# Better Aligning Text-to-Image Models with Human Preference

Xiaoshi Wu1, Keqiang Sun1, Feng Zhu2, Rui Zhao2,3, Hongsheng Li1,4
1Multimedia Laboratory, The Chinese University of Hong Kong
2SenseTime Research 3Qing Yuan Research Institute, Shanghai Jiao Tong
University
4Centre for Perceptual and Interactive Intelligence (CPII)
{wuxiaoshi@link, kqsun@link, <EMAIL_ADDRESS>{zhufeng,
<EMAIL_ADDRESS>
###### Abstract

Recent years have witnessed a rapid growth of deep generative models, with
text-to-image models gaining significant attention from the public. However,
existing models often generate images that do not align well with human
aesthetic preferences, such as awkward combinations of limbs and facial
expressions. To address this issue, we collect a dataset of human choices on
generated images from the Stable Foundation Discord channel. Our experiments
demonstrate that current evaluation metrics for generative models do not
correlate well with human choices. Thus, we train a human preference
classifier with the collected dataset and derive a Human Preference Score
(HPS) based on the classifier. Using the HPS, we propose a simple yet
effective method to adapt Stable Diffusion to better align with human
aesthetic preferences. Our experiments show that the HPS outperforms CLIP in
predicting human choices and has good generalization capability towards images
generated from other models. By tuning Stable Diffusion with the guidance of
the HPS, the adapted model is able to generate images that are more preferred
by human users. The project page is available here:
https://tgxs002.github.io/align_sd_web/.

##  1 Introduction

Figure 1: Generated images often do not align well with the human aesthetic
preference and intention. Input prompts are shown below images.

Figure 2: We show that Stable Diffusion v1.4 can be adapted to better align
with the human aesthetic preference and intention when guided by the proposed
human preference classifier. The input prompt is shown below images.

The recent progress in diffusion models [22, 27, 32, 29] has enabled
impressive advancements in text-to-image generation, with many models now
being deployed in real-world applications such as DALL·E [27] and Stable
Diffusion [29]. However, public attention has also highlighted new issues,
such as the awkward combinations of limbs and facial expressions of generated
persons as shown in Fig. 2. The users usually need to cherry-pick results to
avoid these artifacts. In other words, the generated images are misaligned
with the human aesthetic preference.

To further improve the quality of generated images, it is essential to track
the ability of a model to generate human preferable images. However, it is
uncertain whether the existing evaluation metrics, such as Inception Score
(IS) [33] and Fréchet inception distance (FID) [11], are sensitive to the
aesthetic aspect of a generated image. These metrics perceive an image through
a classification-based CNN trained on ImageNet [31], which has been shown to
be biased towards image texture rather than general image contents [8], and
thus may not align well with human perception. Also, both IS and FID are
single modal evaluation metrics, which do not take user intention into
account. Some recent studies [22, 27, 3] use the CLIP [26] model as a proxy
for human judgement to evaluate the alignment between generated images and
text prompts. The CLIP [26] model is trained on a rich dataset and is believed
to capture subtle aspects of human intention better. However, it is uncertain
whether CLIP [26] can measure the quality of generated synthetic images, which
may not adhere to the same constraints as real images, such as the example
shown in Fig. 2.

In this study, we investigate the problem of human preference using a novel,
large-scale dataset of human choices on images generated by Stable Diffusion
[29] using the same prompt. The dataset comprises 98,807 diverse images
generated from user-provided prompts, along with 25,205 human choices. By
evaluating on this dataset, we find that the Inception Score (IS) [33], the
Fréchet Inception distance (FID) [11] and the CLIP score do not fully match
the human choice, which means that the human preference is a missing dimension
of image quality that is not well tracked by existing mainstream metrics.

We further train a human preference classifier on this dataset by fine-tuning
the CLIP [26] model, and define human preference score (HPS) based on it. We
validate HPS’s alignment with human choices and its generalization capability
towards other generative models through user studies. HPS can be utilized to
guide generative models towards producing human-preferred images. To this end,
we devise a simple yet effective method to adapt Stable Diffusion [29] by LoRA
[15] with aware of human preference. We conduct user studies to validate the
effectiveness of our approach. The results show that the adapted model can
better capture human intentions, and generate more preferable images, which
significantly mitigates the kind of artifact shown in Fig. 2.

Our contributions are as follows: (1) We create a large-scale dataset for
studying human preferences. To our best knowledge, this dataset is the first
of its kind that contains massive human choices on images generated with the
same prompt. (2) We find that the human choice cannot be accurately predicted
by the existing mainstream evaluation metrics, while it can be better
predicted via fine-tuning CLIP on the proposed dataset. (3) We propose a
simple yet effective method to guide the Stable Diffusion model towards
generating images with better aesthetic quality and better aligning with human
intention.

Figure 3: Examples of the collected data. The images are generated by Stable
Diffusion with the prompts shown below each row of images. The preferred
images are highlighted with red borders. More examples can be found in the
appendix.

##  2 Related Works

Text-to-image generative models. Text-to-image generative model has long been
an active research area. Mansimov _et al_. [20] show that Deep Recurrent
Attention Writer (DRAW) [9] can be conditioned on captions to generate novel
scene compositions. Generative Adversarial Networks (GANs) improve image
fidelity by training a discriminator to provide supervision for the generative
model. DALL·E [28] firstly achieves open-domain text-to-image synthesis with
the help of massive image-text pairs.

Diffusion model formulates the generative process as the inverse of the
diffusion process [36], which was improved by Song and Ermon [37] and Ho _et
al_. [13]. Dhariwal _et al_. firstly show the superiority of diffusion model
over GANs on image generation. Several following works, including DALL·E 2
[27], GLIDE [22], Imagen [32], ERNIE-ViLG [7, 41], Stable Diffusion [29],
bring the magic of text-to-image generation to the public attention. Among
these models, Stable Diffusion is an open-source model with active user
community.

Several recent works improve Stable Diffusion on different aspects. DreamBooth
[30] and ELITE [40] explore customizing Stable Diffusion to a certain object.
Feng _et al_. [6] propose a training-free method to guide diffusion model for
better compositional capabilities. It has been discovered that prompt
engineering plays an important role in generating high quality images. Hao _et
al_. [10] devise an automatic prompt engineering scheme via reinforcement
learning. Our method focus on the misalignment between the generated image and
human preference, which is orthogonal to the above mentioned topics.

Datasets of generated images. Thanks to active user communities of text-to-
image models, several databases of images generated by diffusion models have
been introduced. Lexica (lexica.art) is a large database of images generated
by Stable Diffusion and Lexica Aperture. It also provides related information
about the image, such as the prompt and guidance scale. However, the database
is closed-source, and only allows online browsing. DiffusionDB [39] is a
large-scale open-source database collected from the Stable Foundation Discord
channel, containing the text prompt and parameters for each image. SAC [25] is
a dataset of images generated from Stable Diffusion and GLIDE [22], along with
user ratings from a aesthetic survey. However, SAC only contains limited user
choices compared to our dataset.

