11institutetext: Rochester Institue of Technology, NY. USA

# Distributionally Robust Optimization and Invariant Representation Learning
for Addressing Subgroup Underrepresentation:
Mechanisms and Limitations

Nilesh Kumar∗ Ruby Shrestha∗ Zhiyuan Li  Linwei Wang

###### Abstract

Spurious correlation caused by subgroup underrepresentation has received
increasing attention as a source of bias that can be perpetuated by deep
neural networks (DNNs). Distributionally robust optimization has shown success
in addressing this bias, although the underlying working mechanism mostly
relies on upweighting under-performing samples as surrogates for those
underrepresented in data. At the same time, while invariant representation
learning has been a powerful choice for removing nuisance sensitive features,
it has been little considered in settings where spurious correlations are
caused by significant underrepresentation of subgroups. In this paper, we take
the first step to better understand and improve the mechanisms for debiasing
spurious correlation due to subgroup underrepresentation in medical image
classification. Through a comprehensive evaluation study, we first show that
1) generalized reweighting of under-performing samples can be problematic when
bias is not the only cause for poor performance, while 2) naive invariant
representation learning suffers from spurious correlations itself. We then
present a novel approach that leverages robust optimization to facilitate the
learning of invariant representations at the presence of spurious
correlations. Finetuned classifiers utilizing such representation demonstrated
improved abilities to reduce subgroup performance disparity while maintaining
high average and worst-group performance.

###### Keywords:

Spurious correlations DRO Invariant representations.

##  1 Introduction

**footnotetext: These authors contributed equally to this work

As deep neural networks (DNNs) continue to demonstrate successes in various
tasks [7, 18], their potential to generate and perpetuate bias also received
growing attention [16, 14]. One source of biases being increasingly discussed
is the presence of spurious correlation due to subgroup underrepresentation,
where features irrelevant to a task happen to co-exist with a decision label
in the majority of the samples (bias-aligning samples) whereas samples that do
not exhibit such correlation are underrepresented (bias-conflicting samples).
Examples may include the presence of treatment features in positive disease
samples (e.g., the presence of drain as a treatment was observed in the
majority of pneumothorax samples in CXR-14 chest X-rays [16]), or a higher
prevalence of disease in certain demographic subgroups (e.g., malignant skin
cancer was more commonly reported in lighter skin tones [9]). With the
standard empirical risk minimization (ERM), [23], the DNN is trained to
exploit such spurious correlation for reducing the average loss of all
training samples [14]. Such DNN will struggle with bias-conflicting subgroups.

Most works have approached this bias from the lens of distributionally robust
optimization (DRO), a classic optimization concept that focuses on minimizing
worst-case losses over possible test distributions [1]. The recent integration
of DRO into DNN was done by describing possible test distributions with
importance sampling of the observed subgroups [20]: the group DRO (GDRO) is
then solved by iteratively optimizing the importance weights of all subgroups
to maximize the DNN loss while optimizing the DNN to minimize such worst-case
loss. This min-max formulation enables GDRO to pay equal or even greater
attention to underrepresented subgroups, although it requires the subgroup
labels to be known a priori. To remove this need, various approaches have
followed to identify bias-conflicting samples by, for instance, considering
high-loss samples in a biased model [12, 15] or clustering the biased
representations [22]. At the core of all of these approaches is a generalized
re-weighting scheme to achieve DRO, i.e., minimizing worst-case losses by
upweighting the contribution from bias-conflicting subgroups. A fundamental
working mechanism behind these approaches, however, assumes that high training
loss of the ERM model can be used to determine the upweighting of bias-
conflicting samples. While this was shown to be empirically effective on
carefully designed benchmark datasets where spurious correlation is the
dominating cause for poor performance, it is not clear how this working
mechanism may generalize in real-world datasets where the high loss of a
training sample may be due to other challenges such as the difficulty to
extract discriminative semantics for certain disease categories.

In parallel, invariant representation learning has been a mainstream approach
for removing confounding semantics not responsible for a task, especially for
training DNNs that can adapt or generalize across domains [4]. It has also
been widely used in fair representation learning to obfuscate sensitive group
attributes from latent representations [3, 13, 17, 21, 27]. These works
however either do not deal with spurious correlations due to underrepresented
subgroups, deal with limited spurious correlation resulting from insubstantial
bias ratio, or make the assumption that there exists a curated subset of data
(for example, a training class) that does not exhibit subgroup
underrepresentation. Within limited attempts of using invariant
representations to handle spurious correlation, there are often specific
assumptions of what may be the spurious factors (e.g. texture of the image)
and thus lacks generality [24]. To the best of our knowledge, there is very
limited work on general purpose methods to learn representations invariant to
spurious correlations [26]. This leaves another intriguing open question: what
may be the role and challenges of invariant representation learning in
addressing spurious correlation caused by underrepresentation?

In a nutshell, despite increasing interest in DNN fairness and bias [25],
approaches to specifically address spurious correlation caused by subgroup
representation have been limited in medical image classification [8, 22] and
their working debiasing mechanisms and limitations are not well understood. In
this paper, we take the first step to better understand and improve the
debiasing mechanism for addressing spurious correlation in medical image
classification. To understand the potential mechanisms for debiasing and
corresponding limitations (Section 2), we conduct a comprehensive evaluation
study in which we examine the aforementioned families of approaches to address
spurious correlation on two skin lesion datasets. Results suggested that 1)
generalized reweighting based on under-performing samples can be problematic
in real-world datasets where bias is not the only cause for low performance,
and 2) invariant representation learning suffers from spurious correlation and
subgroup underrepresentation itself. As a first step to address these observed
limitations, we further show that – while GDRO in itself has little influence
on representation learning – its combination with domain adversarial loss can
enable invariant representation learning in the presence of spurious
correlation (Section 3). Fine-tuning the classifier using such representation
on balanced validation dataset, as inspired by deep feature reweighting (DFR)
[10], further improved the reduction of subgroup disparity while achieving
high average performance.

##  2 Assessing Debiasing Mechanisms

###  2.1 Methodology

####  2.1.1 Data:

We consider two skin lesion datasets that exhibit different levels of
complexity in the underlying tasks, biases, and the number of subgroups.

ISIC [2]: The spurious correlation in ISIC was originally caused by the
presence of bandages in benign examples and the absence of bandages in
malignant examples [5]. Here we introduce a small subgroup of such malignant
samples in the training to assess bias due to such underrepresentation. This
is achieved by

artificially adding bandages to malignant examples using the segmentation
masks for bandages from [19]. Table 1 shows the distribution of training data.

Fitzpatrick [6]: As summarized in Table 1, in Fitzpatrick, there is a higher
prevalence of malignant skin cancers in lighter skin tones (Fitzpatrick skin
type 1-2) versus a higher prevalence of non-neoplastic lesions in darker skin
tones (Fitzpatrick skin type 5-6). This shift of label distribution among
skin-type subgroups may create a bias of under-diagnosis of malignant skin
cancers in individuals with darker skin tones. To focus on this potential
bias, we reduce overall skin-type imbalance by downsampling samples from skin
types 1-4, while maintaining the same label distribution within each skin-type
subgroup.

Table 1: Subgroup distribution in training. Bias-conflicting subgroups shaded
gray.

Dataset |  | Subgroup Distribution in Training Data
---|---|---
ISIC |  | Benign-no bandage | Benign-bandage | Malignant-no bandage | Malignant-bandage
|  | 4843 | 4890 | 5205 | 100

Fitzpatrick |  Class Skin type | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
| Benign | 15.07% | 13.96% | 14.36% | 13.20% | 10.37% | 6.93%
| Malignant | 15.37% | 15.43% | 13.78% | 10.82% | 9.59% | 9.61%
| Non-neoplastic | 69.56% | 70.61% | 71.86% | 75.98% | 80.04% | 83.46%

####  2.1.2 Models:

We consider three families of models to provide complete coverage of existing
strategies focused on DRO and/or representation learning.

Generalized reweighting approaches: Representing approaches using known
subgroup labels, we consider simple importance weighting (with fixed subgroup
weights) and GDRO [20] (with dynamic weights). Representing approaches with
unknown subgroups, we consider JTT [12] where bias-conflicting samples are
identified as under-performing samples in ERM, followed by upweighting.

Invariant representation learning: We consider a classic invariant
representation learning strategy that removes domain information from the
latent representation by a reversal gradient layer (known as domain
adversarial training of neural networks/DANN) [4]. We use the bias-inducing
factor as the domain, i.e. the presence and absence of bandage in ISIC and the
skin types in Fitzpatrick. We also consider the only existing general-purpose
invariant representation learning approach reported for spurious correlation,
where contrastive learning is used to push representations of samples of the
same class closer (CnC) [26].

Separated representation & decision-boundary learning: We include deep feature
reweighting (DFR) [10] that first obtains biased ERM representation, and then
re-trains a classifier using such representations on a balanced validation
set.

####  2.1.3 Evaluation Metrics:

We consider three types of evaluation metrics.

To evaluate bias in representations, we leverage self-organizing maps (SOM)
[11] to cluster and visualize the latent representations. We also adopt a
quantitative metric of cluster purity as a surrogate measure of biases in the
latent representations. This is motivated by the concept that, in an ideal
scenario, an unbiased representation should discard and thus not be separable
by spuriously-correlated features.

