#  i-MAE: Are Latent Representations in Masked Autoencoders Linearly
Separable?

Kevin Zhang†,∗, Zhiqiang Shen‡,§,♮,
†Peking University ‡Hong Kong University of Science and Technology
§Mohamed bin Zayed University of Artificial Intelligence ♮Carnegie Mellon
University
{kevinzyz6<EMAIL_ADDRESS>
The two authors have equal contribution to this work. Zhiqiang Shen is the
corresponding author.

###### Abstract

Masked image modeling (MIM) has been recognized as a strong and popular self-
supervised pre-training approach in the vision domain. However, the
interpretability of the mechanism and properties of the learned
representations by such a scheme are so far not well-explored. In this work,
through comprehensive experiments and empirical studies on Masked Autoencoders
(MAE), we address two critical questions to explore the behaviors of the
learned representations: (i) Are the latent representations in Masked
Autoencoders linearly separable if the input is a mixture of two images
instead of one? This can be concrete evidence used to explain why MAE-learned
representations have superior performance on downstream tasks, as proven by
many literature impressively. (ii) What is the degree of semantics encoded in
the latent feature space by Masked Autoencoders? To explore these two
problems, we propose a simple yet effective Interpretable MAE (i-MAE)
framework with a two-way image reconstruction and a latent feature
reconstruction with distillation loss to help us understand the behaviors
inside MAE’s structure. Extensive experiments are conducted on CIFAR-10/100,
Tiny-ImageNet and ImageNet-1K datasets to verify the observations we
discovered. Furthermore, in addition to qualitatively analyzing the
characteristics of the latent representations, we examine the existence of
linear separability and the degree of semantics in the latent space by
proposing two novel metrics. The surprising and consistent results across the
qualitative and quantitative experiments demonstrate that i-MAE is a superior
framework design for interpretability research of MAE frameworks, as well as
achieving better representational ability. 111Code is available at
https://github.com/vision-learning-acceleration-lab/i-mae.

##  1 Introduction

Self-supervised learning aims to learn representations from abundant unlabeled
data for benefiting various downstream tasks. Recently, many self-supervised
approaches have been proposed in the vision domain, such as pre-text based
methods (Doersch et al., 2015; Zhang et al., 2016; Gidaris et al., 2018),
contrastive learning with Siamese networks (Oord et al., 2018; He et al.,
2020; Chen et al., 2020; Henaff, 2020), masked image modeling (MIM) (He et
al., 2022; Bao et al., 2022; Xie et al., 2022), and etc. Among them, MIM has
shown a preponderant advantage in performance, and the representative method
Masked Autoencoders (MAE) (He et al., 2022) has attracted much attention in
the field. A natural question is then raised: Where is the benefit of the
transferability to downstream tasks from in MAE-based training? This motivates
us to develop a framework to shed light on the reasons for the superior latent
representation from MAE. Also, as the interpretability of the MAE framework is
still under-studied in this area, it is crucial to explore this in a specific
and exhaustive way.

Intuitively, a good representation should be separable and contain enough
semantics from its input, so that it can have a qualified ability to
distinguish different classes with better performance on downstream tasks.
Nonetheless, how to evaluate the separability and the degree of semantics on
the latent features is not clear thus far. Moreover, the mechanism of an
Autoencoder compressing the information from input by reconstructing itself
has been a well-established self-supervised learning architecture, but the
explanation of the features learned from such approaches is still under-
explored.

To address the difficulties of identifying separability and semantics in
latent features, we first propose a novel framework, i-MAE, upon vanilla MAE.
It consists of a mixture-based masked autoencoder branch for disentangling the
mixed representations by linearly separating two different instances, and a
pre-trained vanilla MAE as the guidance to distill the disentangled
representations. An illustration of the overview framework architecture is
shown in Fig. 2. This framework is designed for answering two interesting
questions: (i) Are the latent representations in Masked Autoencoders linearly
separable? (ii) What is the degree of semantics encoded in the latent feature
space by Masked Autoencoders? These two questions can reveal the fact that MAE
learned features are good at separating different classes. We attribute the
superior representation of MAE to it learning separable features for
downstream tasks with enough semantics.

In addition to qualitative studies, we also develop two metrics to address the
two questions quantitatively. In the first metric, we employ $\ell_{2}$
distance from high-dimensional Euclidean spaces to measure the similarity
between i-MAE’s disentangled feature and the “ground-truth” feature from pre-
trained MAE on the same image. In the second metric, we control different
ratios of semantic classes as a mixture within a mini-batch and evaluate the
finetuning and linear probing results of the model to reflect the learned
semantic information. More details will be provided in Section 3.

We conduct extensive experiments on different scales of datasets: small
CIFAR-10/100, medium Tiny-ImageNet and large ImageNet-1K to verify the linear
separability and the degree of semantics in the latent representations. We
also provide both qualitative and quantitative results to explain our
observations and discoveries. The characteristics we observed in latent
representations according to our proposed i-MAE framework are: (I) i-MAE
learned feature representation has great linear separability for its input
data, which is proven beneficial for downstream tasks. (II) Though the
training scheme of MAE is different from instance classification pre-text in
contrastive learning, its representation still encodes sufficient semantic
information from input data. Moreover, mixing the same-class images as the
input training samples substantially improves the quality of learned features.
(III) We can reconstruct an image from a mixture by i-MAE effortlessly, even
if the it is the subordinate part. To the best of our knowledge, this is the
pioneering study to explicitly explore the separability and semantics inside
MAE’s features with extensive well-designed qualitative and quantitative
experiments.

Figure 1: Reconstruction results of i-MAE on ImageNet-1K validation images
with different mixing coefficients $\bm{\alpha}$ (listed on the left). i-MAE
is pre-trained with linearly mixed input and the subordinate image
${\bm{I}_{s}}$ as the only target with a 0.5 mask ratio. Visually, i-MAE
predictions reflect features of $\bm{I}_{s}$ even at 0.1 and still
reconstructs the individual image well, whereas at 0.45 reconstructions show
the appearance of dominant image (hence the green patches). More
visualizations are provided in the Appendix.

Our contributions in this work are:

  * •

We propose an i-MAE framework with two-way image reconstruction and latent
feature reconstruction with a distillation loss, to explore the
interpretability of mechanisms and properties inside the learned
representations of the MAE framework.

  * •

We introduce two metrics to examine the linear separability and the degree of
semantics quantitatively on the learned latent representations.

  * •

We conduct extensive experiments on different scales of datasets:
CIFAR-10/100, Tiny-ImageNet and ImageNet-1K and provide sufficient qualitative
and quantitative results.