Learning from human feedback. Human feedback has long been used in a wide
range of deep learning tasks. Christiano _et al_. [5] and Arakawa _et al_. [1]
incorporate human feedback into RL training, which is proved to accelerate the
model convergence. Krishna _et al_. [17] propose “socially situated AI”, which
significantly improves image recognition performance via interacting with
human users on Instagram. InstructGPT [23] fine-tunes GPT via a reward
function trained on human feedback, establishing the foundation for the
success of ChatGPT. [10] and [18] use similar methodology to improve text-to-
image models, which are highly related to our work. In [10], this is achieved
by augmenting the text prompt. [18] is a concurrent work that focuses more on
the exact alignment between text and image, while our work shows that the
potential of human feedback is far beyond the exact alignment when the
feedback takes into account the aesthetic preference of human.

Figure 4: Interactions in the Discord channel. The human choice is highlighted
in orange.

##  3 Human Preference Dataset

In order to get a better understanding of the human preferences on the images
generated from prompts, and to improve text-to-image generation quality, we
start by collecting a dataset of human choices.

Data collection. We utilize the “dreambot” channel on the Stable Foundation
Discord server to gather human choice data. The chat history of these channels
is obtained using the DiscordChatExporter [14] tool, which downloads the full
chat history of a Discord channel and stores it in JSON format. Among the chat
messages, a discernible pattern of interaction is observed, as depicted in
Fig. 4, which reveals human preferences. In this pattern, a user initiates a
session by sending a text prompt to the bot, which generates several images in
response. Then, the user selects a preferred image and sends it back to the
bot, along with the original text prompt. The bot will return several refined
images. This interaction follows a pre-defined grammar, which allows us to
extract human choice and related images using simple pattern matching
techniques.

Data format and statistics. Finally, we obtain a total of 98,807 images
generated from 25,205 prompts. Each prompt corresponds to several images,
among which one image is chosen by the user as the preferred one, while others
are non-preferred negatives. Each prompt corresponds with varying number of
images. 23,722 prompts have four images, 953 prompts have three images, and
530 prompts have two images. The number of images for each prompt depends on
the user’s specifications in the generation request. Notably, the dataset
exhibits a high level of diversity, with images generated across a broad range
of themes. The dataset consists choices made by 2,659 different users, and
each user contributes at most 267 choices. Examples of the collected dataset
can be found in Fig. 3. For further details on the dataset, we refer the
readers to Fig. 10 in the appendix.

Privacy and NSFW contents. We observe that a small portion of images is
generated with image condition (the condition image may be either generated or
uploaded by the user). Since user-uploaded images may contain sensitive
information or privacy, we do not include them in our dataset. For the images
with potential NSFW contents, we use the channel bot’s NSFW detector to filter
them out.

In this work, we utilize this dataset to study the existing metrics’
correlation with human preferences, which will be introduced in Sec. 4. The
dataset also serves as the training data for our human preference classifier,
which is to be introduced in Sec. 5.

##  4 Existing Metrics

In this section, we show that the current mainstream evaluation metrics are
not well correlated with human preferences on our dataset.

| IS | FID
---|---|---
Preferred | $16.27\pm 0.56$ | 38.2
Non-preferfed | $16.23\pm 0.53$ | 37.7
Table 1: The IS and FID of both preferred images and non-preferred images in
our collected dataset. The FID is computed by comparing against a subset of
images from the LAION-5B dataset that corresponds to the text inputs.

###  4.1 Metrics by Inception Net

Inception Score (IS) [33] and Fréchet inception distance (FID) [11] are two
popular metrics used to evaluate the quality of generated images. Both of them
perceive an image through an Inception Net [38] trained on ImageNet [31]. In
this section, we investigate their correlation with human choices.

Inception Score (IS) measures the quality of generated images by computing the
expected KL-divergence between the marginal class distribution over all
generated images and the conditional distribution for a particular generated
image, using the class probability predicted by the Inception Net. This metric
is expected to capture both the fidelity and diversity of generated images. To
determine the correlation between IS and human preferences, we compute IS for
both the set of preferred and non-preferred images in our dataset. For each
setting, we divide 20,000 images into 10 splits and reported the mean and
standard deviation of IS computed on them. Our results, as shown in Tab. 1,
indicate no significant difference between the preferred and non-preferred
images.

Fréchet Inception Distance (FID) measures the similarity between the embedding
feature of generated and real images. This is achieved by fitting the
embedding features into a multivariate Gaussian distribution and computing
their Fréchet distance. To define the target distribution, FID requires a set
of real images. However, in the case of images generated from user-provided
prompts, such as in our dataset, the target distribution is defined by users’
intention, which can only be inferred from text prompts. To address this, we
randomly sample 10,000 text prompts from our dataset, and for each prompt, we
query the LAION [35] dataset via the official api to find the closest image,
which is taken as a “pseudo ground truth” for that prompt. This provides a set
of real images aligned with the users’ intentions. We randomly sample 10,000
images from both the preferred and non-preferred split of the collected
dataset to compute FID with the real images. Our results, as shown in Tab. 1,
reveals no significant difference between the preferred and non-preferred
images in terms of FID. This suggests that FID may not be a reliable metric
for evaluating human preference.

Discussion. IS and FID may suffer from the following three issues when
evaluating human preference. Firstly, generated images often contain shape
artifacts, as shown in Fig. 2. However, classification-based CNNs tend to be
biased towards image texture rather than shape [8], making them be likely to
ignore shape artifacts in generated images. Additionally, domain gap can pose
a problem. While the evaluation model is trained on real images from ImageNet
[31], the generated images in our dataset exhibit a wide range of styles and
themes, from oil painting portraits to digital art of cyborgs. As a result,
the ImageNet-trained model may not have meaningful representations for these
diverse images [2]. Furthermore, these metrics are limited by their single-
modal nature, which means that they cannot infer user intentions by accessing
prompts, unless the target images are known or provided, like we do.

Figure 5: Left: training human preference classifier to derive HPS. Right:
adapting Stable Diffusion to generate preferable images. During training, the
Stable Diffusion is tuned to associate the concept of non-prefer with the
prompt prefix [Identifier]. During inference, [Identifier] is used as the
negative prompt in classifier free guidance.

###  4.2 Metrics by CLIP

Model | Text | Preference Acc. (%)
---|---|---
Random guess | ✗ | 26.1
Human | ✓ | 42.0
CLIP ViT-L/14 | ✓ | 32.9
CLIP RN50x64 | ✓ | 33.1
Aesthetic Classifier | ✗ | 31.4
HP classifier (ours) | ✓ | 43.5
Table 2: Preference Acc. refers to the human choice prediction accuracy on
5,000 user choices. The Aesthetic Classifier makes prediction without seeing
the text prompt.

Thanks to the large and diverse set of training data, CLIP is better at
encoding images from various domains compared to ImageNet-trained models.
Moreover, it can capture users’ intention by encoding text prompts, making it
a plausible choice for evaluating the alignment between a prompt and a
generated image [22, 27, 3, 29]. Aesthetic Score Predictor [34] is another
CLIP-based tool for image quality evaluation, which has been utilized to
filter the training data for Stable Diffusion [29]. In this section, we
evaluate the capability of these tools in predicting the human choice, which
is done by counting the accuracy of the human choice prediction task conducted
on a split of 5,000 samples from our dataset.