To evaluate bias in the decision boundary, we consider two general types of
quantitative metrics: 1) test performance for the worst-performing subgroup,
the anticipated bias-conflicting subgroups, and averages across all subgroups;
2) The performance disparity ($\Delta$) among all subgroups: in ISIC, this is
measured as performance difference between worst-performing subgroups to best-
performing subgroups ($\Delta_{best-worst}$) and to group-average performance
($\Delta_{avg-worst}$), respectively. In Fitzpatrick, because the number of
subgroups is large ($3\times 6$), this is measured by $\Delta$ between best-
and worst-performing subgroups in each class: malignant, non-neoplastic, and
benign ($\Delta_{mn},\Delta_{nn},\Delta_{bg}$) as well as their average
($\Delta_{avg}$).

To assess the debiasing mechanisms, for generalized reweighting with and
without known subgroups, we track their importance weights, or the subgroups
identified for up-weighting, respectively. For learning invariant
representations, we track the accuracy of the domain classifier in removing
spurious features.

###  2.2 Experiments and Results

| Avg | Worst | Malignant Bandage | $\Delta_{best-worst}$ | $\Delta_{avg-worst}$
---|---|---|---|---|---
ERM | $0.803\pm 0.005$ | $0.556\pm 0.072$ | $0.556\pm 0.072$ | $0.443\pm 0.072$ | $0.246\pm 0.066$
Important weighting | $0.793\pm 0.011$ | $0.593\pm 0.040$ | $0.593\pm 0.040$ | $0.406\pm 0.040$ | $0.200\pm 0.030$
JTT | $0.790\pm 0.011$ | $0.550\pm 0.051$ | $0.550\pm 0.051$ | $0.450\pm 0.051$ | $0.240\pm 0.041$
DANN | $0.843\pm 0.005$ | $0.730\pm 0.017$ | $0.730\pm 0.017$ | $0.266\pm 0.023$ | $0.113\pm 0.011$
DFR | $\pagecolor{shadecolor}0.871\pm 0.004$ | $0.758\pm 0.008$ | $\pagecolor{shadecolor}0.865\pm 0.010$ | $0.232\pm 0.010$ | $0.112\pm 0.010$
GDRO | $0.846\pm 0.011$ | $0.760\pm 0.017$ | $0.763\pm 0.015$ | $0.240\pm 0.017$ | $0.086\pm 0.005$
GDRO with group adjustment | $\mathbf{0.866\pm 0.005}$ | $0.786\pm 0.011$ | $0.803\pm 0.020$ | $0.213\pm 0.011$ | $0.080\pm 0.005$
Proposed method | $\mathbf{0.863\pm 0.005}$ | $\pagecolor{shadecolor}0.800\pm 0.000$ | $\mathbf{0.813\pm 0.023}$ | $\pagecolor{shadecolor}0.190\pm 0.000$ | $\pagecolor{shadecolor}0.063\pm 0.005$

Table 2:  Accuracy of benign vs. malignant classification on ISIC. Malignant
samples with a bandage is the underrepresented bias-conflicting group as
expected from Table 1. Columns 2-4 list accuracy of averaged and individual
subgroups. Columns 5-6 measure the performance disparity among subgroups. Best
performance is shaded gray, with bolded performance closely behind.

Following convention [20, 26], we used ResNet-50 from torch-vision with pre-
trained weights from Imagenet. We used held-out validation set to choose the
best model unless the method is using the validation set for fine-tuning
(e.g., DFR).

Figure 1: SOM plots and their averaged subgroup purities.

####  2.2.1 ISIC:

As summarized in Table 2, with ERM, the DNN suffered from a significant
performance drop for the bias-conflicting subgroup (malignant samples with
bandage). This bias is also evident in the SOM (Fig 1A), which shows clearly
separated subgroups and a relatively high purity score.

Generalized reweighting approaches: Simple importance weighting (to equally
weight all subgroups) improved the test accuracy on the bias-conflicting group
but to a limited extent. GDRO more significantly improved the test performance
of the bias-conflicting subgroup. Interestingly, a closer inspection on the
subgroup weights obtained by the GDRO (Figure. 2A) showed that GDRO actually
upweighted the two non-bandage subgroups. Evidently, with group-based average
loss, the underrepresented group is already upweighted in GDRO which quickly
resulted in a lower loss at the early stage of training. As such, the other
two non-bandage subgroups unexpectedly became the higher-loss subgroups
(Figure. 2B) and were upweighted. The separation of subgroups in the SOM
remained similar to ERM with slightly reduced purity (Fig 1B).

Facing this unexpected working mechanism of GDRO, we further experimented with
a version of GDRO with group adjustment [20]: this adds an additional term of
$\frac{1}{\sqrt{N_{s}}}$ to the subgroup loss where $N_{s}$ is the size of a
subgroup. As expected, this term dominates the loss used to optimize the
weights (Figure. 2D) and resulted in a heavy upweighting of the bias-
conflicting group (Figure. 2C). This resulted in further improvement on this
subgroup. Clustering of the latent representation and its purity however
remained similar to the ERM (Fig 1C).

These results revealed that the use of group adjustment played a significant
role in the weighting mechanism of GDRO. Without this, the upweighting was
influenced by other factors contributing to low performance and the overall
results appeared to be adversely affected.

Invariant representation approaches: DANN was trained with upsampling of the
underrepresented subgroup. Its performance was better than simple importance
weighting, but inferior to GDRO. A closer look at the obtained representation
clusters (Fig 1D) and the behaviour of the domain classifier (Figure 3A)
suggested that DANN was not successful in removing bandage information from
the bias-aligning subgroup (benign with bandage). This failure suggested that
domain-invariant representation learning also suffers from spurious
correlations.

Approaches without subgroup labels: For the remaining models that do not
require the use of subgroup labels, the performance was suboptimal. JTT showed
minor improvement over ERM, while CnC under-performed than ERM (results thus
not included). Common to both approaches was the reliance on a biased ERM
model to identify bias-conflicting examples. A closer inspection showed that,
among the under-performing samples selected from the biased ERM, only
1.05-2.75% belonged to the bias-conflicting subgroup. This explained why the
subsequent learning approaches may not be successful. Finally, DFR achieved an
impressive average performance as well as on the bias-conflicting subgroup,
albeit at the expense of creating a new worst-performing subgroup.

Figure 2:  Weight and loss plots for GDRO with and without group adjustment.
Figure 3: Subgroup accuracy of domain classifier in A) DANN and B) as
proposed.

####  2.2.2 Fitzpatrick:

Table 3 shows the average AUC, worst-AUC among all 18 subgroups, and AUCs for
four selected subgroups that are most indicative of the bias present: with
ERM, the subgroups with darker skin tones are disadvantaged in malignant
cancer detection (MN-56, AUC of 0.81 vs., 0.88 for MN-12), whereas the
subgroups with lighter skin tones are disadvantaged in non-neoplastic lesion
detection (NN-12, AUC of 0.78 vs., 0.86 for NN-56). This disparity in
performance is perfectly associated with the subgroup label distributional
shifts shown in Table 1, an intriguing non-artificial bias that may be
challenging to address.

The trend of performance of all models was similar to that observed in ISIC
with some distinction. JTT and CnC continued to under-perform and their
results are not included. Unlike on ISIC, here both DANN and DFR obtained
negligible or even negative gains over ERM. In comparison, various reweighting
strategies were all effective in addressing the bias against MN-56, albeit 1)
at the expense of decreased average AUC, and 2) ineffective in addressing the
bias against NN-12. The measures of performance disparity within and averaged
across classes (Table 4) suggested the same trend.

Note that the worst-performance subgroups from all models were different from
the bias-conflicting subgroups, which may be why this particular bias was
challenging to remove.

##  3 Improving the Debiasing of Spurious Correlations

####  3.0.1 Methodology:

The investigations in Section 2 provided two important findings. First, the
reliance on low training performance for identifying or upweighting bias-
conflicting samples may face challenges where bias is not the only cause for
under-performing samples. Second, while invariant representation in concept
appears a natural candidate for removing confounders, its learning ironically
also suffers from spurious correlation caused by underrepresentation. This
raises a critical question: is our best hope upweighting known bias-
conflicting subgroups?

We re-examine DANN’s failure to confuse the domain classifier on the bias-
aligned subgroups (Fig. 3A). A possible explanation is that the main task
classifier was exploiting spurious correlations and overpowers the domain
classifier. Based on this, we hypothesize that a robust main classifier with a
reduced tendency to exploit spurious correlation may better support DANN to
learn representations invariant to spurious correlations. To this end, we
propose to optimize the main classifier in DANN with GDRO and, with the
learned invariant representation, fine-tune only the classifier on a small
validation set.