##  2 Related Work

Masked image modeling. Motivated by masked language modeling’s success in
language tasks (Devlin et al., 2018; Radford & Narasimhan, 2018), Masked Image
Modeling (MIM) in the vision domain learn representations from images
corrupted by masking. State-of-the-art results on downstream tasks are
achieved by several approaches. BEiT (Bao et al., 2022)proposes to recover
discrete visual tokens, whereas SimMIM (Xie et al., 2022) addresses the MIM
task as a pixel-level reconstruction. In this work, we focus on MAE (He et
al., 2022), which proposes to use a high masking ratio and a non-arbitrary ViT
decoder. Despite the great popularity of MIM approaches and their conceptual
similarity to language modeling, the question of why these representations are
stronger has not been thoroughly addressed. Furthermore, as revealed by MAE,
pixels are semantically sparse; therefore, we utilize a novel semantics mixing
method to investigate semantic-level information quantitatively.

Image mixtures. Widely adopted mixture methods in visual supervised learning
include Mixup (Zhang et al., 2017) and Cutmix (Yun et al., 2019). However,
these methods require ground-truth labels for calculating mixed labels; in
this work, we adapt Mixup to our unsupervised framework by formulating losses
on only one of the two input images. On the other hand, in very recent visual
SSL, joint embedding methods and contrastive learning approaches such as MoCo
(He et al., 2020), SimCLR (Chen et al., 2020), and more recently UnMix (Shen
et al., 2022) have acquired success and predominance in mixing visual inputs.
These approaches promote instance discrimination by aligning features of
augmented views of the same image. However, unlike joint embedding methods,
i-MAE does not heavily rely on data augmentation and negative sampling.
Moreover, whereas most MIM methods are generative tasks, i-MAE learns more
separable representations for enhanced instance discrimination performance.

Invariance and disentangling representation learning in Autoencoders.
Representation learning focuses on the properties of the features learned by
the layers of deep models while remaining agnostic to the particular
optimization process. Variance and entanglement are two commonly discussed
factors that occur in data distribution for representation learning. In this
work, we focus on the latent disentanglement that one feature is correlated or
connected to other vectors in the latent space. Autoencoder is a classical
generative unsupervised representation learning framework based on image
reconstruction as a loss function. Specifically, autoencoders learn both the
mapping of inputs to latent features and the reconstruction of the original
input. Denoising autoencoders reconstruct the original input from a corrupted
input, and most MIM methods are categorized as denoising autoencoders that use
masking as a noise type. We notice that recent work in the literature (He et
al., 2022; Bao et al., 2022) performs many experiments in masking strategies,
but to the best of our knowledge, we are the first to introduce image mixtures
in the pre-training of MIM.

##  3 i-MAE

In this section, we first introduce an overview of our proposed framework.
Then, we present each component in detail. Ensuing, we elaborate on the
metrics we proposed evaluating linear separability and degree of semantics, as
well as broadly discuss the observations and discoveries.

###  3.1 Framework Overview

As shown in Fig 2, our framework consists of three submodules: (i) a mixture
encoder module that takes the masked mixture image as the input and output
mixed features; (ii) a disentanglement module that splits the mixed feature to
the individual ones; (iii) a MAE teacher module that provides the pre-trained
embedding for guiding the splitting process in the disentanglement module.

Figure 2: Framework overview of our i-MAE. ① is the main branch that consists
of a mixture encoder, a disentanglement module, and a two-way image
reconstruction module. ② is the encoder part of a pre-trained vanilla MAE for
distillation purposes (i.e., latent reconstruction). ③ is the decoder part in
MAE and is discarded in training.

####  3.1.1 Components

Input Mixture with MAE Encoder. Inspired by Mixup, we use an unsupervised
mixture of inputs formulated by $\alpha*{\bm{I}}_{1}$ and
$(1-\alpha)*{\bm{I}}_{2}$, ${\bm{I}}_{1},{\bm{I}}_{2}$ are the input images.
Essentially, our encoder extrapolates mixed features from a tiny fraction
(e.g., 25%) of visible patches, which we then tune to only represent the
subordinate image. The mixed image will be:

${\bm{I}}_{\bm{m}}=\alpha*{\bm{I}}_{1}+(1-\alpha)*{\bm{I}}_{2}$ (1)

where $\alpha$ is the coefficient to mix two images following a Beta
distribution.

Two-branch Masked Autoencoders with Shared Decoder. Although sufficient
semantic information from both images is embedded in the mixed representation
to reconstruct both images, the vanilla MAE cannot by itself associate
separated features to either input. The MAE structure does not retain
identification information about the two mixed inputs (e.g., order or
positional information), i.e., the model cannot tell which of the two images
to reconstruct to, since both are sampled from the same distribution and mixed
randomly. The consequence is that both reconstructions look identical to each
other and fail to look similar to either original input.

Similar to how positional embeddings are needed to explicitly encode spatial
information, i-MAE implicitly encodes the semantic difference between the two
inputs by using a dominant and subordinate mixture strategy. Concretely,
through an unbalanced mix ratio and a reconstruction loss targeting only one
of the inputs, our framework encodes sufficient information for i-MAE to
linearly map the input mixture to two outputs.

Two-way Image Reconstruction Loss. Formally, we build our reconstruction loss
to recover individual images from a mixed input, which is first fed into the
encoder to generate mixed features:

$\bm{h}_{m}=\bm{E}_{\text{i-MAE}}(\bm{I}_{m})$ (2)

where $\bm{E}_{\text{i-MAE}}$ is i-MAE’s encoder, $\bm{h}_{m}$ is the latent
mixed representation. Then, we employ two non-shareable linear embedding
layers to separate the mixed representation from the individual ones:

$\displaystyle\bm{h}_{1}=\bm{f}_{1}(\bm{h}_{m})$ (3)
$\displaystyle\bm{h}_{2}=\bm{f}_{2}(\bm{h}_{m})$

where $\bm{f}_{1}$, $\bm{f}_{2}$ are two linear layers with different
parameters for disentanglement and $\bm{h}_{1}$ and $\bm{h}_{2}$ are
corresponding representations. After that, we feed the individual
representations into the shared decoder with the corresponding reconstruction
losses:

$\begin{array}[]{l}\mathcal{L}_{\text{recon}}^{\bm{I}_{1}}=\mathbb{E}_{\bm{I}_{1}\sim
p\left(\bm{I}_{1}\right)}[\left\|\bm{D}_{\text{shared}}\left(\bm{h}_{1}\right)-\bm{I}_{1}\right\|_{2}].\\\
\mathcal{L}_{\text{recon}}^{\bm{I}_{2}}=\mathbb{E}_{\bm{I}_{2}\sim
p\left(\bm{I}_{2}\right)}[\left\|\bm{D}_{\text{shared}}\left(\bm{h}_{2}\right)-\bm{I}_{2}\right\|_{2}].\end{array}$
(4)

In practice, we train the linear separation layers to distinguish between the
dominant input $\bm{I}_{d}$ (higher mix ratio) and the subordinate input
$\bm{I}_{s}$ (lower ratio). Showing that our encoder learns to embed
representations of both images, we intentionally choose to reconstruct only
the subordinate image $\bm{I}_{s}$ to prevent the $\bm{I}_{d}$ from guiding
the reconstruction. Essentially, successful reconstructions from only the
$\bm{I}_{s}$ prove that representations of both images can be learned and that
the subordinate image is not filtered out as noise.