CLIP score is derived as the cosine similarity between the prompt embedding
and the image embedding computed by CLIP. We evaluated the performance on
ViT-L/14 and RN50x64 models, which are the largest open-source CLIP models for
transformer and CNN architecture. Our results, presented in Tab. 2,
demonstrate that both CLIP models exhibit superior performance over random
guessing. However, we will show in Sec. 7.1 that the CLIP score is not
sensitive to the aesthetic aspect of an image. Nevertheless, we will also show
that it can be further fine-tuned on our dataset to better align with human
preferences.

Aesthetic score  is based on a pre-trained ViT-L/14 CLIP image encoder, which
is adapted to the task of aesthetic score prediction by adding a MLP layer on
top of the CLIP image encoder. The MLP is trained on several aesthetic
datasets, including both real images and generated images (e.g., AVA [21], SAC
[25]) to predict aesthetic scores ranging from 1 to 10. Unlike CLIP, the
aesthetic classifier does not condition on the prompt, so the image with the
highest predicted score is taken as the model choice. As shown in Tab. 2, the
aesthetic classifier also exhibits better-than-chance accuracy in predicting
user choice, indicating the importance of the aesthetic aspect of an image in
human decision-making.

##  5 Human Preference Score

We firstly train a human preference classifier to predict the human choice
based on the prompt text, and then derive HPS based on the trained classifier,
which complements image quality assessments by incorporating human aesthetic
preferences.

Human preference classifier We fine-tune the ViT-L/14 version of CLIP on our
dataset to better align with human preferences. Each sample in the training
set contains one prompt along with $n\in\\{2,3,4\\}$ images, among which only
one image is preferred by the user. The model is trained to maximize the
similarity between the embedding of the text prompt computed by the CLIP text
encoder and the embedding of the preferred image computed by the CLIP visual
encoder, while minimize the similarity for non-preferred images. By fine-
tuning on human choices of generated images, the model is encouraged to better
align with human preferences.

Human preference score (HPS) is derived from the human preference classifier.
We define HPS as:

$\mathrm{HPS}(\mathrm{img},\mathrm{txt})=100\cos(\mathit{enc}_{v}(\mathrm{img}),\mathit{enc}_{t}(\mathrm{txt})),$

where $\mathit{enc}_{v}$ and $\mathit{enc}_{t}$ are the visual encoder and the
text encoder of the human preference classifier. We multiply the cosine
similarity by a factor of 100 for better visualization.

##  6 Better Aligning Stable Diffusion with Human Preferences

HPS can be used to guide diffusion-based generative models to better align
with human users. We argue that the misalignment between generated images and
human preferences is a problem of missing “awareness” rather than model
capacity. To address this issue, we propose to adapt the generative model by
explicitly distinguishing preferred images from non-preferred ones. Our
solution is straightforward and intuitive. We construct another dataset
consisting of prompts and their newly generated images, which we categorize as
either preferred or non-preferred using our previously trained human
preference classifier. For the non-preferred images, we modify their
corresponding prompts by prepending a special prefix. By adapting Stable
Diffusion on this dataset via LoRA [15], we enhance the model’s ability to
learn the concept of non-preferred images, which can subsequently be avoided
during inference.

Constructing training data. We construct the training data from the
“large_first_1m” split of DiffusionDB [39], and a subset of the pre-train
dataset of Stable Diffusion (LAION-5B) for regularization. DiffusionDB [39] is
a large-scale dataset of generated images along with their text prompts. For
images from DiffusionDB, we firstly compute HPS for each image-prompt pair.
After that, we group the images by their prompts, and for each prompt $T$, we
add the image $I^{*}$ with highest HPS into the training data if it passes the
following criteria:

$p>\frac{\alpha}{n},$

where $n$ is the number of images with the same prompt, $\alpha$ is a hyper-
parameter that controls the selectivity. $p$ is given by:

$p=\frac{\exp(HPS(I^{*},T))}{\sum_{I\in B}\exp(HPS(I,T))},$

where $B$ is the set of images with the same prompt. Similarly, we construct
the non-preferred subset by the same criteria, but using negative HPS.
Finally, we get a mixed dataset of generated images and real images, where the
non-preferred generated images are identified by their prompt prefix.

Adapting Stable Diffusion. We adopt LoRA [15] to adapt Stable Diffusion to the
training data, in which the parameters of the original model are kept frozen,
and the {key, query, value, out} projection matrices are augmented with a low-
rank residual. LoRA does not add new parameters to the model, since the
learned projection matrices can be merged into the base model once trained.
During training, we use the prompt as caption for generated images. For non-
preferred images, we prepend a special identifier before each of their
captions (we choose “Weird image.” as the special identifier in our case).
During inference, the special identifier is used as the negative prompt for
classifier-free guidance [12] to avoid generating non-preferred images.

Figure 6: Correlation between HPS and CLIP score. While the CLIP score
emphasises more on the direct matching between the image content and the text
prompt, HPS emphasises more on the aesthetic quality of images.

##  7 Experiments

In this section we firstly validate the reliability of HPS in Sec. 7.1, and
then in Sec. 7.2, we introduce our experiments of adapting Stable Diffusion.

Figure 7: Comparison of images generated by the original model, the
regularization only model, and our adapted model. “Regularization Only” refers
to a head-to-head setting against “Adapted Model”, where Stable Diffusion is
adapted without HPS-labeled images. Images in the same row are generated with
the same prompt and random seed. The prompts are sampled from DiffusionDB. The
adapted model can better capture the user intention from the prompt, and
generate more preferable images with less artifacts.

###  7.1 HPS

Implementation details of human preference classifier.  We use 20,205 samples
from our dataset during training, which contains 20,205 prompts and 79,167
images. We use the ViT-L/14 version of CLIP in our experiments. We fine-tune
the last 10 layers of the CLIP image encoder and the last 6 layers of the text
encoder. The model is trained by the AdamW optimizer [16] with a learning rate
of $1.7\times 10^{-5}$ for 1 epoch. The batch size is 5. The learning rate
decays with a cosine learning rate schedule. Weight decay is set as $3.1\times
10^{-3}$. Instead of using the original data augmentation of random resized
crop, we directly resize the longest edge of the image to 224, and then pad
zeros to make the shorter edge increase to 224. We empirically find that
fixing the aspect ratio of the image is beneficial. The hyper parameters are
tuned via Bayesian optimization.

| Agreement (%)
---|---
Human vs. Human | $63.5\pm 4.3$
CLIP vs. Human | $56.8\pm 1.7$
HPS vs. Human | $61.5\pm 1.1$
Table 3: Agreement on comparing images generated by Stable Diffusion and
DALL·E.

Alignment with human.  As shown in Tab. 2, the trained model significantly
outperforms CLIP in the human choice prediction task. Due to the strong
diversity of human preference, the accuracy is even higher than our human
participants.

Generalization.  We evaluate HPS’ generalization capability towards other
generative models by user studies. In this experiment, we let the human
preference classifier and several human participants to evaluate 398 pairs of
images. In each pair, the images are generated by DALL·E [27] and Stable
Diffusion [29] with the same text prompt. The prompts are randomly sampled
from DiffusionDB [39], which is a large database of images and prompts sourced
from the Stable Foundation Discord channel. We filter out the NSFW prompts by
the indicator provided in DiffusionDB [39].