####  3.0.2 Experiments and Results:

On ISIC, the proposed model was successful in extracting invariant
representations, evidenced both by the confused DANN domain classifier (Figure
3B) and the substantially reduced SOM purity (Fig 1E). After fine-tuning this
invariant representation on a small balanced validation set as used in DFR,
the proposed approach achieved the highest worst-performance among all models
considered (Table 2). Its performance on the bias-conflicting subgroup was
better than GDRO with group adjustment, and its average performance on par.
Note that while DFR exhibited a stronger average performance on the bias-
conflicting subgroup, its worst-group performance was much lower. In
comparison, the proposed method was the most successful in reducing subgroup
disparity (last two columns in Table 2) with strong average performance.

| Average | Worst | MN-12 | MN-56 | NN-12 | NN-56
---|---|---|---|---|---|---
ERM | $\pagecolor{shadecolor}0.810\pm 0.000$ | $0.617\pm 0.040$ | $0.878\pm 0.008$ | $0.810\pm 0.018$ | $0.777\pm 0.010$ | $\pagecolor{shadecolor}0.855\pm 0.005$
Importance weighting | $0.803\pm 0.015$ | $0.673\pm 0.035$ | $0.878\pm 0.010$ | $0.853\pm 0.008$ | $0.780\pm 0.001$ | $0.853\pm 0.010$
DANN | $0.787\pm 0.006$ | $0.607\pm 0.015$ | $0.857\pm 0.010$ | $0.758\pm 0.014$ | $0.785\pm 0.005$ | $0.800\pm 0.015$
DFR | $0.790\pm 0.010$ | $0.660\pm 0.036$ | $0.863\pm 0.003$ | $0.813\pm 0.020$ | $0.778\pm 0.008$ | $0.828\pm 0.010$
GDRO | $0.800\pm 0.010$ | $0.640\pm 0.030$ | $0.868\pm 0.008$ | $0.848\pm 0.055$ | $0.788\pm 0.012$ | $0.825\pm 0.020$
GDRO with group adjustment | $0.780\pm 0.010$ | $0.647\pm 0.006$ | $0.852\pm 0.008$ | $\pagecolor{shadecolor}0.877\pm 0.016$ | $0.782\pm 0.015$ | $0.788\pm 0.022$
Proposed method | $\pagecolor{shadecolor}0.810\pm 0.000$ | $\pagecolor{shadecolor}0.680\pm 0.026$ | $\pagecolor{shadecolor}0.883\pm 0.003$ | $0.837\pm 0.003$ | $\pagecolor{shadecolor}0.803\pm 0.003$ | $0.830\pm 0.009$

Table 3:  AUC of benign, malignant, and non-neoplastic skin lesion
classification on Fitzpatrick. MN-12/-56: malignant with skin type 1-2 / 5-6.
NN-12/56: non-neoplastic with skin type 1-2 / 5-6. MN-56 and NN-12 are the
primary bias-conflicting subgroups.

| $\Delta_{mn}$ | $\Delta_{nn}$ | $\Delta_{bg}$ | $\Delta_{avg}$
---|---|---|---|---
ERM | $0.140\pm 0.026$ | $0.083\pm 0.015$ | $0.223\pm 0.032$ | $0.149\pm 0.020$
Importance Weighting | $0.073\pm 0.021$ | $0.080\pm 0.020$ | $0.153\pm 0.006$ | $0.102\pm 0.010$
DANN | $0.250\pm 0.036$ | $0.040\pm 0.010$ | $0.233\pm 0.015$ | $0.174\pm 0.018$
DFR | $0.157\pm 0.040$ | $0.053\pm 0.006$ | $0.150\pm 0.030$ | $0.120\pm 0.021$
GDRO | $0.090\pm 0.087$ | $0.053\pm 0.025$ | $0.197\pm 0.055$ | $0.113\pm 0.055$
GDRO with group adjustment | $\pagecolor{shadecolor}0.053\pm 0.015$ | $\pagecolor{shadecolor}0.030\pm 0.010$ | $0.163\pm 0.021$ | $\mathbf{0.082\pm 0.004}$
Proposed method | $0.090\pm 0.010$ | $\mathbf{0.037\pm 0.006}$ | $\pagecolor{shadecolor}0.117\pm 0.031$ | $\pagecolor{shadecolor}0.081\pm 0.008$

Table 4:  Difference in AUCs between best- and worst-performing subgroups in
each class, from left to right: malignant, non-neoplastic, benign, and
averaged.

Evidence of improved invariant representations was similar on Fitzpartrick
(results shown in supplemental materials).

As summarized in Tables 3-4, compared to reweighting methods, the fine-tuned
classifier as proposed was the only one that was able to reduce the bias
against NN-12, and delivered the best worst-group as well as average
performance. Overall, its performance in removing subgroup disparity was on
par with GDRO with group adjustment (Table 4) while delivering significantly
higher average and worst-group AUCs (Table 3).

####  3.0.3 Conclusions and Discussion:

We presented an evaluation study that derived important new insights into the
working mechanisms and limitations of DRO and invariant representation
learning to address spurious correlation caused by underrepresentation. The
findings motivated us to present a novel approach that leverages robust
optimization to facilitate the learning of invariant representations at the
presence of spurious correlations. Finetuned classifiers utilizing such
representation demonstrated an improved ability to reduce subgroup performance
disparity while maintaining high average and worst-group performance. Future
investigations will include a broader spectrum of approaches including those
utilizing data augmentation, as well as extending to a wider range of medical
image datasets exploring potential hidden biases.

##  4 Acknowledgments

This work is supported by the National Institute of Nursing Research (NINR) of
the National Institutes of Health (NIH) under Award Number R01NR018301.

## References

  * [1] Ben-Tal, A., Den Hertog, D., De Waegenaere, A., Melenberg, B., Rennen, G.: Robust solutions of optimization problems affected by uncertain probabilities. Management Science 59(2), 341–357 (2013)
  * [2] Codella, N., Rotemberg, V., Tschandl, P., Celebi, M.E., Dusza, S., Gutman, D., Helba, B., Kalloo, A., Liopyris, K., Marchetti, M., et al.: Skin lesion analysis toward melanoma detection 2018: A challenge hosted by the international skin imaging collaboration (isic). arXiv preprint arXiv:1902.03368 (2019)
  * [3] Deng, W., Zhong, Y., Dou, Q., Li, X.: On fairness of medical image classification with multiple sensitive attributes via learning orthogonal representations. In: International Conference on Information Processing in Medical Imaging. pp. 158–169. Springer (2023)
  * [4] Ganin, Y., Ustinova, E., Ajakan, H., Germain, P., Larochelle, H., Laviolette, F., Marchand, M., Lempitsky, V.: Domain-adversarial training of neural networks (2015). https://doi.org/10.48550/ARXIV.1505.07818, https://arxiv.org/abs/1505.07818
  * [5] Goel, K., Gu, A., Li, Y., Ré, C.: Model patching: Closing the subgroup performance gap with data augmentation. arXiv preprint arXiv:2008.06775 (2020)
  * [6] Groh, M., Harris, C., Soenksen, L., Lau, F., Han, R., Kim, A., Koochek, A., Badri, O.: Evaluating deep neural networks trained on clinical images in dermatology with the fitzpatrick 17k dataset. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 1820–1828 (2021)
  * [7] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 770–778 (2016)
  * [8] Jiménez-Sánchez, A., Juodelye, D., Chamberlain, B., Cheplygina, V.: Detecting shortcuts in medical images-a case study in chest x-rays. arXiv preprint arXiv:2211.04279 (2022)
  * [9] Kinyanjui, N.M., Odonga, T., Cintas, C., Codella, N.C., Panda, R., Sattigeri, P., Varshney, K.R.: Fairness of classifiers across skin tones in dermatology. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2020: 23rd International Conference, Lima, Peru, October 4–8, 2020, Proceedings, Part VI. pp. 320–329. Springer (2020)
  * [10] Kirichenko, P., Izmailov, P., Wilson, A.G.: Last layer re-training is sufficient for robustness to spurious correlations. arXiv preprint arXiv:2204.02937 (2022)
  * [11] Kohonen, T.: The self-organizing map. Proceedings of the IEEE 78(9), 1464–1480 (1990)
  * [12] Liu, E.Z., Haghgoo, B., Chen, A.S., Raghunathan, A., Koh, P.W., Sagawa, S., Liang, P., Finn, C.: Just train twice: Improving group robustness without training group information. In: Meila, M., Zhang, T. (eds.) Proceedings of the 38th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 139, pp. 6781–6792. PMLR (18–24 Jul 2021), https://proceedings.mlr.press/v139/liu21f.html
  * [13] Louizos, C., Swersky, K., Li, Y., Welling, M., Zemel, R.: The variational fair autoencoder (2015). https://doi.org/10.48550/ARXIV.1511.00830, https://arxiv.org/abs/1511.00830
  * [14] McCoy, T., Pavlick, E., Linzen, T.: Right for the wrong reasons: Diagnosing syntactic heuristics in natural language inference. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. pp. 3428–3448. Association for Computational Linguistics, Florence, Italy (Jul 2019). https://doi.org/10.18653/v1/P19-1334, https://aclanthology.org/P19-1334
  * [15] Nam, J., Cha, H., Ahn, S., Lee, J., Shin, J.: Learning from failure: De-biasing classifier from biased classifier. Advances in Neural Information Processing Systems 33, 20673–20684 (2020)
  * [16] Oakden-Rayner, L., Dunnmon, J., Carneiro, G., Ré, C.: Hidden stratification causes clinically meaningful failures in machine learning for medical imaging. In: Proceedings of the ACM conference on health, inference, and learning. pp. 151–159 (2020)
  * [17] Park, S., Hwang, S., Kim, D., Byun, H.: Learning disentangled representation for fair facial attribute classification via fairness-aware information alignment. In: Proceedings of the AAAI Conference on Artificial Intelligence. vol. 35, pp. 2403–2411 (2021)
  * [18] Rajpurkar, P., Irvin, J., Zhu, K., Yang, B., Mehta, H., Duan, T., Ding, D., Bagul, A., Langlotz, C., Shpanskaya, K., et al.: Chexnet: Radiologist-level pneumonia detection on chest x-rays with deep learning. arXiv preprint arXiv:1711.05225 (2017)
  * [19] Rieger, L., Singh, C., Murdoch, W., Yu, B.: Interpretations are useful: penalizing explanations to align neural networks with prior knowledge. In: International conference on machine learning. pp. 8116–8126. PMLR (2020)
  * [20] Sagawa, S., Koh, P.W., Hashimoto, T.B., Liang, P.: Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. arXiv preprint arXiv:1911.08731 (2019)
  * [21] Sarhan, M.H., Navab, N., Eslami, A., Albarqouni, S.: Fairness by learning orthogonal disentangled representations. In: Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXIX 16\. pp. 746–761. Springer (2020)
  * [22] Sohoni, N., Dunnmon, J., Angus, G., Gu, A., Ré, C.: No subclass left behind: Fine-grained robustness in coarse-grained classification problems. Advances in Neural Information Processing Systems 33, 19339–19352 (2020)
  * [23] Vapnik, V.: Principles of risk minimization for learning theory. Advances in neural information processing systems 4 (1991)
  * [24] Wang, H., He, Z., Lipton, Z.C., Xing, E.P.: Learning robust representations by projecting superficial statistics out. arXiv preprint arXiv:1903.06256 (2019)
  * [25] Wu, Y., Zeng, D., Xu, X., Shi, Y., Hu, J.: Fairprune: Achieving fairness through pruning for dermatological disease diagnosis. In: Wang, L., Dou, Q., Fletcher, P.T., Speidel, S., Li, S. (eds.) Medical Image Computing and Computer Assisted Intervention – MICCAI 2022. pp. 743–753. Springer Nature Switzerland, Cham (2022)
  * [26] Zhang, M., Sohoni, N.S., Zhang, H.R., Finn, C., Re, C.: Correct-n-contrast: a contrastive approach for improving robustness to spurious correlations. In: Chaudhuri, K., Jegelka, S., Song, L., Szepesvari, C., Niu, G., Sabato, S. (eds.) Proceedings of the 39th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 162, pp. 26484–26516. PMLR (17–23 Jul 2022), https://proceedings.mlr.press/v162/zhang22z.html
  * [27] Zhao, Q., Adeli, E., Pohl, K.M.: Training confounder-free deep learning models for medical applications. Nature communications 11(1), 6010 (2020)