Patch-wise Distillation Loss for Latent Reconstruction. With the linear
separation layers and an in-balanced mixture, the i-MAE encoder is presented
with sufficient information about both images to perform visual
reconstructions. However, information is inevitably lost during the mixing
process, harming the value of the learned features in downstream tasks such as
classification. To mitigate such an effect, we propose a knowledge
distillation module for not only enhancing the learned features’ quality, but
also demonstrating that a successful distillation can evidently prove the
linear separability of our features.

Intuitively, MAE’s features can be regarded as “ground-truth” and i-MAE learns
features distilled from the original MAE. Specifically, our loss function
computes $\ell_{2}$ loss between disentangled representations and original
representations to help our encoder learn useful features of both inputs.

Our Patch-wise latent reconstruction loss can be formulated as:

$\displaystyle\mathcal{L}_{\text{recon }}^{h_{1}}$
$\displaystyle=\mathbb{E}_{\bm{h}_{1}\sim
q\left(\bm{h}_{1}\right)}[\left\|\bm{E}_{\mathrm{p}-\mathrm{MAE}}\left(\bm{I}_{1}\right)-\bm{h}_{1}\right\|_{2}].$
(5) $\displaystyle\mathcal{L}_{\text{recon }}^{\bm{h}_{2}}$
$\displaystyle=\mathbb{E}_{\bm{h}_{2}\sim
q\left(\bm{h}_{2}\right)}[\left\|\bm{E}_{\mathrm{p}-\mathrm{MAE}}\left(\bm{I}_{2}\right)-\bm{h}_{2}\right\|_{2}].$

where $\bm{E}_{\text{p-MAE}}$ is the pre-trained MAE encoder.

###  3.2 Linear Separability

For i-MAE to reconstruct the subordinate image from a potentially linear
mixture, not only should the encoder be general enough to retain information
from both inputs, but it also needs to generate embeddings that are specific
enough for the decoder to distinguish them into their pixel-level forms. A
straightforward interpretation of how i-MAE fulfills both conditions is that
the latent mixture ${{\bm{h}}_{m}}$ is a linear combination of features that
closely relate to ${\bm{h}_{1}}$ and ${\bm{h}_{2}}$, e.g., in a linear
relationship. Our distillation module mitigates the information loss. To
verify this explanation, we employ a linear separability metric to
experimentally observe such a behavior.

Metric of Linear Separability. A core contribution of our i-MAE is the
quantitative analysis of features. In general, linear separability is a
property of two sets of features that can be separated into their respective
sets by a hyperplane. In our example, the set of latent representations
$\bm{H}_{1}$ and $\bm{H}_{2}$ are linearly separable if there exists $n+1$
real numbers $\bm{w}_{1},\bm{w}_{2},...,\bm{w}_{n},\bm{b}$, such that every
$\bm{h}\in\bm{H}_{1}$ satisfies $\sum{{\bm{w}}_{i}{\bm{h}}_{i}>\bm{b}}$ and
every $\bm{h}\in\bm{H}_{2}$ satisfies $\sum{{\bm{w}}_{i}{\bm{h}}_{i}<\bm{b}}$.
It is a common practice to train a classical linear classifier (e.g., SVM) and
evaluate if two sets of data are linearly separable.

However, to quantitatively measure the separation of latent representations,
we devised a more intuitive yet effective metric. Our metric computes the Mean
Squared Error (MSE) distance between the disentangled feature of the
subordinate image $I_{s}$ and the vanilla MAE feature of a single input
$I_{s}$. Since the disentangled feature without constraints will unlikely
resemble the vanilla feature, we utilize a linear layer to transform the
disentangled feature space to the vanilla feature space. Note that this is
similar to knowledge distillation, but happens after the pre-training process
without finetuning the parameters and conceptually measures the distance
between the two latent representations, and thus the linear transformation
will not be needed for i-MAE with distillation. The detailed formulation of
the metric is:

$\mathcal{M}_{\text{ls}}=\frac{1}{N}\sum_{n=1}^{\bm{N}}\left\|{\bm{h}}^{n}_{s}-\bm{f}_{\bm{\theta}}\left(\bm{I}^{n}_{s}\right)\right\|_{2}^{2}$
(6)

where $N$ is the total number of samples. $\bm{f}_{\bm{\theta}}$ is the
encoder of vanilla MAE. $\bm{I}_{s}$ is the subordinate image and
$\bm{I}_{s}\in\\{\bm{I}_{1},\bm{I}_{2}\\}$.

###  3.3 Semantics

Metric of Semantics. Vanilla MAE exhibits strong signs of semantics
understanding (He et al., 2022). However, studying the abstract concept of
semantics in the visual domain is difficult due to its semantic sparsity and
repetitiveness. Addressing this problem, we propose a metric unique to i-MAE
that is readily available for examining the degree of semantics learned in the
model. Apart from straightforwardly evaluating classification accuracy to
measure the quality of latent representation, i-MAE utilizes the mixing of
semantically similar instances to determine to what degree the disentangled
latent representations can reflect image-level meaning.

Naturally, segmenting different instances from the same class is more
difficult than segmenting different classes; intra-class separation
necessitates knowledge of high-level visual concepts, such as semantic
differences, rather than lower-level patterns, such as shape or color.

Moreover, when mixing images of the same class, their latent features are
naturally more similar and their two-way loss functions will be updated in the
same direction. This means that an intra-class mixture’s latent features will
encode more information that is more robust about a specific class than an
inter-class mixture, where the two latent features may confuse or conflict
with each other. Consequently, when the mixed representations have semantics
more closely aligned, the information propagated into the two branches
contains more information about a specific class; when the mixed
representations are from different classes, the disentangled features may not
have semantics perfectly resembling their classes, thus containing a lower
degree of semantics for a single class.

Our semantics-controllable mixture scheme is another data augmentation that
introduces significantly more mixtures of the same class into the training
process.