In Tab. 3, we evaluate the agreement between the predictions from human, CLIP
and HPS. The agreement is computed by averaging the similarity of prediction
of each participant. HPS is better aligned with human preference compared to
CLIP socre, and its agreement with human approaches the agreement between
humans. It shows that HPS can generalize towards images generated by other
models. We refer the readers to the supplementary material for a full list of
images and choices made in this user study.

Correlation with CLIP score.  In Fig. 6, we visualize the correlation between
HPS and CLIP score. The text prompts are randomly sampled from the COCO
Captions [4] dataset, and the images are generated by Stable Diffusion [29].
We can see that HPS has a positive correlation with CLIP score, but emphasises
more on the aesthetic quality of an image. However, HPS put less importance on
the direct matching between image contents and text prompts, which can be
interpreted as a visual analogy of “alignment tax” introduced in [24].

###  7.2 Better Aligning Stable Diffusion with Human Preferences

Implementation details.  We use the Stable Diffusion [29] v1.4 for all our
experiments. $\alpha$ is set to 2.0 for both preferred images and non-
preferred images when constructing the training set. The constructed training
set contains 37,572 preferred generated images, 21,108 non-preferred generated
images. The regularization images are from a 625k subset of LAION-5B filtered
by the aesthetic score predictor with a threshold of 6.5. 200,231
regularization images participates training. We only fine-tune the UNet of
Stable Diffusion, while keeping the VAE and the text encoder frozen during
training. The rank is set to 32 in LoRA [15]. The LoRA weights are trained for
10k iterations with the AdamW [16] optimizer with a learning rate of $1\times
10^{-5}$ and a weight decay of $1\times 10^{-2}$, which is kept constant
during training. We use a batch size of 40 in our experiments. For inference,
we run the diffusion process by 50 steps for each image with PNDM [19] noise
scheduler. We use the default guidance scale of 7.5 for classifier-free
guidance [12].

Figure 8: Human evaluation results on images generated from 100 randomly
sampled prompts. The color represents the number of positive votes received
from 5 participants. Figure 9: The adapted model can hardly improve images
that are already well aligned with human aesthetics. The left/right image is
generated by the original/adapted model. Both images are generated using the
same prompt and random seed.

Human evaluation.  We compare our trained model with the original Stable
Diffusion by conducting user studies. In this study, we randomly sample 100
user-provided prompts from DiffusionDB [39]. For each prompt, we generate an
image from both models with the same random seed for fair comparison. By doing
this, we get 100 pairs of generated images as the test set. We ask 5
participants to read the prompt, and then choose between the image generated
by our trained model and the original Stable Diffusion based on to what extent
the generated image satisfies the prompt writers’ intention.

In Fig. 3, we visualize our result by showing the percentages of images with
different numbers of positive votes. The adapted model significantly
outperforms the original model. 72% of the images generated by the adapted
model has more than 3 votes, while the number is 28% for the original model.
It should be noted that a large portion of images generated by the original
model is already good enough, the adapted model struggles to further improve
them, as shown in Fig. 9. The image quality improves the most when there is
obvious artifacts. We provide a full list of images and user choices in this
user study in the supplementary material.

Qualitative Evaluation In Fig 7, we show some typical cases of improvement. We
compare the original model, the regularization only model, and the adapted
model. The adapted model is trained with both real regularization images and
generated images with HPS preference labels. The regularization only model is
a head-to-head comparison with the adapted model, which is trained by removing
the generated images from the training set, and is trained exclusively on
regularization images for the same number of steps. The results show that the
adapted model can better capture the user intention from the prompt, as shown
in the first row. The last three rows show that training with generated images
mitigates the problem of unnatural limbs. We refer the readers to Fig. 7 and
Fig. 12 in the appendix for more examples.

##  8 Limitations

There are several limitations about the dataset. The collected dataset
contains generated prompts and images of public figures. We choose to mark
them out instead of removing them to keep the diversity of the dataset.
Despite the diversity of the dataset, we are also aware that it only
represents the aesthetic preference of a small portion of people in the world,
and it may be biased towards the certain group of people that are active in
the Stable Foundation Discord channel. Another potential bias about this
dataset is that a large portion of text prompts are written by experienced
Stable Diffusion users. These prompts are very likely to be tweaked to
activate the potential of Stable Diffusion, and deviates from normal language
habits.

##  9 Conclusion

In this work, we study the human preference on a large-scale dataset of
generated images. We find that the previous evaluation metrics for generative
models are not well aligned with human preferences, but the CLIP model can be
fine-tuned into a human preference classifier to better align with the human
choice. Then, we show a simple yet effective method to adapt the generative
model to generate more preferable images with the guidance of human preference
score. We hope our work can inspire the community to explore new possibilities
of the human-aligned AI research.

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##  Appendix A Datasheet

###  A.1 Motivation

### Why was the dataset created?

The dataset was created to facilitate future academic Computer Vision research
about human aesthetic preference.

### Who created this dataset (_e.g_. which team, research group) and on behalf
of which entity (_e.g_. company, institution, organization)?

The dataset was created by researchers at MMLab, The Chinese University of
Hong Kong.

###  A.2 Composition

### What do the instances that comprise the dataset represent (_e.g_.
documents, photos, people, countries)? Are there multiple types of instances?
(_e.g_. movies, users, ratings; people, interactions between them; nodes,
edges)

The instances are prompts and generated images, along with human preference
choices among the images generated by the same prompt.

### Are relationships between instances made explicit in the data (_e.g_.
social network links, user/movie ratings, etc.)?

Yes, instances generated by the same user are identified by the same user id,
which is anonymized for privacy.

### How many instances are there? (of each type, if appropriate)?

There are 25,205 instances in the dataset.

### What data does each instance consist of? “Raw” data (_e.g_. unprocessed
text or images) or Features/attributes? Is there a label/target associated
with instances? If the instances related to people, are sub-populations
identified (_e.g_. by age, gender, etc.) and what is their distribution?

Each instance consists of $n\in{2,3,4}$ image, one prompt and one human
choice.

### Is any information missing from individual instances? If so, please
provide a description, explaining why this information is missing (_e.g_.
because it was unavailable). This does not include intentionally removed
information, but might include, _e.g_. redacted text.

Yes, we omit the specific parameters for generating the images, such as
diffusion steps and guidance scale. They are omitted because we are more
interested in the users’ preference about the generated images, rather than
how they are created. Also, since the same batch of images (among which users
make comparisons) are always generated with the same set of parameters except
the random seed, they are irrelevant variables when studying human
preferences.

### Is everything included or does the data rely on external resources?

The dataset is self-contained.

### Are there recommended data splits and evaluation measures? (_e.g_.
training, development, testing; accuracy or AUC)

In our experiments, we use a training set of 20,205 instances and validation
set of 5,000 images, which will be made public. We recommend using accuracy
(%) with one decimal place.

### Are there any errors, sources of noise, or redundancies in the dataset?

No.

### Is the dataset self-contained, or does it link to or otherwise rely on
external resources (_e.g_. websites, tweets, other datasets)?

The dataset is self-contained.

### Does the dataset contain data that might be considered confidential
(_e.g_. data that is protected by legal privilege or by doctorpatient
confidentiality, data that includes the content of individuals non-public
communications)?