=2016

# An approach to directly probe simultaneity00footnotetext: This is an Open
Access article published by World Scientific Publishing Company. It is
distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY)
License. Further distribution of this work is permitted, provided the original
work is properly cited.

Edward T. Kipreos∗ and Riju S. Balachandran† [ [ University of Georgia, 120
Cedar Street, Athens, GA 30602, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(28 September 2015; 15 April 2016; 27 June 2016)

###### Abstract

The theory of special relativity derives from the Lorentz transformation. The
Lorentz transformation implies differential simultaneity and light speed
isotropy. Experiments to probe differential simultaneity should be able to
distinguish the Lorentz transformation from a kinematically-similar alternate
transformation that predicts absolute simultaneity, the absolute Lorentz
transformation. Here, we describe how published optical tests of light speed
isotropy/anisotropy cannot distinguish between the two transformations. We
show that the shared equations of the two transformations, from the
perspective of the “stationary” observer, are sufficient to predict null
results in optical resonator experiments and in tests of frequency changes in
one-way light paths. In an influential 1910 exposition on differential
simultaneity, Comstock described how a “stationary” observer would observe
different clock readings for spatially-separated “moving” clocks. The
difference in clock readings is an integral aspect of differential
simultaneity. We derive the equation for the difference in clock readings and
show that it is equivalent to the Sagnac correction that describes light speed
anisotropies in satellite communications. We describe an experimental strategy
that can measure the differences in spatially-separated clock times to allow a
direct probe of the nature of simultaneity.

###### keywords:

Special relativity; Lorentz transformation; light speed isotropy.

2 August 2016

PACS No.: 03.30.+p

##  1 Introduction

The theory of special relativity (SR) is based on the Lorentz transformation
(LT), which describes how changes in velocity are linked to time dilation and
length contraction. The LT implies that length contraction and time dilation
are reciprocally observed between two inertial reference frames (IRFs).
Embedded in the LT is differential simultaneity and the isotropic speed of
light, denoted $c$. In order to experimentally assess differential
simultaneity, it is necessary to be able to distinguish the LT from alternate
transformations that predict absolute simultaneity. Here, we will show that
current optical tests are unable to distinguish between the LT and a
kinematically-similar transformation theory that predicts absolute
simultaneity, absolute simultaneity theory (AST). We will clarify an equation
for a key attribute of differential simultaneity and describe an experimental
strategy that measures this attribute to probe the nature of simultaneity.

###  1.1 Comparison of the Lorentz transformation and ALT

The only alternate transformation that is compatible with classical tests of
SR is the absolute Lorentz transformation (ALT).[1, 2] ALT was named by
Tangherlini, who described the transformation in 1958.[3] ALT was initially
published by Eagle in 1938.[4] In the modern era, Eagle’s work does not appear
to have been recognized as the first publication of the ALT equations until it
was reported in two 2009 reviews.[5, 6]

The LT and ALT share the same length transformation equation:

$x^{\prime}=\frac{x-vt}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\,.$ (1)

The two transformations differ in their time transformation equations. The LT
time transformation equation is:

$t^{\prime}=\frac{t-\frac{vx}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\,.$ (2)

The ALT time transformation equation from the stationary perspective is:

$t^{\prime}=t\sqrt{1-\frac{v^{2}}{c^{2}}}\,.$ (3)

In the LT framework, $x^{\prime}$ and $t^{\prime}$ are “moving” coordinates
for distance and time, respectively; and $x$ and $t$ are “stationary”
coordinates. “Moving” and “stationary” coordinate systems are interchangeable,
and the velocity $(v)$ is calculated between two IRFs. ALT implies the
presence of a preferred reference frame (PRF) in which light speed is
isotropic. In the ALT framework, $x$ and $t$ are stationary PRF coordinates,
and the velocity of objects is calculated relative to the PRF.

Experimental tests of time dilation in SR use Eq. (3), the ALT time
transformation equation. This is because experimental tests generally analyze
movements in linear, constant-velocity IRFs for which $x=vt$. Substituting
$x=vt$ in the LT time transformation equation (2) produces the ALT time
transformation equation (3).[7] Therefore, in experimental settings, the two
theories calculate time dilation and length contraction using the same Eqs.
(1) and (3) if the measurements are taken from the “stationary” (PRF)
reference frame.

The ALT equations for the stationary and moving observers are not
reciprocal,[3] and the moving (non-PRF) observer will observe the opposite
effects of time contraction and length extension when viewing clocks and
objects in the PRF. ALT therefore predicts directional and absolute time
dilation and length contraction effects based on the object’s movement
relative to a PRF. In ALT there is no theoretical justification for
preferential treatment of IRFs, and ALT transformations apply equivalently to
all motion relative to a PRF, both inertial and non-inertial. In contrast, SR
predicts reciprocal time dilation and length contraction between objects in
IRFs based on the velocity between IRFs. With ALT, one-way light speeds are
isotropic in the PRF, but are anisotropic in other reference frames; the two-
way speed of light is $c$ in all reference frames (Fig. 1(b)).[8]

(a)

(b)

Figure 1: (a) Minkowski and (b) ALT spacetime diagrams for $v=0.6c$. Time
units are femtoseconds (fs), and distance units are light-fs. Points (A) and
(B) are events that occur simultaneously in the “stationary” frame at $t$ = 1
fs (dashed horizontal line). In (a), (A) occurs at $t^{\prime}=+2.0$ fs and
(B) occurs at $t^{\prime}=-1.0$ fs (dashed diagonal lines). In (b), (A) and
(B) occur simultaneously at $t^{\prime}=+0.8$ fs. To illustrate light speeds,
light (solid black line) is sent at $t^{\prime}=0$ from $x^{\prime}=1$ towards
$x^{\prime}=0$, where it is reflected back towards $x^{\prime}=1$. In (a), the
“moving” observer calculates isotropic light speeds. In (b), the moving
observer calculates one-way light speeds of $2.5c$ in the backward direction,
and $0.625c$ in the forward direction; and a two-way light speed of $c$.

(a)

(b)

Figure 2: (a) Minkowski and (b) ALT spacetime diagrams for $v=0.6c$
illustrating how the spatial separation of clocks introduces a RTO for SR but
not for ALT. For both diagrams, at $t^{\prime}=0$, two clocks (small diamond
and large circle) are at $x^{\prime}=0$ light-fs. At $t^{\prime}=2$ fs, two
spatially-separated synchronized clocks are shown at positions $x^{\prime}=0$
and $x^{\prime}=1$. In (a), a “stationary” intelligent observer can determine
the RTO for the separated clocks at $t=2.0$ by noting that the diamond clock
reads $t^{\prime}=1.6$, and the circle clock reads $t^{\prime}=1.0$, a
difference of 0.6 fs.

