(r,\theta,t)}{\partial
r}{\dot{\phi}}^{2}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{d}{ds}[\gamma_{22}(r,\theta,t){\dot{\theta}}]-\frac{1}{2}\frac{\partial\gamma_{11}(r,\theta,t)}{\partial\theta}{\dot{r}}^{2}-\frac{1}{2}\frac{\partial\gamma_{22}(r,\theta,t)}{\partial\theta}{\dot{\theta}}^{2}-\frac{1}{2}\frac{\partial\gamma_{33}(r,\theta,t)}{\partial\theta}{\dot{\phi}}^{2}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{d}{ds}[\gamma_{33}(r,\theta,t){\dot{\phi}}]$
$\displaystyle=$ $\displaystyle 0$ (46)

For a full radial motion we can set $\theta=\pi/2$ yielding

$\displaystyle\frac{d}{ds}[\alpha^{2}(r,t){\dot{t}}]-\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
t}{\dot{t}}^{2}+\frac{1}{2}\frac{\partial\gamma_{11}(r,t)}{\partial
t}{\dot{r}}^{2}+\frac{1}{2}\frac{\partial\gamma_{33}(r,t)}{\partial
t}{\dot{\phi}}^{2}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{d}{ds}[\gamma_{11}(r,t){\dot{r}}]+\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
r}{\dot{t}}^{2}-\frac{1}{2}\frac{\partial\gamma_{11}(r,t)}{\partial
r}{\dot{r}}^{2}-\frac{1}{2}\frac{\partial\gamma_{33}(r,t)}{\partial
r}{\dot{\phi}}^{2}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{d}{ds}[\gamma_{33}(r,t){\dot{\phi}}]$ $\displaystyle=$
$\displaystyle 0$ (47)

The last equation of the set can be integrated out to give

$\dot{\phi}=\frac{A}{\gamma_{33}(r,t)}$ (48)

being $A$ an integration constant. This can be substituted in the other twos
to give

$\displaystyle\frac{d}{ds}[\alpha^{2}(r,t){\dot{t}}]-\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
t}{\dot{t}}^{2}+\frac{1}{2}\frac{\partial\gamma_{11}(r,t)}{\partial
t}{\dot{r}}^{2}+\frac{1}{2}\frac{\partial\gamma_{33}(r,t)}{\partial
t}\frac{A^{2}}{\gamma_{33}^{2}(r,t)}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{d}{ds}[\gamma_{11}(r,t){\dot{r}}]+\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
r}{\dot{t}}^{2}-\frac{1}{2}\frac{\partial\gamma_{11}(r,t)}{\partial
r}{\dot{r}}^{2}-\frac{1}{2}\frac{\partial\gamma_{33}(r,t)}{\partial
r}\frac{A^{2}}{\gamma_{33}^{2}(r,t)}$ $\displaystyle=$ $\displaystyle 0$ (49)

Now, we know from eq.(IV) that

$\displaystyle\gamma_{11}(r,t)$ $\displaystyle=$ $\displaystyle
U(r,t)\gamma_{11}^{(0)}$ $\displaystyle\gamma_{33}(r,t)$ $\displaystyle=$
$\displaystyle U(r,t)\gamma_{33}^{(0)}$ (50)

and then

$\displaystyle\frac{d}{ds}[\alpha^{2}(r,t){\dot{t}}]-\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
t}{\dot{t}}^{2}+\frac{1}{2}\frac{\partial U(r,t)}{\partial
t}\left[{\dot{r}}^{2}\gamma_{11}^{(0)}+\frac{A^{2}}{U^{2}(r,t)\gamma_{33}^{(0)}}\right]$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{d}{ds}[\gamma_{11}(r,t){\dot{r}}]+\frac{1}{2}\frac{\partial\alpha^{2}}{\partial
r}{\dot{t}}^{2}-\frac{1}{2}\frac{\partial U(r,t)}{\partial
r}\left[{\dot{r}}^{2}\gamma_{11}^{(0)}+\frac{A^{2}}{U^{2}(r,t)\gamma_{33}^{(0)}}\right]$
$\displaystyle=$ $\displaystyle 0.$ (51)

This set can be solved only numerically. So, this approach does not lend
itself for a straightforward computation of the radial velocity.