No, the dataset is collected from the Stable Foundation Discord server, which
is publicly available for any user with an account.

### Does the dataset contain data that, if viewed directly, might be
offensive, insulting, threatening, or might otherwise cause anxiety? If so,
please describe why.

We collect images and their prompts from the Stable Foundation discord server.
Even though the discord server has rules against users sharing any NSFW (not
suitable for work, such as sexual and violent content) and illegal images, our
dataset still contains some NSFW images and prompts that were not removed by
the server moderators.

### Does the dataset relate to people?

Yes, the prompts are written by users and the choices are made by users.

### Does the dataset identify any subpopulations (_e.g_. by age, gender)?

No.

### Is it possible to identify individuals (_i.e_. one or more natural
persons), either directly or indirectly (_i.e_. in combination with other
data) from the dataset?

No.

### Does the dataset contain data that might be considered sensitive in any
way (_e.g_. data that reveals racial or ethnic origins, sexual orientations,
religious beliefs, political opinions or union memberships, or locations;
financial or health data; biometric or genetic data; forms of government
identification, such as social security numbers; criminal history)?

The dataset may contain sensitive data, because the prompts written by users
may contain sensitive information, such as public figures and religious
beliefs.

### What experiments were initially run on this dataset? Have a summary of
those results.

It has been used to validate the correlation between human preference and
several popular image quality evaluation metrics, and serve as the training
data for a human preference classifier. The results show that the tested
metrics do not correlate well with human preference, and the correlation of
the ViT-L/14 version of CLIP can be improved via fine-tuning on the dataset.

###  A.3 Data Collection Process

### How was the data associated with each instance acquired?

The prompts, images and human choices are directly observable from the Stable
Foundation Discord server.

### What mechanisms or procedures were used to collect the data (_e.g_.
hardware apparatus or sensor, manual human curating, software program,
software API)?

Automatic scraping procedures were used to collect the data.

### If the dataset is a sample from a larger set, what was the sampling
strategy (_e.g_. deterministic, probabilistic with specific sampling
probabilities)?

The dataset is not a sample of a larger set.

### Who was involved in the data collection process (_e.g_. students, crowd-
workers, contractors) and how were they compensated (_e.g_. how much were
crowdworkers paid)?

The authors of this paper were solely involved in the data collection process.

### Over what time-frame was the data collected?

The dataset covers the chat history of dreambot channels between Dec. $2^{nd}$
2022 and Jan. $18^{th}$ 2023.

### Were any ethical review processes conducted (_e.g_. by an institutional
review board)?

No official processes were conducted, due to the public nature of the data on
Discord channel.

### Does the dataset relate to people?

No.

### Did you collect the data from the individuals in question directly, or
obtain it via third parties or other sources (_e.g_. websites)?

The data was obtained from public messages in the Discord server.

### Has an analysis of the potential impact of the dataset and its use on data
subjects (_e.g_. a data protection impact analysis)been conducted?

No analysis has been conducted.

###  A.4 Data Preprocessing

### What preprocessing/cleaning was done? (_e.g_. discretization or bucketing,
tokenization, part-of-speech tagging, SIFT feature extraction, removal of
instancess, processing of missing values)?

No preprocessing is done on the images and prompts.

###  A.5 Uses

### Has the dataset been used for any tasks already? If so, please provide a
description.

As described in the paper, this dataset has been used for analysis about
several image quality evaluation metrics and training the proposed human
preference classifier.

### Is there a repository that links to any or all papers or systems that use
the dataset?

No.

### What (other) tasks could the dataset be used for?

It can be used for tasks related to human preference on generated images.

### Is there anything about the composition of the dataset or the way it was
collected and preprocessed/cleaned/labeled that might impact future uses?

Yes. As discussed in Sec. 8, the dataset is biased towards the preference of
the certain group of people that are active in the Stable Foundation Discord
server.

### Are there tasks for which the dataset should not be used?

No.

###  A.6 Data Distribution

### Will the dataset be distributed to third parties outside of the entity
(_e.g_. company, institution, organization) on behalf of which the dataset was
created? If so, please provide a description

Yes. Researchers at academic institutions will be able to request access to
the dataset.

### How will the dataset be distributed? (_e.g_. tarball on website, API,
GitHub; does the data have a DOI and is it archived redundantly?)

We will provide download links for researchers on a GitHub repository.

### When will the dataset be distributed?

Before April 15, 2023.

### Will the dataset be distributed under a copyright or other intellectual
property (IP) license, and/or under applicable terms of use (ToU)?

We will provide a terms of use agreement with the dataset. The dataset as a
whole will be distributed under a non-commercial license.

### Have any third parties imposed IP-based or other restrictions on the data
associated with the instances? If so, please describe these restrictions, and
provide a link or other access point to, or otherwise reproduce, any relevant
licensing terms, as well as any fees associated with these restrictions.

No.

### Do any export controls or other regulatory restrictions apply to the
dataset or to individual instances? If so, please describe these restrictions,
and provide a link or other access point to, or otherwise reproduce, any
supporting documentation.

Unknown.

###  A.7 Dataset Maintenance

### Who is supporting/hosting/maintaining the dataset?

The authors of this paper are maintainers of this dataset.

### How can the owner/curator/manager of the dataset be contacted (_e.g_.
email address)?

By email<EMAIL_ADDRESS>.

### Is there an erratum?

At this time, we are not aware of errors in our dataset. However, we will
create an erratum as errors are identified.

### Will the dataset be updated? If so, how often and by whom? How will
updates be communicated? (_e.g_. mailing list, GitHub)

The dataset will be updated by the authors on an at-will basis (but no more
than once a month).

### If the dataset relates to people, are there applicable limits on the
retention of the data associated with the instances (_e.g_. were individuals
in question told that their data would be retained for a fixed period of time
and then deleted)? If so, please describe these limits and explain how they
will be enforced.

No such limits are established.

### Will older versions of the dataset continue to be
supported/hosted/maintained?

N/A

### If others want to extend/augment/build on this dataset, is there a
mechanism for them to do so? If so, is there a process for tracking/assessing
the quality of those contributions. What is the process for
communicating/distributing these contributions to users?

There will not be a mechanism to build on top of the dataset.

Figure 10: More examples of the collected data. The images are generated by
Stable Diffusion with the prompts shown below each group of images. The
preferred images are highlighted with red borders.

Figure 11: The adapted model generates images with less artifacts. Images in
the same group are generated with the same prompt and random seed.

Figure 12: The adapted model generates images that better capture user
intentions. Images in the same group are generated with the same prompt and
random seed.

##  Appendix B More Dataset Examples

See Fig. 10 for more examples.