SR is generally considered to be a unique solution to the theory of general
relativity (GR) with zero curvature. Tangherlini reported that both ALT and SR
are compatible with GR.[3] He showed that obtaining SR as a unique solution
involves applying the restriction that all uniformly translating frames are
equivalent in all respects. In the absence of this restriction, ALT is also a
solution to GR with zero curvature.[3]

###  1.2 In an absolute simultaneity framework, ALT cannot be resynchronized
to SR

It has been proposed that SR and ALT are equivalent because ALT coordinates
can be resynchronized to Minkowski coordinates.[9] However, resynchronization
would obscure the two theories, as ALT would then predict differential
simultaneity rather than absolute simultaneity (Fig. 1).[1] Further, it would
not be possible to resynchronize the clocks properly in an absolute
simultaneity framework. The $-vx/c^{2}$ term in the LT time transformation
equation (2) indicates that different observers will see different times on a
given clock based on their distance and velocity relative to the clock. For
resynchronization to occur in an absolute simultaneity framework, a clock
would need to be resynchronized to simultaneously match the times expected by
different observers. However, absolute simultaneity implies that all observers
will observe the same time on the clock at a given instant. Therefore, within
an absolute simultaneity framework, it would not be possible to resynchronize
a clock to accommodate multiple moving observer’s expectations for different
times on the clock.

ALT is similarly incompatible with Einstein synchronization, which imposes
isotropic one-way light speeds.[7] While Einstein synchronization is
compatible with ALT if performed in the PRF, where light speed is isotropic,
the implementation of Einstein synchronization in non-PRF moving reference
frames would transform ALT coordinates into Minkowski coordinates.[1]
Conversely, synchronizing clocks with instantaneous signals would transform
Minkowski coordinates into ALT coordinates.[3, 5] Therefore, experiments to
distinguish differential simultaneity from absolute simultaneity cannot use
Einstein synchronization or instantaneous synchronization. Both theories are
compatible with “slow clock transport”,[1] in which clocks are synchronized in
one location and then spatially-separated equivalent distances. For both
theories, the clocks maintain their synchrony within the “moving” reference
frame, and the synchronization does not alter the two transformations’
respective coordinate systems (Fig. 2).

With ALT, velocities are calculated relative to PRFs. Therefore, the choice of
PRF affects the predicted time dilation and length contraction. The cosmic
microwave background (CMB) radiation has been considered as the PRF for
ALT.[1, 10] However, as the Earth is moving relative to the CMB by $\sim$ 368
km/s,[11] that velocity would have to be added to velocities calculated on the
Earth, which would make ALT predictions incompatible with multiple tests of
SR. It has been suggested that ALT can be resynchronized so that the isotropic
“privileged” inertial system is transferred to another IRF. The resynchronized
IRF becomes the isotropic privileged reference frame, and all other reference
frames are then not isotropic (including the original privileged reference
frame).[12] It was suggested that the resynchronization would make it
impossible to detect the true privileged reference frame.[12] In regards to
this idea, we note that there is a lack of a plausible, real-world mechanism
that would mediate the clock resynchronizations required to shift the PRF;
additionally, the arbitrary nature of the PRFs would preclude testing ALT
because PRFs could be assigned ex post facto.

For the reasons described above, we think that the proper approach to consider
the validity of ALT is in the absence of resynchronization. Without
resynchronization, ALT is only compatible with current experimental results in
the situation that PRFs are locally associated with centers of gravitational
mass.[13] This scenario provides velocities relative to the PRF that allow ALT
to match experimental results for time dilation.[13] In this work, we will
discuss how this type of PRF allows ALT to match current experimental data for
light speed isotropy/anisotropy.

###  1.3 AST encompasses ALT with PRFs that are locally associated with
centers of gravitational mass

The combination of ALT with gravity-centered PRFs has been termed AST.[13] The
extent of AST-defined PRFs are potentially similar to the spheres of influence
(SOIs) defined in the field of astrodynamics. SOIs are mathematically defined
regions around a celestial body where the primary gravitational influence is
from the celestial body.[14] The patched conic approximation, based on SOIs,
has been used to design interplanetary and lunar trajectories.[14] AST
proposes that an object near a planet would have as its PRF the relevant
planet-centered non-rotating IRF, similar to the SOI of the planet. When
leaving a planet-centered PRF, the object would enter the Sun’s PRF, which is
the heliocentric inertial (HCI) reference frame. If AST were to be found to be
valid, the extent to which PRFs approximate SOIs and the interactions between
PRFs would need to be experimentally determined.

In the vicinity of the Earth, the relevant PRF for AST is the non-rotating
Earth-centered inertial (ECI) reference frame. Notably, the ECI is often used
as the “stationary” reference frame for calculating relativistic
transformations and certain quantum mechanical effects.[15, 16, 17, 18, 19,
20, 21, 22] The ECI is used as the “stationary” reference frame for
experimental tests of SR because it is an IRF.[18] Therefore, in many
experimental settings, SR and AST utilize the same “stationary” reference
frame with the same Eqs. (1) and (3) to predict results, thereby producing
equivalent predictions that preclude the experiments from distinguishing the
two theories.

AST is an incomplete theory. It is unclear how or why PRFs are linked to local
centers of gravitational mass, or how PRFs interact. This has implications for
the velocity of objects relative to the PRF as they approach the boundaries of
gravitational influence, e.g. the Lagrangian point. Despite this, it is
worthwhile to experimentally distinguish AST from SR, because AST is the only
potential alternate transformation theory that is compatible with current
experiments. If AST were invalidated by experimental results, then it would
remove the last viable alternate to SR.

##  2 Optical Experiments that are Incapable of Distinguishing SR and AST

###  2.1 Two-way light path and optical resonator experiments

When considered from the perspective of a PRF observer, ALT shares a host of
kinematics with SR, including that the velocity of light is independent of the
velocity of its source,[3] and kinematic equations for: length contraction
(1); time dilation for a constant-velocity IRF (3); relativistic Doppler
shift;[8, 23] and relativistic stellar aberration angle.[8, 23]

The shared kinematics of SR and ALT when considered from the “stationary”
perspective are sufficient to produce null results in two-way light path
experiments. The null results of Michelson–Morley type experiments are
explained by the effects of length contraction and the relativistic stellar
aberration angle.[1, 24] The null results of Kennedy–Thorndike type
experiments are explained by the effects of length contraction, time dilation
and relativistic stellar aberration angle.[1, 24]

Here, we will demonstrate that the shared kinematics of SR and ALT (from the
“stationary” perspective) are sufficient to produce null results in optical
resonator experiments. These are two-way light path experiments that utilize
orthogonally-arranged optical resonators.[25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37] The frequencies of light required to maintain standing optical
waves in the two cavities are compared to determine if they change with
motion.

To demonstrate that SR kinematic equations are sufficient to produce a null
result, one demonstrates that a “stationary” observer will observe a null
result based on the effects of the kinematics. To demonstrate this for optical
resonator experiments, consider orthogonal cavities in which one cavity is
aligned in the direction of motion. A standing wave requires that the cavity
length $(l)$ is a multiple of 1/2 the wavelength $(\lambda)$. Therefore, to
maintain the same number of standing waves per cavity length when in motion,
the ratio of the “moving” wavelength $(\lambda^{\prime})$ to the translatory
distance that the light travels according to the “stationary” observer
$(l_{t}^{\prime})$ must maintain the ratio $\lambda/l$.

We will first consider the optical resonator parallel to the direction of
motion. The relativistic Doppler shift equation defines $\lambda^{\prime}$. In
the forward direction, the “stationary” observer observes a $l_{t}^{\prime}$
of $cl^{\prime}/(c-v)$; substituting $l(1-v^{2}/c^{2})^{0.5}$ for $l^{\prime}$
produces $cl(1-v^{2}/c^{2})^{0.5}/(c-v)$. The ratios
$\lambda^{\prime}/l_{t}^{\prime}$ and $\lambda/l$ are therefore equivalent:

$\frac{\lambda^{\prime}}{l_{t}^{\prime}}=\frac{\lambda\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}}{\frac{cl\sqrt{1-\frac{v^{2}}{c^{2}}}}{c-v}}=\frac{\lambda}{l}\,.$
(4)

The ratios are also equivalent when light is sent in the backwards direction:

$\frac{\lambda^{\prime}}{l_{t}^{\prime}}=\frac{\lambda\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}}{\frac{cl\sqrt{1-\frac{v^{2}}{c^{2}}}}{c+v}}=\frac{\lambda}{l}\,.$
(5)

We will now consider the optical resonator aligned at a right angle to the
direction of motion. The transverse light path $(l_{t}^{\prime})$, as viewed
by the “stationary” observer, can be calculated using the relativistic stellar
aberration angle equation as $l_{t}^{\prime}=l/(1-v^{2}/c^{2})^{0.5}$. By
definition, $\lambda=c/f$, where $f$ is the frequency of the light; and
therefore $\lambda^{\prime}=c/f^{\prime}$. Because of the effect of time
dilation, $c/f^{\prime}=c/f(1-v^{2}/c^{2})^{0.5}$. Substituting $\lambda$ for
$c/f$ gives $\lambda^{\prime}=\lambda/(1-v^{2}/c^{2})^{0.5}$. The ratio
$\lambda^{\prime}/l_{t}^{\prime}$ is therefore equivalent to $\lambda/l$:

$\frac{\lambda^{\prime}}{l_{t}^{\prime}}=\frac{\frac{\lambda}{\sqrt{1-\frac{v^{2}}{c^{2}}}}}{\frac{l}{\sqrt{1-\frac{v^{2}}{c^{2}}}}}=\frac{\lambda}{l}\,.$
(6)

Thus, the shared kinematics of length contraction, time dilation, relativistic
Doppler shift and relativistic stellar aberration angle (from the “stationary”
perspective) are sufficient to produce null results in optical resonator
experiments.