##  VI Radial Velocity

We consider a particle of mass $m$ moving in our metric. The definition of
momenta is given by

$p_{\alpha}=g_{\alpha\beta}p^{\beta}.$ (52)

This yields the dispersion relation

$p_{\alpha}p^{\alpha}=-m^{2}.$ (53)

Similarly, we can derive the 4-velocity from this and is given by

$u_{\alpha}=(-\alpha^{2}{\dot{t}},\gamma_{11}{\dot{r}},\gamma_{22}{\dot{\theta}},\gamma_{33}{\dot{\phi}}).$
(54)

Then, the radial motion will be characterized by

$v_{r}=\frac{\gamma_{11}}{\alpha}\frac{dr}{dt}=\frac{U(r,\theta,t)}{\alpha(r,\theta,t)}\gamma_{11}^{(0)}\frac{dr}{dt}.$
(55)

One gets a warp factor, arising from the applied perturbation,

$U(r,t)=\exp\left[r_{k}\alpha_{0}^{-1}\int_{0}^{t}dt^{\prime\prime}\frac{\alpha(t^{\prime\prime})}{\frac{3}{2}r_{k}\alpha_{0}^{-1}\int_{0}^{t^{\prime\prime}}\alpha(t^{\prime})dt^{\prime}+\sqrt{\gamma(0)}}\right]$
(56)

and we realize that, with this geometry, we can have an exponential growth of
the radial velocity depending on the applied perturbation.

We can provide a closed form solution for a very simple case, a toy model. We
take for a perturbation

$\alpha_{1}(t)=\frac{t}{\eta}$ (57)

being $\eta$ a constant. This is a linear time increasing term. Then,

$\alpha^{2}(r,t)=\alpha_{0}^{2}(r)+\frac{t}{\eta}.$ (58)

Then,

$U(r,t)=\exp\left[r_{k}\alpha_{0}^{-1}(r)\int_{0}^{t}dt^{\prime\prime}\frac{\sqrt{\alpha_{0}^{2}(r)+\frac{t^{\prime\prime}}{\eta}}}{\frac{3}{2}r_{k}\alpha_{0}^{-1}(r)\int_{0}^{t^{\prime\prime}}\sqrt{\alpha_{0}^{2}(r)+\frac{t^{\prime}}{\eta}}dt^{\prime}+\sqrt{\gamma(0)}}\right].$
(59)

This yields

$U(r,t)=\exp\left[r_{k}\alpha_{0}^{-1}(r)\int_{0}^{t}dt^{\prime\prime}\frac{\sqrt{\alpha_{0}^{2}(r)+\frac{t^{\prime\prime}}{\eta}}}{\frac{3}{2}r_{k}\alpha_{0}^{-1}(r)\eta\left[\frac{2}{3}\left(\alpha_{0}^{2}(r)+\frac{t^{\prime\prime}}{\eta}\right)^{\frac{3}{2}}-\frac{2}{3}\alpha_{0}(r)\right]+\sqrt{\gamma(0)}}\right].$
(60)

and

$U(r,t)=\exp\left[r_{k}\alpha_{0}^{-1}(r)\frac{3}{2}\eta\int_{\frac{2}{3}\alpha_{0}(r)}^{\frac{2}{3}\left(\alpha_{0}^{2}(r)+\frac{t}{\eta}\right)^{\frac{3}{2}}}dx\frac{1}{\frac{3}{2}r_{k}\alpha_{0}^{-1}(r)\eta\left[x-\frac{2}{3}\alpha_{0}(r)\right]+\sqrt{\gamma(0)}}\right].$
(61)

Final result is

$U(r,t)=\frac{a\left(\alpha_{0}^{2}(r)+\frac{t}{\eta}\right)^{\frac{3}{2}}-b}{a\alpha_{0}^{3}(r)-b},$
(62)

being $a=r_{k}\eta\alpha_{0}^{-1}(r)$ and $b=r_{k}\eta-\sqrt{\gamma(0)}$. It
is to see that this factor is always greater than one (this value is taken for
$t=0$) and increasing as time increases.