##  Appendix C More Visualization

See Fig. 12 and Fig. 12 for more visualizations. We show that the adapted
model generates images with less artifacts and are better aware of users’
intentions.


those in Theorem 2.1), thus extending the results of [BGW19] and [Gal20]? We
note that the colored six-vertex model can be viewed as a discrete time
version of the colored ASEP (Asymmetric Simple Exclusion Process), which is a
generalization of the colored TASEP, such that when the Poisson clock on edge
$(x,x+1)$ rings, and the color on site $x$ is larger than the color on site
$x+1$, we swap the particles with some probability $p\in[0,1]$. The colored
TASEP then corresponds to the colored six-vertex model with the parameter
$b_{2}=0$. By adapting our arguments to the discrete time setting, and using
invertability of the transition matrices instead of analyticity of the
transition distribution functions, we can get extensions of the height
function shift-invariance of the colored six-vertex model, when $b_{2}=0$.
However, for the general case where $b_{2}>0$, one might need to prove some
shift-invariance for the multi-time distribution of the colored ASEP with
$p>0$. Our arguments for the colored TASEP rely on the following observation:
for the trajectories of the $i$-th rightmost particle in some $\mu^{A}$, and
of the $j$-th leftmost hole in some $\mu^{A^{\prime}}$, they would be
independent once they cross each other; but this is not true for the colored
ASEP. Some new ideas are needed to prove (or disprove) possible extensions to
the ASEP setting.

##  7 Joint equal in distribution between OSP and LPP

In this section we prove Theorem 1.4 and its implications.

Recall that for the colored TASEP, we encoded it with a family of TASEPs
$\mu^{A}$, for $A\in\mathbb{Z}$, with different initial conditions and coupled
using the same Poisson clocks. We similarly encode OSP on $\\{1,\ldots,N\\}$,
with a family of of TASEPs on $\\{1,\ldots,N\\}$, denoted as $\nu^{N,A}$, for
$A\in\\{1,\ldots,N-1\\}$, such that $\nu^{N,A}_{0}(x)=0$ for $x\leq A$, and
$\nu^{N,A}_{0}(x)=\infty$ for $x>A$ (as before we use $0$ to denote particles
and $\infty$ to denote holes). Then we have

$U_{N}(A)=\inf\\{t>0:\nu^{N,A}_{t}(x)=0,\forall N-A+1\leq x\leq
N;\nu^{N,A}_{t}(x)=\infty,\forall 1\leq x\leq N-A\\},$

i.e., $U_{N}(A)$ is the time for the last jump of $\nu^{N,A}$.

We use the following input from [AHR09]. To state it, we consider two kinds of
operators on particle-hole configurations on $\mathbb{Z}$ introduced there.
For each $k\in\mathbb{N}$, let $R_{k}$ be the _cut-off operator_ , such that
for any configuration, it keeps the $k$ rightmost particles, and change all
other particles to holes (unless there are less than $k$ particles, in which
case the configuration keeps the same). For each $n\in\mathbb{Z}$ we let
$B_{n}$ be the _push-back operator_ , which pushes all particles onto
$(-\infty,n]\cap\mathbb{Z}$: for the $j$-th rightmost particle, it is moved to
site $x\wedge n+1-j$, if it is originally at site $x$.

We couple the colored TASEP with OSP on $\\{1,\ldots,N\\}$, such that the
Poisson clocks on $(k,k+1)$ for each $1\leq k\leq N-1$ are the same.

######  Lemma 7.1 ([AHR09, Lemma 3.3]).

Under the above coupling, we have that $B_{N}R_{A}\mu^{A}$ on
$\\{1,\cdots,N\\}$ is the same as $\nu^{N,A}$, for each $1\leq A\leq N-1$.

We can now prove the joint equal in distribution between $\mathbf{U}_{N}$ and
passage times in LPP, from Theorem 2.4 and using this connection.

###### Proof of Theorem 1.4.

By Lemma 7.1, for each $1\leq A\leq N-1$, if and only if $t\leq
T^{A}_{N-A,A}$, there are at least $A$ particles on or to the right of site
$N-A+1$ in $\mu^{A}_{t}$, thus in $\nu^{N,A}_{t}$; and this is equivalent to
that $\nu^{N,A}_{t}(x)=0,\forall N-A+1\leq x\leq N$ and
$\nu^{N,A}_{t}(x)=\infty,\forall 1\leq x\leq N-A$. Thus we must have that
$T^{A}_{N-A,A}=U_{N}(A)$.

Via the connection between LPP and TASEP, as stated in Section 1, and symmetry
between the two coordinates in LPP, we have that
$\\{T^{1}_{N-A,A}\\}_{A=1}^{N-1}$ has the same distribution as
$\\{L_{(1,1),(A,N-A)}\\}_{A=1}^{N-1}$. It remains to show that it also has the
same distribution as $\\{T^{A}_{N-A,A}\\}_{A=1}^{N-1}$. For this we apply
Theorem 2.4: for each $1\leq i<N-1$, we have that $\\{T^{A\wedge
i}_{N-A,A}\\}_{A=1}^{N-1}$ has the same distribution as
$\\{T^{A\wedge(i+1)}_{N-A,A}\\}_{A=1}^{N-1}$. Thus $\\{T^{A\wedge
i}_{N-A,A}\\}_{A=1}^{N-1}$ has the same distribution for each $1\leq i\leq
N-1$. By taking $i=1$ and $i=N-1$ we get the conclusion. ∎

We now deduce the asymptotic results from Theorem 1.4, using existing results
of LPP.

###### Proof of Theorem 1.5 and 1.6.

By Theorem 1.4, it suffices to consider the function

$x\mapsto\frac{(y(1-y))^{1/6}}{(1+2\sqrt{y(1-y)})^{2/3}N^{1/3}}\\\
\times(L_{(1,1),(\lfloor yN+xN^{2/3}\rfloor,N-\lfloor
yN+xN^{2/3}\rfloor)}-(1+2\sqrt{y(1-y)})N-\frac{1-2y}{\sqrt{y(1-y)}}xN^{2/3}).$

We have that it converges to $\mathcal{A}_{2}(x)-x^{2}$ weakly, in the
topology of uniform convergence on compact sets. The finite dimensional
convergence follows from e.g. [BF08, BP08], and it can be upgraded to uniform
convergence on compact sets using a tightness result from [Pim18]. See [BGZ21,
Theorem 3.8] for a proof in the $y=1/2$ case, which also goes through
essentially verbatim for any fixed $y\in(0,1)$.

For Theorem 1.6, we note that $k_{*}=\operatorname{argmax}_{1\leq k\leq
N-1}L_{(1,1),(k,N-k)}$. Then we just need the following two additional facts:
the process $\mathcal{A}_{2}(x)-x^{2}$ attains its maximum at a unique point,
and that $\frac{k_{*}-N/2}{N^{2/3}}$ is tight as $N\to\infty$. The first fact
is proved in [CH14]. The tightness follows from a transversal estimate in LPP,
e.g. [BGZ21, Proposition 4.2], which implies the following: there exist
$c_{1},c_{2}>0$, such that for any $\phi$ large enough and any
$N\in\mathbb{N}$,

$\mathbb{P}\left[\max_{1\leq k\leq N-1,|k-N/2|>\phi
N^{2/3}}L_{(1,1),(k,N-k)}>2N-c_{1}\phi^{2}N^{1/3}\right]<e^{-c_{2}\phi^{3}}.$