###  2.2 One-way light path experiments

Reviews have described five types of experiments that have been used to search
for anisotropies in the speed of light using one-way light paths.[38, 39] We
will describe how these experiments are unable to distinguish between SR and
AST.

Vessot et al. conducted an experiment that compared the hydrogen maser clock
frequencies sent between a rocket and Earth-surface locations, and calculated
the one-way speed of light during the rocket’s trajectory.[19, 20] In this
experiment,the ECI was used as the coordinate system. AST predicts isotropic
light speedsin the ECI, and therefore both theories predict isotropic light
speeds.

Experiments that used the Compton scattering of laser photons on high-energy
electrons to determine the one-way speed of light, analyzed changes in light
speed as the Earth rotates and orbits the Sun.[40, 41, 42] The motion of the
Earth alters the orientation of the experimental apparatus relative to a
potential external PRF. However, the direction and speed of the apparatus did
not change relative to the ECI, and therefore no changes with time are
expected in the AST framework.

Rotating Mössbauer experiments also analyze changes during Earth’s rotational
and orbital motion,[43, 44, 45] but as there is no differential movement
relative to the ECI, no changes with time are predicted in the AST framework.
Additionally, Ruderfer, who was the first to propose using rotating Mössbauer
experiments as a test of SR,[46] published an erratum that stated that
theories that incorporate relativistic effects, such as ALT, would also
predict null results.[47, 48]

Riis et al. conducted an experiment analyzing the first-order Doppler shift of
light emitted by an atomic beam using fast-beam laser spectroscopy.[49] This
was subsequently shown to be incapable of distinguishing SR from ALT.[50, 51]

Krisher et al. sought to determine changes in the one-way speed of light in
response to the Earth’s rotational and orbital motion by analyzing changes in
maser frequencies after one-way travel over a 21 km fiber optics cable.[52]
This experiment also did not incorporate movement relative to the ECI, and so
would not engender changes with time in an AST framework. Additionally, the
experimental design is incapable of distinguishing SR and AST because, as
described below, the shared kinematics of the two theories ensure null results
from the “stationary” perspective.

From the perspective of the “stationary” observer, both the emitter and the
detector are moving at the same velocity. The “stationary” observer would
observe a blueshifted emission of light from the emitter and a redshift upon
detection. The blueshift and redshift exactly cancel each other, so that no
change in frequency with motion would be detected. The redshifted wavelength
at the detector $(\lambda^{\prime}_{\rm det})$ is calculated by the
relativistic Doppler shift equation with the input wavelength being the
blueshifted wavelength from the emitter $(\lambda^{\prime}_{\rm bl})$. Thus,
the wavelength detected is the original wavelength:

$\lambda^{\prime}_{\rm det}=\lambda^{\prime}_{\rm
bl}\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}=\left(\lambda\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}\,\right)\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}=\lambda\,.$
(7)

For similar reasons, a test of light speed anisotropy that analyzed frequency
changes in a one-way light path,[53] would be unable to distinguish SR and
AST.

Additional optical experiments have been recognized to produce equivalent
results for SR and ALT: an interferometer experiment with a three-angle closed
light path through vacuum and glass;[54, 55] a one-way interferometer
experiment through air and water;[56, 57] and an experiment on the relative
frequency of two ammonia masers arrayed in opposite directions.[58]

##  3 Derivation of the Relativistic Time Offset Equation

In 1910, Comstock published an influential thought experiment that illustrated
differential simultaneity.[59] In the thought experiment, a platform moves
uniformly past a “stationary” observer with velocity $v$. A light flashes from
the center of the “moving” platform of length $l$. An observer on the
platform, who considers the platform “at rest”, observes the light reach both
ends of the platform at the same time, with clocks at each end of the platform
recording the same clock readings when the light arrives. In contrast, the
“stationary” observer observes that the light strikes the back of the platform
first because the back of the platform moves towards the light and the front
moves away from the light. Significantly, the “stationary” observer would
observe that clocks at each end of the “moving” platform record the same time
when the light strikes, even though the clocks are struck sequentially. This
is because the “stationary” observer observes that the back clock is offset
forward in time and the front clock is offset backward in time relative to
each other. We will refer to the difference in time between the two clocks
(that is observed by the “stationary” observer) as the “relativistic time
offset” (RTO). For another description of the RTO see Ref. 60.

The RTO can be viewed on Minkowski spacetime diagrams when using the
convention of “intelligent observers”,[61] who take into account the transit
times of light signals when determining event timing (Fig. 2(a)).

Comstock calculated the non-relativistic times that the light would take to
reach the back of the platform, $(l/2)/(c+v)$, and the front of the platform,
$(l/2)/(c-v)$. He stated that the difference in the two times,
$(l/c)(v/c)/(1-v^{2}/c^{2})$, is the amount by which the clocks disagree for
the “stationary” observer,[59] i.e. the RTO.

Comstock’s derivation does not incorporate the fact that the “stationary”
observer would view a length-contracted platform on which the clocks run
slower. These relativistic effects need to be incorporated, as they affect the
clock readings on the “moving” platform that a “stationary” observer would
observe. Here, we derive the RTO equation taking the relativistic effects into
consideration.

As viewed by the “stationary” observer, the time for the light to travel from
the center to the back of the platform in the “moving” reference frame is the
time transformation equation (3) in which the input for $t$ is the time that
the light takes to cross the length-contracted platform,
$(l(1-v^{2}/c^{2})^{0.5}/2)/(c+v)$:

$t^{\prime}_{\rm
back}=\frac{l\sqrt{1-\frac{v^{2}}{c^{2}}}}{2(c+v)}\sqrt{1-\frac{v^{2}}{c^{2}}}=\frac{l\\!\left(1-\frac{v}{c}\right)}{2c}\,.$
(8)

The corresponding time for the light to reach the front clock is:

$t^{\prime}_{\rm
front}=\frac{l\sqrt{1-\frac{v^{2}}{c^{2}}}}{2(c-v)}\sqrt{1-\frac{v^{2}}{c^{2}}}=\frac{l\\!\left(1+\frac{v}{c}\right)}{2c}\,.$
(9)

The RTO is obtained by subtracting $t^{\prime}_{\rm back}$ from
$t^{\prime}_{\rm front}$:

${\rm RTO}=\frac{vl}{c^{2}}\,.$ (10)

It is not unexpected that the derivation of the RTO gives $vl/c^{2}$, as this
is equivalent to the $vx/c^{2}$ term in Eq. (2), which alters clock times with
distance to allow differential simultaneity and isotropic light speeds.

The RTO equation is equivalent to the difference in timing between the
anisotropic light propagation time predicted by ALT and the calculated
isotropic light speed in the moving frame. To derive this, consider light sent
in the direction of motion: subtracting the isotropic light propagation time,
$l/c$, from the anisotropic light propagation time, $l(1+v/c)/c$, gives
$vl/c^{2}$, which is the RTO equation. This equivalence is expected, as the
RTO is implicit in the isotropic light propagation time but is absent from the
ALT anisotropic light propagation time, so the difference between the two
times is the RTO value.

The RTO equation is also equivalent to the equation for the “Sagnac
correction” (after substituting for the corresponding angular values).[62]
This allows AST to be compatible with light speed anisotropies observed in
satellite communications, as described below.

##  4 AST is Compatible with Light Speed Anisotropies Observed in Satellite
Communications

In 1913, Sagnac discovered that light traveling in a rotating frame strikes a
rotating receiver moving towards the light earlier than a rotating receiver
moving away from the light. This was interpreted by Sagnac to mean that the
light was traveling in a reference frame independent of the rotating frame,
thereby providing evidence for an “ether”.[63] In 1914, Witte argued that the
observation of anisotropic light speeds within the rotating reference frame
did not necessitate the existence of an “ether” because the motion was
rotational and therefore not an IRF in which the second postulate of SR would
be expected to hold.[64]

In 1925, Michelson and Gale demonstrated that light speed is anisotropic on
the surface of the rotating Earth.[65, 66] An around-the-world Sagnac
experiment by Allan et al., 1985, showed that light propagates isotropically
in the ECI but not in the rotating earth-centered, earth-fixed (ECEF)
reference frame.[67] The experiment demonstrated that using the distance
measured in the rotating ECEF reference frame and the time of one-way light
propagation around the Earth produces a calculation of light speed anisotropy.
If instead, the distance was calculated from the non-rotating ECI perspective
then the light speed would be isotropic. Consistent with the Allan et al.
result, communications between satellites of the global navigation satellite
system (GNSS) and receivers linked to atomic clocks on the Earth’s surface
routinely demonstrate that light propagates isotropically in the ECI but
anisotropically if distances are measured in the rotating ECEF reference
frame.[16, 68] The Sagnac correction describes the difference between the
light propagation times in the ECI versus in rotating reference frames, such
as the ECEF.[69] The application of the Sagnac correction allows a rotating
observer (who is measuring distances in the rotating frame) to convert their
calculated “isotropic” light propagation times to the actual, observed
anisotropic values.