From the formula for radial velocity we can derive the force. This will be
obtained by the first derivative of eq.(55). This yields

$\frac{dv_{r}}{d\tau}=\frac{d}{d\tau}\left[\gamma_{11}\frac{dr}{d\tau}\right].$
(63)

This gives, for a mass $M$,

$F=M\frac{dv_{r}}{d\tau}=M\frac{dt}{d\tau}\frac{d\gamma_{11}}{dt}\frac{dr}{d\tau}+\gamma_{11}\frac{d^{2}r}{d\tau^{2}}.$
(64)

This gives,

$F=M\frac{dv_{r}}{d\tau}=M\frac{1}{\alpha}\frac{d\gamma_{11}}{dt}\frac{dr}{d\tau}+\gamma_{11}\frac{d^{2}r}{d\tau^{2}}.$
(65)

In our toy model, we consider $\alpha_{0}\approx 1$ and
$\gamma_{11}^{(0)}\approx 1$, so that

$\frac{d\gamma_{11}}{dt}=\frac{\frac{a}{\eta}\left(1+\frac{t}{\eta}\right)^{\frac{1}{2}}-\frac{db}{dt}}{a-b}+\frac{a\left(1+\frac{t}{\eta}\right)^{\frac{3}{2}}-b}{(a-b)^{2}}\frac{db}{dt}.$
(66)

This yields,

$\frac{d\gamma_{11}}{dt}\approx\frac{r_{k}\left(1+\frac{t}{\eta}\right)^{\frac{1}{2}}+2r\frac{dr}{dt}}{r^{2}}+\frac{r_{k}\eta\left(1+\frac{t}{\eta}\right)^{\frac{3}{2}}-r_{k}\eta+r^{2}}{r^{4}}2r\frac{dr}{dt}.$
(67)

Then, we get

$\displaystyle F$ $\displaystyle\approx$ $\displaystyle
M\left(1+\frac{t}{\eta}\right)^{-\frac{1}{2}}\left[\frac{r_{k}}{r^{2}}\left(1+\frac{t}{\eta}\right)^{\frac{1}{2}}+\frac{2}{r}\frac{dr}{dt}+\frac{2r_{k}\eta\left(1+\frac{t}{\eta}\right)^{\frac{3}{2}}-2r_{k}\eta+2r^{2}}{r^{3}}\frac{dr}{dt}\right]\frac{dr}{dt}$
(68)
$\displaystyle+M\frac{r_{k}\eta\left(1+\frac{t}{\eta}\right)^{\frac{3}{2}}-r_{k}\eta+r^{2}}{r^{2}}\frac{d^{2}r}{d\tau^{2}}.$

with the simple kinematic law of motion $r(t)=r_{0}+v_{0}t$ it is easy to get

$\displaystyle F$ $\displaystyle\approx$ $\displaystyle
Mv_{0}\frac{r_{k}}{r^{2}(t)}+2M\left(1+\frac{t}{\eta}\right)^{-\frac{1}{2}}\frac{v_{0}^{2}}{r(t)}+$
(69) $\displaystyle
M\left(1+\frac{t}{\eta}\right)^{-\frac{1}{2}}\frac{2r_{k}\eta\left(1+\frac{t}{\eta}\right)^{\frac{3}{2}}-2r_{k}\eta+2r^{2}(t)}{r^{3}(t)}v_{0}^{2}.$

Force is non-null and dependent on the initial velocity and the sphere radius.
It is interesting to note that the force tend to 0 as time increases but this
corresponds to the unphysical case of a perturbation never turned off. This
equation simplifies a lot if we can neglect the terms dependent on $r_{k}$.
One has

$F\approx
3M\left(1+\frac{t}{\eta}\right)^{-\frac{1}{2}}\frac{v_{0}^{2}}{r(t)}.$ (70)

This result is independent on the sphere geometry or the Schwarzschild radius.
Such a perturbation is not completely physical. So, we considered some others
having the characteristic to be practically realizable. Considering the
interior solution, for a perturbation like $\alpha_{1}=At^{2}$ we get

Figure 1: Warp factor for a $t^{2}$ perturbation with an equation of motion
$r(t)=h_{0}+v_{0}t$.

,

and for a sinusoidal perturbation

Figure 2: Warp factor for a $\sin(\omega t)$ perturbation with frequency 1 MHz
and equation of motion $r(t)=h_{0}+v_{0}t+kt^{2}$ .

As expected from the toy model, the warp factor is always greater than one and
can reach significantly large values depending on the applied perturbation.

##  VII Conclusions

We have solved the Einstein equations for a strong perturbation in the case of
a spherical symmetry solution. In this case, the perturbation series reduces
to the case of a gradient expansion and the equations are amenable to an exact
analytical treatment. We were able to show that, when a perturbation is
properly applied, there appears a multiplicative warp factor on the radial
velocity that can, in this way, increase exponentially in time. This warp
effect does not require exotic energy and everything is completely in the
realm of positive energy solutions of the Einstein equations, even if as a
perturbation series.

We hope these results will find some application in the near future.

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