We find our method to boost the semantics of features learned by this
semantics-controllable mixture scheme. Specifically, we choose training
instances from the same or different classes following different distributions
to constitute an input mixture, so as to examine the quality of learned
features as follows:

$\bm{p}=\bm{f}_{\bm{m}}(\bm{I}_{c_{a}}+\bm{I}_{c_{b}})$ (7)

where ${\bm{f}}_{\bm{m}}$ is the backbone network for mixture input and
${\bm{p}}$ is the corresponding prediction. ${\bm{I}}_{c_{a}}$ and
${\bm{I}}_{c_{b}}$ are the input samples and $c_{a}$, $c_{b}$ have a certain
percentage ${\bm{r}}$ that belongs to the same category. For instance, if
${\bm{r}}=0.1$, it indicates that 10% images in a mini-batch are mixed with
the same class. When ${{\bm{r}}}=1.0$, all training images will be mixed with
another one from the same class, which can be regarded as a semantically
enhanced augmentation. During training, ${\bm{r}}$ is fixed for individual
models, and we study the degree of semantics that the model encodes by
changing the percentage value ${\bm{r}}$. After the model is trained by i-MAE
using such kind of input data, we finetune the model with Mixup strategy (both
baseline and our models) and cross-entropy loss. We use accuracy as the metric
of semantics under this percentage of instance mixture:

$\mathcal{M}_{\text{sem}}=-\sum_{i=1}^{n}\bm{t}_{i}\log\left(\bm{p}_{i}\right)$
(8)

where $\bm{t}_{i}$ is the ground-truth. The insight behind this is that: if
the input mixture is composed of two images or instances with the same
semantics (i.e., the same category), it will confuse the model during training
and i-MAE will struggle to disentangle them. Thus, the encoded information and
semantics may be weakened in the training process, and it can be reflected by
the quality of learned representation. It is interesting to see whether this
conjecture is supported by the empirical results. We use the representation
quality through finetuned accuracy to monitor the degree of semantics with
this semantics-controllable mixture scheme. Since in the metric we involve
additional prior knowledge of same or different classes for the mixture
samples, this is only for evaluating purposes in our framework. An alternative
way to avoid using prior label information is clustering the samples to
identify the same or different classes in a mini-batch.

##  4 Empirical Results and Analysis

In this section, we analyze the properties of i-MAE’s disentangled
representations through empirical studies on an extensive range of datasets.
First, we provide the datasets used and our implementation details. Then, we
thoroughly ablate our experiments, focusing on the properties of linear
separation, and controllable-semantic mixture. Lastly, we give our final
evaluation of the results.

###  4.1 Datasets and Training Implementation for Baseline and i-MAE.

Settings: We conduct experiments of i-MAE on CIFAR-10/100, Tiny-ImageNet, and
ImageNet-1K. On CIFAR-10/100, we adjust MAE’s structure to better fit the
smaller datasets during unsupervised pre-training: ViT-Tiny (Touvron et al.,
2021) in the encoder and a lite-version of ViT-Tiny (4 layers) as the decoder.
Our pre-training lasts 2,000 epochs with a learning rate $1.5\times 10^{-4}$
and 200 warm-up epochs. On Tiny-ImageNet, i-MAE’s encoder is ViT-small and
decoder is ViT-Tiny, trained for 1,000 epochs with a learning rate $1.5\times
10^{-4}$. Additionally, we apply warm-up for the first 100 epochs, and use
cosine learning rate decay with AdamW optimizer as in vanilla MAE.

Supervised Finetuning: In the finetuning process, we apply Mixup for all
experiments to fit our pre-training scheme, and compare our results with
baselines of the same configuration. On CIFAR-10/100, we finetune 100 epochs
using the AdamW optimizer and a learning rate of $1.5\times 10^{-3}$.

Linear Probing: For linear evaluation, we follow MAE (He et al., 2022) to
train with no extra augmentations and use zero weight decay. Similarly, we
adopt an additional BatchNorm layer without affine transformation.

###  4.2 Ablation Study

In this section, we perform ablation studies on i-MAE to concretely examine
the property of linear separability and its existence at different mix-levels.
Then, we analyze the effects of semantics-controllable mixture on i-MAE
learned representations.

####  4.2.1 Ablation for Linear Separability

To begin, we thoroughly perform our ablation experiments on a diverse group of
datasets (ImageNet-1K is performed for final evaluation) and demonstrate how
i-MAE’s learned features display linear separability with different settings.
Specifically, we experiment with the separability of the following aspects of
our methods: (i) constant and probability mix factors; (ii) masking ratio of
input mixtures; (iii) different ViT architectures. Unless otherwise stated,
the default settings used in our ablation experiments are ViT-Tiny, masking
ratio of 75%, fixed mixing ratio of 35%, and reconstructing only the
subordinate image for a harder task.

Mix Ratio. To demonstrate the separable nature of the input mixtures for
subordinate reconstruction, we compared different mixture factors between 0
and 0.5, and random mixture ratios from a Beta distribution. Intuitively,
lower mixing ratios contain less meaning information that the encoder may
easily confuse with noise, whereas higher ratios destroy the subordinate-
dominant relationship. Experimentally, we observe matching visual results
shown in the Appendix (Fig. 1 and Fig. 11). The better separation performance
near 0.3 indicates that i-MAE features are better dichotomized when the mix
factor is balanced between noise and useful signals. Whereas below 0.15, the
subordinate image is noisy and reconstructions are not interpretable, mixing
ratios above 0.45 break the subordinate relationship between the two images,
and the two features are harder to distinguish from each other. Moreover, Fig.
1 presents a problematic case where a mix factor of 0.45 reconstructs dominant
features (hence the green patches in the background).

Mask Ratio. In i-MAE, visible information of the subordinate image is
inherently limited by the unbalanced mix ratio, in addition to masking. Hence,
a high masking ratio (75% (He et al., 2022)) may not be necessary to suppress
the amount of information the encoder sees, and we consequently attempt lower
ratios of 50%, 60% to introduce more information about the subordinate target.
As shown in Fig. 3, a lower masking ratio of 0.5 or 0.6 can significantly
improve reconstruction quality.

Combining our findings in mix and mask ratios, we empirically find that i-MAE
can compensate for the information loss at low ratios with the additional
alleviation of more visible patches (lower mask ratio). Illustrated in Fig. 1,
we display a case of i-MAE’s reconstruction succeeding in separating the
features an input with $\alpha=0.1$ mix factor and 0.5 masking ratio. Through
studying the mix ratio and masking ratio, we reveal that i-MAE can learn
linearly separable features under two conditions: (i) enough information about
both images must be present (determined by the trade-off between mask ratio
and mix ratio). (ii) the image-level distinguishing relationship between
minority and majority (determined by mix ratio) is potent enough for i-MAE to
encode the two images separately.

ViT Backbone Architecture. We studied whether different scales of ViT effect
linear separation in the Appendix of Fig. 5. Our results show that larger
backbones are not necessary for i-MAE to disentangle features on small
datasets, as the insufficient training data cannot fully utilize the
capability; however, large ViTs are crucial to the large-scale ImageNet-1K.

####  4.2.2 Ablation for Degree of Semantics

Semantic Mixes. Depending on the number of classes and their overall size,
datasets in pristine states usually contain around 10% (e.g., CIFAR-10) to
$<$1% (e.g. ImageNet-1K) samples pertaining to the same class, meaning that by
default, uniformly random sampling mixtures will most likely be of different
objects. On the other hand, the semantics-controllable mixture scheme examines
whether the introduction of semantically homogeneous mixtures affects the
classification performance. That is, we intentionally test to see if similar
instances during pre-training negatively influence the classification
performance.