Light speed anisotropy is also observed for satellite-to-satellite
communications if distances are calculated relative to the satellites. This is
illustrated by the two GRACE satellites that follow each other in the same
orbit at a distance of $\sim$ 220 km.[70] The GRACE satellites use USO
(ultrastable oscillators) to send microwave signals with encoded time
information between the satellites.[71] The comparison of the time-stamped
signal when it is received relative to the USO time on the receiving satellite
is used to calculate the time of flight of the microwave signal. This time
difference is used to calculate the distance between the satellites.[71] The
distance is also independently determined by geodetic-quality GPS receivers on
the GRACE satellites.[71, 72] Viewed from the ECI perspective, the leading
satellite moves away from the light signal sent by the lagging satellite,
while the lagging satellite moves towards the light signal sent by the leading
satellite. Calculating the speed of light based on the instantaneous distance
between the satellites and the observed light propagation times produces
anisotropic values that are described by the Sagnac correction.[73]

Light speed anisotropy has also been observed for motion that has both
rotational and non-rotational components. Vessot et al. calculated the speed
of light as isotropic in the ECI reference frame for communications between
Earth-based stations and a rocket.[19, 20] In the Vessot et al. experiment,
the timing of light signals were made from the “moving” Earth-based stations
and the rocket, using their “moving” clocks. Within the limits of the
experiment, the results support isotropic light speeds when calculated in the
ECI reference frame, providing differences relative to the isotropic light
speed of $1.9\times 10^{-8}$; $-3.2\times 10^{-9}$ and $-5.6\times 10^{-8}$
for different parts of the trajectory. The small differences between the
observed one-way light speeds and the expected isotropic light speed when
calculated in the ECI, implies that if the calculations were made from the
rotating ECEF, the light speed would be found to be anisotropic in the ECEF
reference frame.

The experiments described above utilized observations made in the “moving”
reference frame because both the Earth’s surface and the satellites are
“moving” relative to the ECI, which is considered the isotropic “stationary”
reference frame for the calculations of light speed. When light signals are
sent in IRFs, SR predicts that the RTO would affect clock timing so that the
“moving” observers would calculate isotropic light speeds of $c$. The
satellite data indicates that there is no appreciable RTO affecting the
“moving” clocks. Within experimental limits, the light speed anisotropy
calculated by the “moving” observers matches that predicted by the Sagnac
correction, which does not incorporate a RTO.[67] Even a significant partial
RTO would manifest itself in the GNSS communications that transfer high
resolution universal coordinate time (UTC) information.[74] Therefore, one can
infer that clocks do not experience a detectable RTO in rotating reference
frames. This observation is consistent with AST, which predicts the absence of
a RTO for both IRFs and non-IRFs.

Both SR and AST are considered to be compatible with light speed anisotropy in
rotating reference frames,[75, 76, 77, 78, 79] which makes it difficult to use
measurements of light speed in rotating frames to distinguish the two
theories. In contrast, SR predicts light speed isotropy for measurements made
within an IRF, while AST predicts light speed anisotropy. Critically, this
distinction applies only to IRFs that are not considered to be a PRF in the
AST framework, such as the ECI or HCI. Therefore, experiments to distinguish
the two transformations must utilize IRFs that are not AST-defined PRFs.
Additionally, measurements of distance and time must be carried out by
“moving” observers in the IRF. Below, we describe an experimental strategy
that analyzes light speed within a linear, constant velocity IRF to directly
measure the RTO, and thereby distinguish the two theories.

##  5 A Strategy to Directly Test Differential Simultaneity

Comstock discussed his thought experiment from the perspective of the
“stationary” observer. However, by utilizing “moving” observers to measure the
clock readings in the “moving” reference frame, a similar experimental
strategy can be used to distinguish SR and AST. The SR “moving” observer will
observe the light strike the front and back clocks at identical clock readings
of $(l/2)/c$ (Fig. 3(a)). The ALT moving observer will observe the light
strike the back and front clocks with times described by Eqs. (8) and (9),
respectively. The difference in the two times is equivalent to the RTO,
$vl/c^{2}$, and will be referred to as the “clock difference” for ALT,
$CD_{\rm ALT}$ (Fig. 3(b)).

(a)

(b)

Figure 3: (a) Minkowski and (b) ALT spacetime diagrams for $v=0.6c$
illustrating clock differences. For both diagrams, the initial position of a
vehicle 1 light-fs in length is marked by a thick dark-gray line and the final
position of the vehicle is marked by a thick light-gray line. At
$t^{\prime}=0$, light beams (black lines) are sent from the middle of the
vehicle towards the two ends of the vehicle. In (a), the “moving” observer
observes that the two light beams strike the ends of the vehicle
simultaneously, producing a null $CD$ value that implies a RTO. In (b), the
moving observer observes a $CD_{\rm ALT}$ of 0.6 fs that implies no RTO.

An experimental strategy that would be capable of measuring $CD$ values is
described here. The components include an elongated chamber under vacuum with
light detectors at each end, and a laser capable of femtosecond pulses aligned
at a right angle to the center of the chamber (Fig. 4(a)). Across from the
laser is a knife-edge prism that is mounted with a piezo-based picomotor that
allows fine longitudinal movements. Light pulses from the laser are split by
the knife-edge prism and directed to each end of the vacuum chamber where
detectors receive the light signals and atomic optical clocks record the time.
The difference in the time readings is the $CD$.

Figure 4: (a) Diagram of the $CD$ measurement apparatus. (b) Diagram of the
Earth’s circumference at the latitude of the flight. Flight trajectories are
shown with arrows.

To synchronize the two clocks, the apparatus is placed in a north–south
orientation on the Earth’s surface. In this orientation, both clocks are at
the same $x$-axis position relative to the rotational motion of the Earth. The
times of the clocks are synchronized in the north–south orientation so that
when the light hits the two detectors, they give identical clock readings. The
apparatus is then shifted to a west–east orientation, which separates the
clocks in the direction of motion relative to the ECI. For SR, this has the
effect of shifting the two clocks to different times relative to a
“stationary” ECI observer, i.e. introducing the RTO (Fig. 2(a)). In the SR
framework, the RTO ensures that no $CD$ value is observed in the “moving”
reference frame, which implies differential simultaneity. AST does not have a
RTO, and so a stationary ECI observer would observe that the two clocks
maintain their synchrony after the shift (Fig. 2(b)). A $CD$ value equivalent
to $CD_{\rm AST}$ (which is the $CD_{\rm ALT}$ value with velocity calculated
relative to an AST-defined PRF) signifies no RTO, which implies absolute
simultaneity.

The synchronization scheme utilizes synchronization at the same location
(relative to the direction of motion) followed by equivalent slow clock
transport to spatially separate the clocks in the direction of motion. While
it may be argued that this synchronization assumes the direction of motion,
this does not affect the ability of the experiment to distinguish between SR
and AST. SR predicts a null $CD$ after synchronization and movement in any
direction, while AST predicts an exact $CD_{\rm AST}$ value for this
experimental strategy.

The experiment can be undertaken in an IRF with a linear, constant velocity
relative to the ECI. For example, an airplane in eastward and westward
trajectories that are linear relative to the ECI, as shown in Fig. 4(b). For
SR, flights in both directions are predicted to produce null $CD$ values. For
AST, the eastward flight would have a greater velocity relative to the ECI.
Consider an airplane traveling at 250 m/s at the latitude of Boulder, CO, USA
(which has an angular velocity of 356 m/s due to the Earth’s rotation). The
velocity relative to the ECI would be $\sim$ 606 m/s when flying eastward and
$\sim$ 106 m/s when flying westward. The $CD_{\rm AST}$ for a 10 m-long
apparatus on the eastward flight would be 67.4 fs, while the $CD_{\rm AST}$
for the westward flight would be 11.8 fs.

The experimental strategy described above is currently not technically
feasible using velocities available with airplanes because it requires
distinguishing timing differences in the range of 10–50 fs. Optical atomic
clocks operate at 0.9–2.3 fs per frequency cycle, which would, in principle,
be sufficient to detect such timingdifferences.[80, 81, 82, 83] However, it is
currently not possible to count individual frequency cycles with these optical
clocks.[84] Time-to-digital converters can measure 0.1 picosecond (ps) time
intervals,[85] but this does not provide sufficient resolution.

The experimental strategy is technically feasible if carried out in a space-
based setting where greater velocities are possible. A suitable IRF could be a
spacecraft traveling linearly near a planet in a direction opposite to the
planet’s orbital motion. For example, a spacecraft that was traveling at 12
km/s (in the HCI) in a direction opposite to the Earth’s orbital motion of 30
km/s would have a velocity of 42 km/s relative to the ECI. In this setting,
the $CD_{\rm AST}$ for a 10 m-long experimental apparatus is predicted to be
4.67 ps, which is detectable with current time-to-digital converters. Notably,
when the spacecraft is distant from planetary SOIs, the $CD_{\rm AST}$ is
predicted to be 1.34 ps based on a velocity of 12 km/s relative to the HCI.
These timing differences can be distinguished using current time-to-digital
converters.

##  6 Controls and Statistical Tests for the Experimental Strategy

Control tests can be used to assess random and systematic errors. Random and
non-directional errors can be identified by determining if the clock readouts
accurately reflect the time of light propagation. This can be checked by
moving the knife-edge prism closer to one end of the apparatus so that the
difference in distance that the light travels in each arm of the apparatus
should produce a defined, positive reading. For example, moving the knife-edge
prism 20 $\mu$m from the center with the piezo-based picomotor would be
expected to produce a timing difference of 67 fs in a 10 m-long apparatus when
in a north–south orientation.