We also have that $\mathbb{P}[L_{(1,1),(\lfloor N/2\rfloor,\lceil
N/2\rceil)}-2N<-\phi N^{1/3}]<c^{-1}e^{-c\phi^{3}}$ for some constant $c>0$.
This is due to that $L_{(1,1),(\lfloor N/2\rfloor,\lceil N/2\rceil)}$ has the
same distribution as the largest eigenvalue of $X^{*}X$, where $X$ is an
$\lfloor N/2\rfloor\times\lceil N/2\rceil$ matrix with i.i.d. standard complex
Gaussian entries (see [Joh00, Proposition 1.4]); and the estimate is from
[LR10, Theorem 2]. With these bounds on the passage times we get tightness of
$\frac{k_{*}-N/2}{N^{2/3}}$. ∎

###### Proof of Theorem 1.7.

It is proved in [BBS21, Theorem 2.1] that for the function

$x\mapsto L_{(1,1),(\lfloor yN\rfloor+x,N-\lfloor
yN\rfloor-x)}-L_{(1,1),(\lfloor yN\rfloor,N-\lfloor yN\rfloor)},$

its total variation distance to $g_{N}$ decays to zero (actually it is stated
for the total variation distance between the difference of the passage times,
and the so-called Busemann function, which has the distribution of a two-sided
random walk). Then the conclusion immediately follows from Theorem 1.4. ∎

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# Rotation in the Geometric Scalar Theory of Gravity

Mario Novello  and Vicente Antunes

###### Abstract

We present solutions corresponding to rotational configurations in the
recently proposed Geometric Scalar Gravity (GSG) theory. The solutions
obtained here have the important property that the associated closed time-like
curves are always restricted to a compact domain of space-time. We compare
this property with those of analogous solutions in General Relativity (GR).

##  1 Introduction

In the first attempts to formulate a relativistic theory of gravitation made
by Nordström [1], Einstein and Grossmann [2] in the early 1910’s, the
gravitational field was described by the special relativistic generalization
of the Newtonian potential, which was assumed to propagate in a conformally
flat space-time and to obey a generalised Poisson equation with the trace of
the energy-momentum tensor of matter as its source. The failure of that
program on both theoretical and observational grounds opened the way for the
emergence of General Relativity (GR) and its rapid progression towards its
current status as the paradigmatic theory of gravity. During the last decades,
however, GR itself has been facing a growing pressure in both observational
and theoretical fronts, a situation that has prompted the search of
alternatives to GR. Recently, a theory of gravitation based on a single scalar
field that avoids the drawbacks of those early formulations was proposed by
Novello et al. [3]. In particular, unlike Nordström and Einstein-Grossmann
proposals, the gravitational interaction in this new approach — called
Geometric Scalar Gravity (GSG) — is described in terms of a modified space-
time geometry, in accordance with the fundamental principle of GR.
Furthermore, the source of the scalar field in GSG is not simply the trace of
the energy momentum-tensor of matter. Some cosmological and astrophysical
scenarios within the GSG framework were studied in a series of papers [4, 5].

The purpose of the present work is to present novel geometries in the
framework of GSG corresponding to rotational configurations. It is well known
that many solutions to Einstein field equations displaying rotational effects,
such as Lanczos-van Stockum dust, Gödel’s rotating universe and Taub-NUT
space-time, exhibit closed timelike curves (CTC’s) that are not hidden by any
horizon, a feature that implies (global) causality violations. Furthermore,
most of these solutions cannot be ruled out on theoretical grounds, a fact
that exposes the incompleteness of GR and its need to be complemented by
chronology protection conjectures.

Here, we analyze two rotational configurations in GSG: a vacuum solution
corresponding to a compact domain, called I, and that corresponding to an
exterior domain filled with a perfect fluid, named domain II. It is shown that
in the interior (vacuum) solution, there exists CTCs which are constrained to
stay within the compact domain I ($r<r_{m}.$). Besides, we show that there are
no CTCs in the outside metric, i.e. in domain II. We compare this property
with a similar situation that occurs in Gödel’s solution.

##  2 A brief review of Geometric Scalar Gravity

The GSG theory is based on the following postulates:

  * •

The gravitational interaction is described by a scalar field $\Phi$;

  * •

The field $\Phi$ satisfies a nonlinear dynamics;

  * •

The theory satisfies the principle of general covariance which, in other
words, means that GSG is not a theory restricted to the realm of special
relativity;

  * •

All kinds of matter and energy interact with $\Phi$ only through the pseudo-
Riemannian metric

$q^{\mu\nu}=\alpha\eta^{\mu\nu}+\frac{\beta}{w}\partial^{\mu}\Phi\partial^{\nu}\Phi,$
(2.1)

where $\alpha$ and $\beta$ are functions of $\Phi$ and
$w\equiv\partial_{\mu}\Phi\partial_{\nu}\Phi\eta^{\mu\nu};$

  * •

Test particles follow geodesics relative to the gravitational metric
$q_{\mu\nu};$

  * •

$\Phi$ is related in a nontrivial way with the Newtonian potential $\Phi_{N}$;

  * •

The background Minkowski metric $\eta^{\mu\nu}$ is not observable. Matter and
energy interact gravitationally only through the combination

$\alpha\eta^{\mu\nu}+\frac{\beta}{w}\partial^{\mu}\Phi\partial^{\nu}\Phi$

and its derivatives;

  * •

Electromagnetic waves propagate along null geodesics relative to the metric
$q^{\mu\nu};$

  * •

The contravariant definition of the metric in GSG, that is, equation (2.1) is
an exact expression. The corresponding covariant expression, defined as the
inverse $q_{\mu\nu}q^{\nu\lambda}=\delta^{\lambda}_{\mu}$, is also a binomial
expression

$q_{\mu\nu}=\frac{1}{\alpha}\eta_{\mu\nu}-\frac{\beta}{\alpha(\alpha+\beta)w}\partial_{\mu}\Phi\partial_{\nu}\Phi,$
(2.2)

where $w=\partial_{\mu}\Phi\partial_{\nu}\Phi\eta^{\mu\nu}.$

According to these postulates, the action for the matter fields can be written
as

$S_{m}=\int\sqrt{-q}\,L_{m}\,d^{4}x,$ (2.3)

where $q$ is the determinant of $q_{\mu\nu}$ and we are taking $\kappa=c=1$.
The dynamics of the scalar field (see [3]) is given by

$\sqrt{{V}}\square\Phi=\kappa\chi,$ (2.4)

where

$\chi\equiv\,-\,\frac{1}{2}\left[\,T+\left(2-\frac{V^{\prime}}{2V}\right)E+C^{\lambda}_{~{}\,;\lambda}\right],$
(2.5) $E\equiv\frac{T^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi}{\Omega},$
(2.6)
$C^{\lambda}\equiv\frac{(\alpha^{2}V-1)}{\Omega}\,\left(T^{\lambda\mu}-Eq^{\lambda\mu}\right)\partial_{\mu}\Phi.$
(2.7)

Here, $T$ is the trace of the energy-momentum tensor, $V^{\prime}=dV/d\Phi$
and $\Omega=\partial_{\mu}\Phi\partial_{\nu}\Phi q^{\mu\nu}=(\alpha+\beta)w.$
In the case examined in [3] $\alpha=\exp(-2\Phi)$ and the values of $\beta$
and $V$ are given by

$4\,\alpha^{3}\,V=(\alpha-3)^{2},$ $\alpha+\beta=\alpha^{3}V.$

Note that the vacuum dynamics of $\Phi$ can be written equivalently in the
unobservable Minkowski background by the action