As shown in Tab. 1, after i-MAE pre-training, we perform finetuning and linear
probing on classification tasks to evaluate the degree of semantics learned
given different amounts of intra-class mix $\bm{r}$. From Tab. 1, we discover
that i-MAE overall has a stronger performance in finetuning and linear probing
with a non-zero same-class ratio. Specifically, a high $\bm{r}$ of 1.0
increases the accuracy in linear evaluation most in all datasets, meaning that
the quality of learned features is best and separated, and it gains a strong
prior of category information for semantically enhanced mixtures. On the other
hand, setting $\bm{r}=0.5$ is advantageous during finetuning, as it gains a
balanced prior of separating both intra- and inter-class mixtures.

Table 1: Classification performance of models pre-trained with semantics-controllable mixture with intra-class mix rate $\bm{r}$ from 0.0 to 1.0. Whereas the lower bound represents inputs that are all mixes of different classes, $\bm{r}=1.0$ pre-trains with solely mixtures of same-labeled objects.  | CIFAR-10 | CIFAR-100 | Tiny-ImageNet
---|---|---|---
Same-class Ratio | Finetune | Linear | Finetune | Linear | Finetune | Linear
baseline | 90.78 | 72.47 | 68.66 | 32.57 | 59.28 | 19.62
0.0 | 91.67 | 70.53 | 68.34 | 29.22 | 60.91 | 18.23
0.5 | 92.34 | 72.80 | 69.50 | 30.11 | 60.58 | 18.51
1.0 | 91.60 | 77.61 | 69.33 | 33.39 | 61.13 | 20.40
Figure 3: Comparisons between different mask ratios on Tiny-ImageNet
validation dataset. i-MAE produces enhanced visual reconstructions from lower
masking ratios when reconstructing images.  Table 2: Linear separation metric
using $\ell_{2}$ distance calculated before and after linear regression on
CIFAR-10, CIFAR-100, Tiny-ImageNet, and ImageNet. Reported results are from
100 samples trained for 2000 (5000 on ImageNet-1K) epochs and a fixed mix
ratio of 0.3. i-MAE and i-MAE without distillation are embedding after
disentanglement.

| CIFAR-10 | CIFAR-100 | Tiny-ImageNet | ImageNet-1K
---|---|---|---|---
| Before | After | Before | After | Before | After | Before | After
Baseline | 3.5899 | 0.0584 | 3.0840 | 0.0487 | 10.38 | 0.0204 | 13.91 | 0.2004
i-MAE w/o distill | 0.1384 | 0.0568 | 0.2999 | 0.0475 | 0.0723 | 0.0363 | 0.8799 | 0.1316
i-MAE | 0.0331 | 0.0474 | 0.0312 | 0.0456 | 0.0708 | 0.0352 | 0.2760 | 0.1838

Figure 4: Qualitative comparisons on CIFAR-10. Row (a): baseline vanilla MAE;
(b): MAE with unmixed input; (c): our i-MAE without distillation; and (d):
i-MAE with distillation. Table 3: Finetuning classification acc. on i-MAE with
best semantics-controllable mixture settings.

Method | CIFAR-10 | CIFAR-100 | Tiny-ImageNet
---|---|---|---
Baseline | 90.78 | 68.66 | 59.28
i-MAE | 92.00 | 69.50 | 61.63

Table 4: Linear Evaluation accuracy of i-MAE with best semantics-controllable
mixture settings.

Method | CIFAR-10 | CIFAR-100 | Tiny-ImageNet
---|---|---|---
Baseline | 72.47 | 32.57 | 19.62
i-MAE | 77.61 | 33.39 | 20.40

###  4.3 Results of Final Evaluation

In this section, we provide a summary of our main findings: how separable are
i-MAE embedded features and the amount of semantics embedded in mixed-
representations. Then, we evaluate the quality of our features with
classification and analyze the features.

####  4.3.1 Separability

In this section, we show how i-MAE displays properties of linear separability,
visually and quantitatively, and demonstrate our advantage over baseline
(vanilla MAE).

In a visual comparison of the disentanglement capability in Fig. 4, vanilla
MAE does not perform well out-of-the-box. In fact, the reconstructions
represent the mixed input more so than the subordinate image. Since the
mixture inputs of i-MAE is a linear combination of the two images, and our
results show i-MAE’s powerful ability to reconstruct both images, even at very
low mixture ratios, we attribute such ability to i-MAE’s disentanglement
strongly correlating with the vanilla MAE’s features.

As aforementioned, we previously gave the formal definition of linear
separability; now, we empirically illustrate the strength of the linear
relationship between MAE’s features and i-MAE’s disentangled features with a
linear regressor. We employ $\ell_{2}$ distance as our criterion and results
are reported in Tab. 2. Experimentally, we feed mixed inputs to i-MAE and a
singular image to the target model (vanilla MAE), Before indicates that we
directly calculate the distance between the “ground-truth” features from pre-
trained MAE and our disentangled features. After indicates that we train the
linear regressor’s parameters to fit the “ground-truth”. Baseline is the model
trained without the disentanglement module. It can be observed that our i-MAE
has a significantly smaller distance than the vanilla model, reflecting that
such a scheme can obtain better separability ability.

####  4.3.2 Semantics

Finetune and Linear Evaluation. We evaluate our i-MAE’s performance through
finetuning and linear evaluation of regular inputs and targets. For all
approaches in the finetuning phase, we only use Mixup as augmentation; no
extra augmentations are used for the linear evaluation phase. Classification
performance is outlined in Tab. 4 and Tab. 4. It can be observed that i-MAE
outperforms the baseline by a remarkable margin. As our features are learned
from a harder scenario, they encode more information with a more robust
representation and classification accuracy. Besides, i-MAE shows a
considerable performance boost with both evaluation methods.

Analysis. We emphasize that our enhanced performance comes from i-MAE’s
ability to learn more separable features with the disentanglement module, and
the enhanced semantics learned from training with semantics-controllable
mixture. Our classification results show the cruciality of MAE learning
features that are linearly separable, which can help identify between
different classes. However, to correctly identify features with their
corresponding classes, semantically rich features are needed, which can be
enhanced by the intra-class mix sampling strategy.

##  5 Conclusion

It is non-trivial to understand why Masked Image Modeling (MIM) in the self-
supervised scheme can learn useful representations for downstream tasks
without labels. In this work, we have introduced a novel interpretable
framework upon Masked Autoencoders (i-MAE) to explore two critical properties
in latent features: linear separability and degree of semantics. We identified
that the two specialties are the core for superior latent representations and
revealed the reasons where is the good transferability of MAE from. Moreover,
we proposed two metrics to evaluate these two specialties quantitatively.
Extensive experiments are conducted on CIFAR-10/100, Tiny-ImageNet, and
ImageNet-1K datasets to demonstrate our discoveries and observations in this
work. We also provided sufficient qualitative results and analyses of
different hyperparameters. We hope this work can inspire more studies on the
interpretability of the MIM frameworks in the future.

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## Appendix

In the appendix, we display the detailed configurations of our experiments and
elaborate with more examples of our main text, specifically:

• Section A “Datasets”: specifications of the datasets we used.

• Section B “Implementation Details”: implementation details and configuration
settings for unsupervised pre-training and supervised classification.

• Section C “More Visualizations”: additional reconstruction examples on all
datasets.

• Section D “Pseudocode”: a PyTorch-like pseudocode for our mixture and loss
methods.

##  Appendix A Datasets

CIFAR-10/100 (Krizhevsky, 2009) Both CIFAR datasets contain 60,000 tiny
colored images sized 32$\times$32\. CIFAR-10 and 100 are split into 10 and 100
classes, respectively.

Tiny-ImageNet The Tiny-ImageNet is a scaled-down version of the standard
ImageNet-1K consisting of 100,000 64x64 colored images, categorized into 200
classes.

ImageNet-1K (Deng et al., 2009) The ILSVRC 2012 ImageNet-1K classification
dataset consists of 1.28 million training images and 50,000 validation images
of 1000 classes.

##  Appendix B Implementation Details in self-supervised Pre-training,
Finetuning, and Linear Evaluation

ViT architecture. In our non-ImageNet datasets, we adopt smaller ViT backbones
that generally follow (Touvron et al., 2021). The central implementation of
linear separation happens between the MAE encoder and decoder, with a linear
projection layer for each branch of reconstruction. A shared decoder is used
to reconstruct both images. A qualitative evaluation of different ViT sizes on
Tiny-ImageNet is displayed in Fig. 5; the perceptive difference is not large,
and generally, ViT-small/tiny are sufficient for non-ImageNet datasets.

Pre-training. The default setting for pre-training is listed in Tab. 8. On
ImageNet-1K, we strictly use MAE’s specifications. For better classification
performance, we use normalized pixels (He et al., 2022) and a high masking
ratio (0.75); for better visual reconstructions, we use a lower masking ratio
(0.5) without normalizing target pixels. In CIFAR-10/100, and Tiny-ImageNet,
reconstruct ordinary pixels.

Semantics-controllable mixture. The default settings for our semantics-
controllable mixtures are listed in Tab. 8. We modified the dataloader to mix,
within a mini-batch, $\bm{r}$ percent of samples that have homogenous classes
and $1-\bm{r}$ percent that is different.

Classification. For the classification task, we provide the detailed settings
of our finetuning process in Tab. 8 and linear evaluation process in Tab. 8.

# alpha: mixture ratio

# args.beta: hyperparameter for the Beta Distribution.

#

# args.beta=1.0

for x in loader: # Minibatch x of N samples

alpha = np.random.beta(args.beta, args.beta)

sub_idx = np.argmin(alpha, 1-alpha) # Identifying the subordinate (target)
image

perm = torch.randperm(batch_size) # inner-batch mix

im_1, im_2 = x, x[perm, :]

mixed_images = alpha * im_1 + (1-alpha) * im_2

#

# Subordinate Loss

loss_sub = loss_fn (model(mixed_images), im_2)

#

# update gradients

optimizer.zero_grad()

loss.backward()

optimizer.step()

...

Algorithm 1 PyTorch-style pseudocode for a single subordinate reconstruction
on i-MAE.

##  Appendix C More Visualizations

We provide extra examples of a single-branch trained i-MAE reconstructing the
subordinate image. Fig. 11 are visualizations on CIFAR-100 at mix ratios from
0.1 to 0.45, in 0.05 steps. As shown in Fig. 6 and Fig. 7, we produce finer
ranges of reconstructions from 0.05 to 0.45. Notice that in most cases,
mixture rates above 0.4 tend to show features of the dominant image. This
observation demonstrates that a low mixture rate can better embed important
information separating the subordinate image.

##  Appendix D PyTorch Styled Pseudocode

The pseudocode of our mixture and subordinate reconstruction approach is shown
in Algorithm 1. This is only a simple demonstration of our most basic
framework without distillation losses. In our full-fledged i-MAE, we employ
two additional distillation losses, an additional linear separation branch,
and the semantics-controllable mixture scheme; nonetheless, the key
implementation remains the same as the pseudocode presented here.

Figure 5: Different ViT backbones (tiny, small, and base) on Tiny-ImageNet .
Reconstruction quality is moderately improved when a larger backbone is used.
Table 5: Self-supervised pre-training configurations on CIFAR-10/100, Tiny-
ImageNet, ImageNet-1K. For better visualizations, we use a 0.5 mask ratio on
ImageNet.

Config | CIFAR-10/100 | Tiny-ImageNet | ImageNet-1K
---|---|---|---
base learning rate | 1.5e-4 | 1.5e-4 | 1e-3
batch size | 4,096 | 4,096 | 4,096
Mask Ratio | 0.75 | 0.75 | 0.5
optimizer | AdamW | AdamW | AdamW
optimizer momentum | 0.9, 0.95 | 0.9, 0.95 | 0.9, 0.95
augmentation | None | RandomResizedCrop | RandomResizedCrop

Table 6:  Pre-training Configurations of with the semantics-controllable
mixture scheme.

Config | CIFAR-10/100 | Tiny-ImageNet
---|---|---
Object mix range | 0.0, 0.25, 0.5, 1.0 | 0.0, 0.25, 0.5, 1.0
Image mix ratio | $\text{Beta}(1.0,1.0)$ | $\text{Beta}(1.0,1.0)$
base learning rate | 1.5e-4 | 3.5e-4
batch size | 4,096 | 4,096
Mask Ratio | 0.75 | 0.75
optimizer | AdamW | AdamW
optimizer momentum | 0.9, 0.95 | 0.9, 0.95
augmentation | None | RandomResizedCrop

Table 7: Finetune Classification Configurations.

Config | CIFAR-10/100 | Tiny-ImageNet | ImageNet-1K
---|---|---|---
Object mix range | 0.0 - 1.0 | 0.0 - 1.0 | 0.0, 0.25, 0.5, 1.0
Image mix ratio | $\text{Beta}(1.0,1.0)$ | $\text{Beta}(1.0,1.0)$ | 0.8
base learning rate | 1e-3 | 1e-3 | 1e-3
batch size | 128 | 256 | 1,024
epochs | 100 | 100 | 25
optimizer | AdamW | AdamW | AdamW
optimizer momentum | 0.9, 0.999 | 0.9, 0.999 | 0.9, 0.999
augmentation | Mixup | Mixup, RandomResizedCrop | Mixup, RandomResizedCrop

Table 8: Linear Classification Configurations.