$S_{gravity}=\int\sqrt{-\eta}\,\frac{V(\Phi)}{w}\,\partial_{\mu}\Phi\,\partial_{\nu}\Phi\,\,\eta^{\mu\nu}.$
(2.8)

##  3 Rotation in Vacuum

Let us write the line element associated with the background metric in the
form

$ds^{2}_{M}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=dt^{2}+2h(r)\,dtd\phi-
dr^{2}-dz^{2}+g(r)\,d\phi^{2}.$ (3.1)

In the case where the functions $g(r)$ and $h(r)$ are given by

$h(r)=h_{0},\hskip 14.22636ptg(r)=h_{0}^{2}-w_{0}^{2}\,r^{2},$

where $h_{0}$ and $w_{0}$ are constants, the metric (3.1) is nothing but the
flat Minkowski geometry. The $q$-metric from equation (2.1) leads to the
associated line element for the foreground geometry

$\displaystyle ds^{2}$ $\displaystyle=q_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=\frac{1}{\alpha}\,dt^{2}-\frac{1}{(\alpha+\beta)}\,dr^{2}+\frac{(h_{0}^{2}-w^{2}r^{2})}{\alpha}\,d\phi^{2}-\frac{1}{\alpha}\,dz^{2}$
$\displaystyle+\frac{2h_{0}}{\alpha}\,dt\,d\phi.$ (3.2)

###  3.1 Vacuum solution

Let us consider the vacuum dynamics of the scalar field

$\Box\,\Phi=\frac{1}{\sqrt{-q}}\partial_{\mu}(\sqrt{-q}\,q^{\mu\nu}\partial_{\nu}\Phi)=0,$
(3.3)

In the case in which $\Phi$ depends only on the variable $r$, the field that
satisfies equation (3.2) is given by

$\frac{(\alpha-1)}{\alpha^{3/2}}=b\,\log r$ (3.4)

where $b$ is a constant. For simplicity we take $b=1.$ Note that $r$ as an
inverse function of the field $\alpha$ has a maximum value for the point
$\alpha=3$ which is the singular boundary of this geometry.

Note however that this geometry is restrained to the values of r: $0<r<r_{m}$
where the maximum radius is given by $r_{m}=\exp{(2\sqrt{3}/9)}.$ This value
of $r,$ corresponds to the value $\alpha=3,$ where the metric is singular.
Indeed, the scalar of curvature is given by

$R=\frac{14\,\alpha^{3}+18\alpha}{(\alpha-3)^{2}w_{0}^{2}\,r^{2}}.$

###  3.2 CTCs

Let us consider a curve $x^{\mu}(s)$ such that
$z=constant,t=constant,r=constant,$ that is on the curve we have

$ds^{2}=\frac{1}{\alpha}\,\left(h_{0}^{2}-w^{2}\,r^{2}\right)\,d\phi^{2}.$
(3.5)

Defining $r_{c}^{2}=h_{0}^{2}/w_{0}^{2}$, we can see that this closed curve
has the following property:

  * •

For $r>r_{c}$ the curve is space-like;

  * •

For $r<r_{c}$ the curve is time-like.

We note that this situation is distinct from that encoded by others metrics,
like for instance that of Gödel’s geometry, which allows the presence of CTCs
in an exterior region of a domain defined by a given radius $r_{G}$. Here, on
the contrary, the CTCs are confined in the interior domain of radius $r_{m}.$
This property precludes the application of Gauss’ prescription, by means of
which it is always possible to construct a Gaussian system of coordinates
separating the 3-d space from time (at least, in a small domain).

##  4 Dynamics in the presence of matter

From what we have shown in the previous section, it is natural to try to avoid
the singularity of this metric by joining this geometry to another one in the
exterior domain, before the value $r_{m}$ is reached. In order to do this,
consider a fluid with density $\rho$ without pressure, so that the energy-
momentum tensor admits the expression

$T^{\mu\nu}=\rho\,v^{\mu}\,v^{\nu}.$

We set $v^{\mu}=(A,0,B,0)$, and normalize this vector by imposing the relation
between $A$ and $B$

$\alpha=A^{2}+h_{0}\,AB+(h_{0}^{2}-w_{0}^{2}\,r^{2})\,B^{2}.$ (4.1)

This vector will be a geodesic under the condition

$\frac{d\alpha}{dr}+2w_{0}^{2}\,r\,B^{2}=0,$ (4.2)

where we have used the normalization condition for the vector $v^{\mu}$.

The Geometric Scalar Gravity theory provides the dynamics of the metric as

$\sqrt{V}\,\Box\Phi=\kappa\,\chi,$ (4.3)

where in this case

$\chi=\frac{1}{2}\,\rho.$

It then follows

$\frac{1}{16}\,(\alpha-3)^{2}\,\frac{1}{r}\,\frac{d}{dr}\Delta=-\frac{1}{2}\,\rho,$
(4.4)

where

$\Delta\equiv(\alpha-3)\,\alpha^{-5/2}\,r\,d\alpha/dr.$ (4.5)

Let us consider the case in which $\Delta=2\lambda/r^{2}.$ Thus the density of
energy is given by

$\frac{\lambda}{4}\,\frac{(\alpha-3)^{2}}{r^{3}}=-\,\frac{\kappa}{2}\,\rho.$
(4.6)

The expression of $\Delta$ yields

$\frac{(\alpha-1)}{\alpha^{3/2}}=\frac{\lambda}{r}.$ (4.7)

Note that this geometry is assymptotically flat. Indeed, when $r$ goes to
infinity the density of energy vanishes and the function $\alpha$ goes to $1,$
that is, the q-metric becomes the Minkowski metric. Note that once the
constant $\lambda$ is negative, the value of $\alpha$ in the domain II must be
bounded: $0<\alpha<1.$

Te next step is to join the metrics of domains I and II in order to avoid the
presence of singularity at $\alpha-3=0.$ It then follows for the value
$r_{b}<r_{m}$ the condition

$\exp{(\lambda/r_{b})}=r_{b}.$ (4.8)

##  5 Final comments

The property displayed by the metric of the domain I shows a very unusual
feature: the existence of CTCs in a compact domain. This is not the case in
the most famous solution of GR found by Kurt Gödel [6], [7] . Indeed, Gödel’s
geometry written under the form of the metric (8) has the functions $h$ and
$g$ given by

$h=\sqrt{2}\,\sinh^{2}r,$ $g=2\sinh^{4}r-\sinh^{2}r\cosh^{2}r.$

Define the critical radius $r_{c}$ of Gödel’s metric by setting $\sinh
r_{c}=1.$ For values of $r$ such that $0<r<r_{c}$ the value of $g$ is
negative. However, for values of $r$ such that $r>r_{c}$ the curve is time-
like, that is a CTC. Thus, for Gödel’s metric there are no CTCs in the
interior domain $r<r_{c}$. CTCs appear only in the exterior domain of Gödel’s
metric, $r>r_{c}$. As we have shown above, this behaviour is opposite to the
one obtained for a rotational configuration in GSG theory.

##  6 Acknowledgements

The authors would like to thank the Brazilian agencies Faperj (MN), CNPq, and
the CBPF (VA) for their financial support.

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