Config | CIFAR-10/100 | Tiny-ImageNet | ImageNet-1K
---|---|---|---
Object mix range | 0.0 - 1.0 | 0.0 - 1.0 | 0.0, 0.25, 0.5, 1.0
Image mix ratio | 0.35 | 0.35 | 0.35
base learning rate | 1e-2 | 1e-2 | 1e-2
batch size | 128 | 256 | 1,024
epochs | 200 | 200 | 25
optimizer | SGD | SGD | SGD
optimizer momentum | 0.9, 0.999 | 0.9, 0.999 | 0.9, 0.999
augmentation | None | RandomResizedCrop | RandomResizedCrop

Figure 6: More reconstructions results of i-MAE on ImageNet-1K validation
images with different mixing coefficients $\bm{\alpha}$ (listed on the left)
from models pre-trained with the subordinate image ${\bm{I}_{s}}$ as the only
target, 0.5 mask ratio, and with distillation.

Figure 7: More reconstructions results of i-MAE on ImageNet-1K validation
images with different mixing coefficients $\bm{\alpha}$ (listed on the left)
from models pre-trained with the subordinate image ${\bm{I}_{s}}$ as the only
target, 0.5 mask ratio, and with distillation loss.

Figure 8: Uncurated Tiny-ImageNet reconstructions of different mix ratio, from
0.05 to 0.45, subordinate images.

Figure 9: Visual reconstructions of Tiny-ImageNet validation images using
semantics-controllable mixture pre-trained i-MAE.

Figure 10: CIFAR-100 subordinate reconstruction of different ratios marked on
the left and right side. Similarly, reconstructions at 0.45 are confused with
the dominant image.

Figure 11: Uncurated reconstructions of CIFAR-100 validation images using
semantics-controllable mixture from 0.0 (topmost) to 1.0 (bottom), in 0.1
intervals.


∎

11institutetext: Department of Physics, 22institutetext: Kumaun University,
Nainital
22email<EMAIL_ADDRESS>
# The Schwarzschild Mass in General Relativity

P S Negi

(Received: date / Accepted: date)

###### Abstract

The central (surface) energy-density, $E_{0}(E_{R})$, which appears in the
expression of total static and spherical mass, $M$ (corresponding to the total
radius $R$) is defined as the density measured only by one observer located at
the centre (surface) in the Momentarily Co-moving Reference Frame (MCRF).
Since the mass, $M$, depends only on the central (surface) density for most of
the equations of state (EOSs) and/or exact analytic solutions of Einstein’s
field equations available in the literature, the central (surface) density
measured in the preferred frame (that is, in the MCRF) appears to be not in
agreement with the coordinate invariant form of the field equations that
result for the source mass, $M$. In order to overcome the use of any preferred
coordinate system (the MCRF) defined for the central (surface) density in the
literature, we argue for the first time that the said density may be defined
in the coordinate invariant form, that is, in the form of the average density
($3M/4\pi R^{3})$ of the the configuration which turns out to be independent
of the radial coordinate $`r^{\prime}$ and depends only on the central
(surface) density of the configuration. In this connection, we further argue
that the central (surface) density of the structure should be independent of
the density measured on the other boundary (surface/central) because there
exists no a priori relation between the radial coordinate $`r^{\prime}$ and
the proper distance from the centre of the sphere to its surface Ref1 . In the
light of this reasoning, the various EOSs and analytic solutions of Einstein’s
field equations in which the central and the surface density are
interdependent can not fulfill the definition of central (surface) density
measured only by one observer located in the MCRF at the centre (surface) of
the configuration.

###### Keywords:

Static Spherical Structures Analytic Solutions Neutron Stars Dense Matter:
Equation of State

##  1 Motivation

The metric for the static and spherically symmetric mass distribution can be
written in the curvature coordinates as

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}(d\theta^{2}+{\rm sin}^{2}\theta
d\phi^{2})$ (1)

where $G=c=1$ (we are using geometrized units) and $\nu$ and $\lambda$ are
functions of $`r^{\prime}$ alone.

The Einstein’s field equations for the metric given by eq.(1) can be written
as

$R_{ik}-(1/2)g_{ik}R=-8\pi T_{ik}$ (2)

where $R_{ik}$ is the Ricci tensor, $R$ is the curvature scalar, $T_{ik}$ is
the stress-energy tensor and $g_{ik}$ is the metric tensor. the symbols $i$
and $k$ run from $0$ to $3$ such that $g_{ik}=0$ if $i\neq k$. The $T^{00}$
component of the stress -energy tensor represents the energy-density, $E$, and
$T^{11}=T^{22}=T^{33}=-P$ denote the isotropic pressure of the prefect fluid.

It is well known that in the correspondence limit (the case of weak field and
slow motion) the field equations (eq.2) reduce to the classical Poisson
Equation

$\bigtriangledown^{2}\phi=4\pi\rho$ (3)

Recalling that we have taken $G=c=1,\phi$ is the Newtonian gravitational
potential and $\rho$ is the density of ‘mass’. The solution of eq.(3) for a
‘point’ particle of mass $`M^{\prime}$ is given by

$\phi=-M/r$ (4)

Thus the source of the gravitational field in the Newtonian Gravitation Theory
(NGT) is the mass density, $\rho$, which is approximately equal to the rest-
mass density, $\rho_{0}$. Thus it seems likely that the general relativistic
generalization of this mass density should be the density of total energy
including the rest-mass which is called the energy-density, $E$. However,
since $E$ is the energy-density as measured only by one observer located in
the MCRF which represents the $T^{00}$ component of stress-energy tensor $T$
of the perfect fluid given by eq. (2). This would introduce a preferred
coordinate system (that is, MCRF) in which $T^{00}$ was evaluated. In order to
avoid the use of preferred coordinate systems and to keep intact the
coordinate invariant form of the field equations (eq. 2), it is argued in the
literature Ref1  that the whole of the stress-energy tensor $T$ (all the
components $T^{00}$, $T^{11}$, $T^{22}$ and $T^{33}$) act as the source of
gravitational field (or the curvature of the space-time as given in the left
hand side of eq.2).

##  2 Methodology

The TOV Equations (Ref2 ; Ref3 ) resulting from the field equations (2) for
the metric (eq.1) can be written as

$P^{\prime}=-(P+E)[4\pi Pr^{3}+m]/r(r-2m)$ (5)
$\nu^{\prime}/2=-P^{\prime}/(P+E)$ (6) $m^{\prime}(r)=4\pi Er^{2}$ (7)

where the prime denotes derivative with respect to $`r^{\prime}$ and the mass
function $m(r)$ is given by

$e^{-\lambda}=1-[2m(r)/r]$ (8)

or

$m(r)=\int_{0}^{r}4\pi Er^{2}{\rm d}r$ (9)

The coupled equations (5) - (7) may be solved for an assumed EOS or exact
analytic relation connecting the radial coordinate $`r^{\prime}$ with any of
the four parameters appear in eqs.(5) - (7), subject to the following boundary
conditions at the surface, $r=R$:

$P=P(R)=0;e^{\nu(R)}=e^{-\lambda(R)}=(1-2M/R)=(1-2u)$ that ensures the
continuity of mass at the surface, $m(R)=M$, which appears in the exterior
Schwarzschild solution, viz. ; $e^{\nu}=e^{-\lambda}=(1-2M/r)$, for $r\geq R$.
Thus, the total mass as measured by an external observer is given by

$m(R)=M=\int_{0}^{R}4\pi Er^{2}{\rm d}r$ (10)

Eq. (10) is analogous to the definition of total mass in the NGT. This analogy
is termed as rather ‘deceptive’ in the literature Ref1 , because the energy-
density $E$ is measured locally whereas the integral over the volume element
$4\pi r^{2}{\rm d}r$ is non local. The external observer measures the total
mass-energy which also includes the (negative) gravitational potential energy.
The possibility of the removal of this ‘deception’ is discussed below in the
discussion.

##  3 The Exact Analytic Solutions and EOSs in the Framework of Einstein’s
Field Equations

There are number of EOSs and exact analytic solutions of Einstein’s field
equations available in the literature Ref2 ; Ref4 ; Ref5 ; Ref6 ; Ref7  which
may be categorized among three categories as given below

Category (A): The total mass, $M$, of the configurations in this category
depends only on the central density. The EOSs and analytic solutions have a
positive finite density at the centre ($e^{\lambda}=1$) which decreases
outwards (so called the ‘regular’ solutions) and terminates at the surface
together with pressure. The well known examples of EOS in this category are
polytropic EOSs Ref8  and the analytic solutions in this category correspond
to Tolman’s VII solution with vanishing surface density Ref2  and Buchdahl’s
gaseous model Ref9 . The expression of total mass, for example, for Tolman’s
VII solution with vanishing surface density is given by

$M=8\pi E_{0}R^{3}/15$

, where $E_{0}$ represents the central energy-density of the structure.

Category (B): The total mass, $M$, in this category depends only on the finite
(positive) value of the density at the surface where the pressure vanishes. In
such configurations the central density becomes infinity (together with
pressure and $e^{\lambda}\neq 1$ )which decreases outwards together with
pressure. The total mass, $M$, however remains finite and become independent
of the (infinite) central density. The examples of such exact analytic
solutions are Tolman’s type V and VI solutions Ref2  and the EOS in this
category belongs to the well known case of non-interacting ideal Fermi gas
considered in their pioneering work by Oppenheimer & Volkoff Ref3 . The total
mass $M$, for example, for Tolman’s VI solution is given by the simple
expression:

$M=4\pi E_{R}R^{3}$

, where $E_{R}$ represents the surface density of the structure.

Category (C): The total mass, $M$, in this category depends on the
central/surface density in such a manner that the central and surface
densities have become interdependent. Such structures have a positive finite
density at the centre ($e^{\lambda}=1$) which decreases outwards with pressure
and remains finite positive at the surface where the pressure vanishes. The
examples of exact analytic solutions in this category are Tolman’s IV solution
Ref2  and the solutions obtained in Ref10  \- Ref12  etc., whereas the
examples of EOSs in this category belongs to the case of stiffest EOS Ref13 ,
$\rm dP/\rm dE$ = 1 (in geometrized units) and the EOSs of strange quark
matter (see, for example Ref14  and references therein). The total mass $M$,
for example, in the solution Ref12  is given by the expression:

$M=8X(3+X)R/14(1+X)^{2}$

where $X=CR^{2}$, $C$ is a constant, and the central and surface densities are
connected by the relation

$(E_{R}/E_{0})=(9+2X+X^{2})/9(1+X)^{3}$

This interdependent feature between the central and the surface densities is
clearly in contradiction with the argument mentioned in the literature that
“there is no a priori relation between the radial coordinate ‘$r$’ and the
‘proper’ distance between the centre to the surface of the configuration” Ref1
which was never realized till to date. Because it clearly follows from this
definition that measurements of two local observers located at different
boundaries (centre and surface) can not be connected simply through the radial
coordinate ‘$r$’. However, the author reached to the same conclusion on the
basis of a different argument Ref15 . He showed that if the central and
surface densities have become interdependent, the total mass, $M$, resulting
from the configurations discussed under category (C) will be different from
the total mass $M$ which appears in the exterior Schwarzschild solution.

##  4 Discussion

In the light of the above findings, it follows that the total mass $M$
obtained for the categories (A) and (B) solutions and EOSs depends only on the
central (surface) density ( and turns out to be independent of the surface
(central) density) of the configurations. This feature is fully consistent
with the definition of centre (surface) density mentioned in the literature
that - it is the density measured only by one observer (MCRF) located at the
centre (surface) of the configuration. The total masses $M$ of the
configurations belong to category (C), on the other hand, are found to be
interdependent of the central and the surface densities. This feature,
therefore, can not be considered consistent with the definition of the locally
measured values of the central and the surface densities which should be
independent of each other as argued above in sec. 3. This fact follows from
the definition mentioned above which states that “there is no a priori
relation between the radial coordinate ‘$r$’ and the proper distance between
the centre and the surface of the configuration” Ref1 .

Yet the dependence of the total mass $M$ only on the central (surface) density
further needs explanation, because the central (surface) density which is
measured in the preferred coordinate system (the MCRF at the centre/surface)
would become inconsistent with the coordinate invariant form of the field
equations (5 - 7). Therefore, in order to keep the definition of the central
(surface) density intact which follows from the literature Ref1  and to
express it in the coordinate invariant form consistent with the field
equations, we argue for the first time that the central (surface) density of
the configuration may be defined in the coordinate invariant form of the
‘average (mean) density’, $E_{ave}$, of the structure ($3M/4\pi R^{3}$) so
that the average density depends only on the central (surface) density of the
structure. Except the category (C), this feature is common in the structures
belong to the category (A) and (B). Furthermore, the use of the ‘average’
density in Eq. (10) of the total mass $M$ would also resolve the ‘deception’
related with the analogy of the total mass as defined in the NGT.

The ‘equivalent’ homogeneous density (average density $E_{ave}$) sphere of
total mass $M$ and radius $R$ as measured by an external observer,
corresponding to eq.(10) is given by

$m(R)=M=\int_{0}^{R}4\pi E_{\rm ave}r^{2}{\rm d}r$ (11)

By using the relation connecting the rest-mass density, $\rho_{0}$ to the
energy-density, $E$ Ref16  and remembering that the rest- mass density is
approximately equal to the mass density defined in the NGT Ref1 , Ref5  we get

$\rho_{0}=(P+E)e^{(\nu-\nu_{R})/2}$ (12)

Applying the boundary conditions at the surface ($P=0$,
$e^{\nu}=e^{\nu_{R}}$), for homogeneous density ($E_{ave}$) sphere eq.(12)
yields

$\rho_{0}=E_{ave}$ (13)

The substitution of eq.(13) into eq.(11) yields

$m(R)=M=\int_{0}^{R}4\pi\rho_{0}r^{2}{\rm d}r$ (14)

which is completely analogous to the form of total mass as defined in the NGT.

## References

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