Tests to determine if there are systemic or directional errors can be
accomplished by altering the orientation of the apparatus. Rotating the
apparatus $180^{\circ}$ when it is perpendicular to the direction of motion
should maintain clock synchronization if there are no directional errors.
Rotation of the apparatus $180^{\circ}$ in the direction of motion would be
expected to give the same $CD$ value but with an opposite sign for AST, or
maintain the null $CD$ value for SR. To ensure that the alignment of the
apparatus towards either the front or rear of the vehicle does not affect the
clock readings, the eastward or westward trajectory can be reversed, which
should alter the $CD$ value based on velocity relative to the ECI for AST, or
maintain the null $CD$ value for SR. These tests can ensure that systematic
errors arising from the apparatus are not interpreted as valid results.

The speed of the apparatus is a variable that affects the predicted $CD_{\rm
AST}$. $CD_{\rm AST}$ is predicted to increase by 0.111265 fs for every m/s
increase in velocity relative to the ECI for a 10 m-long apparatus. The speed
of the apparatus during the data collection period should therefore be
recorded and used to calculate the predicted $CD_{\rm AST}$ values.

Experimental tests of SR often use test theories, such as Mansouri and Sexl
(MS), to assess LT parameters.[86] The MS test theory equations are:

$\displaystyle t^{\prime}$ $\displaystyle=$ $\displaystyle at+\epsilon
x^{\prime}\,,$ (11) $\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle
b(x-vt)\,,$ (12) $\displaystyle y^{\prime}$ $\displaystyle=$ $\displaystyle
dy\,,$ (13) $\displaystyle z^{\prime}$ $\displaystyle=$ $\displaystyle dz\,.$
(14)

The parameters that are experimentally analyzed, $a(v)$, $b(v)$ and $d(v)$,
are the same for SR and ALT (when calculated from the “stationary”
perspective): the time dilation factor $a(v)=(1-v^{2}/c^{2})^{0.5}$; the
$x$-axis length contraction factor $b(v)=1/(1-v^{2}/c^{2})^{0.5}$; and the
$z$-axis and $y$-axis length contraction factor $d(v)=1$.[10] The relevant
term in the MS test theory that distinguishes the two transformations is
$\epsilon(v)$, which depends on the method of clock synchronization.
$\epsilon(v)$ is $-v/c^{2}$ for SR and 0 for ALT.[10] An experimentally-
determined $CD$ can be used to determine the corresponding value of
$\epsilon(v)$, as the two functions are related as described below.

Using Eqs. (11) and (12) with the SR/ALT relativistic values of $a(v)$ and
$b(v)$ to produce spacetime diagrams shows that the value of $\epsilon(v)$
determines the slope of the $x^{\prime}$ axis (Fig. 3). In contrast, the slope
of the $t^{\prime}$ axis is not affected by the value of $\epsilon(v)$.
$\epsilon(v)$ can be described in terms of $v$ and the slope of the
$x^{\prime}$ axis, $m$, by solving the relativistic version of Eq. (11) with
$t^{\prime}=0$ (to represent the $x^{\prime}$ axis), and substituting
$x^{\prime}$ with the relativistic version of Eq. (12). Because $t^{\prime}$
is set to 0, $t/x$ is equivalent to $m$. $\epsilon(v)$ in terms of $m$ and $v$
is:

$\epsilon=\frac{-m\\!\left(1-\frac{v^{2}}{c^{2}}\right)}{(1-mv)}\,.$ (15)

The $CD$ value also determines the slope of the $x^{\prime}$ axis on a
spacetime diagram. The expression of $CD$ in terms of the slope of the
$x^{\prime}$ axis can be determined by geometrically solving for $CD$ (Fig.
3(b)). $CD$ in terms of $m$, $v$ and $l$ is:

$CD=\frac{l\\!\left(\frac{v}{c^{2}}-m\right)}{(1-mv)}\,.$ (16)

Subtracting the description of $CD/l$ in terms of $m$ and $v$, shown in Eq.
(16), from the description of $\epsilon(v)$ in terms of $m$ and $v$, shown in
Eq. (15), gives $-v/c^{2}$. Therefore, $\epsilon(v)$ is related to $CD$
through the following equation:

$\epsilon=-\frac{v}{c^{2}}+\frac{CD}{l}\,.$ (17)

Equation (17) allows the value of $\epsilon(v)$ to be calculated from the
experimentally-determined $CD$ value. $z$-statistics can be used to determine
the confidence intervals of the calculated $\epsilon(v)$ values to determine
the extent to which the data matches the predictions of SR $(-v/c^{2})$ or AST
(0).

##  7 Conclusions

This work highlights that AST is the only viable alternate to SR. Malykin and
Vargas have shown that ALT is the only alternate LT that is compatible with
classical tests of SR.[1, 2] We provide proofs that show that AST is
compatible with optical resonator experiments and experiments analyzing
frequency changes in one-way light pathways. We also summarize the literature
to show that other light speed isotropy/anisotropy experiments are compatible
with both AST and SR. Therefore, current light speed experiments fail to
distinguish AST from SR.

It has been proposed that ALT is an alternative form of the LT because clocks
can be resynchronized to convert ALT to the LT.[9] We point out that this
resynchronization of clocks would abolish the absolute simultaneity
characteristic of ALT. Further, we note that this resynchronization of clocks
cannot be implemented in an absolute simultaneity framework because it is not
possible to resynchronize clocks so that a given clock will simultaneously
match the different times expected by different IRF observers. It has also
been proposed that resynchronization would allow the PRF in ALT to switch
seamlessly between different IRFs, so that previous PRFs would become non-
PRFs.[12] While this scenario would allow ALT to match all experimental data
(by choosing the PRF to match the observed data), it lacks a plausible
mechanism that would implement the resynchronization of clocks in a real-world
setting. In the absence of resynchronization, ALT is only compatible with
experimental data if the PRFs proposed by ALT are locally associated with
gravitational centers, similar to the concept of SOIs in astrodynamics.

The RTO is an integral aspect of differential simultaneity, but is absent in
absolute simultaneity. Our derivation of the RTO shows that it is identical to
the Sagnac correction, which describes the extent of light speed anisotropy
calculated in a rotating reference frame relative to the isotropic propagation
of the light in the ECI. The extensive use of the Sagnac correction in
satellite communications highlights that light speed is isotropic in the ECI
but not in the rotating reference frame. This data is compatible with AST
because it shows that there is no appreciable RTO affecting the clock times of
the “moving” observers (satellites and ground-based receivers).

AST predicts light speed isotropy within PRFs that are locally associated with
centers of gravitational mass, such as the ECI, other planetary non-rotating
IRFs and the HCI. The proposed experimental strategy assesses light speed
isotropy within linear, constant velocity IRFs that are not AST-defined PRFs.
Other potential light speed tests that would be capable of distinguishing SR
and AST have been proposed.[87, 88, 89] A major advantage of the experimental
strategy described here is that it provides a direct readout of the RTO, and
therefore a direct measure of differential versus absolute simultaneity. The
proposed experimental strategy is currently technically feasible only in a
space-based experiment. However, we anticipate that continuing technical
advances will make this and similar experimental strategies more accessible,
so that direct experimental tests of the nature of simultaneity can be
accomplished in the near future.

## Acknowledgment

We thank J. Reeves and the University of Georgia Statistical Consulting Center
for advice on statistical analysis; and J. L. Wilmoth and B. E. Kipreos for
helpful discussions.

## References

  * [1] J. G. Vargas, Found. Phys. 11, 235 (1981).
  * [2] G. B. Malykin, Opt. Spectrosc. 107, 592 (2009).
  * [3] F. R. Tangherlini, The velocity of light in uniformly moving frames, Ph.D. thesis, Stanford University (1958).
  * [4] A. Eagle, Philos. Mag. 26, 410 (1938).
  * [5] G. B. Malykin, Phys.-Usp. 52, 263 (2009).
  * [6] G. B. Malykin and E. G. Malykin, Abraham Zelmanov J. 2, 121 (2009).
  * [7] A. Einstein, Ann. Phys.-Leipzig 17, 891 (1905).
  * [8] F. R. Tangherlini, J. Mod. Phys. 5, 230 (2014).
  * [9] G. Rizzi, M. L. Ruggiero and A. Serafini, Found. Phys. 34, 1835 (2004).
  * [10] R. Mansouri and R. U. Sexl, Gen. Relat. Gravit. 8, 497 (1977).
  * [11] G. F. Smoot, Rev. Mod. Phys. 79, 1349 (2007).
  * [12] F. Selleri, Found. Phys. Lett. 18, 325 (2005).
  * [13] E. T. Kipreos, PLoS One 9, e115550 (2014).
  * [14] R. R. Bate, D. D. Mueller and J. E. White, Fundamentals of Astrodynamics (Dover, 1971).
  * [15] C. O. Alley, Proper time experiments in gravitational fields with atomic clocks,aircraft, and laser light pulses, in Quantum Optics, Experimental Gravity, and Measurement Theory, eds. P. Meystre and M. O. Scully (Plenum Press, 1983), pp. 363–427.
  * [16] N. Ashby, Living Rev. Relativ. 6, 1 (2003), http://www.livingreviews.org/lrr-2003-1.
  * [17] J. C. Hafele and R. E. Keating, Science 177, 168 (1972).
  * [18] J. C. Hafele and R. E. Keating, Science 177, 166 (1972).
  * [19] R. F. C. Vessot and M. W. Levine, Gen. Relat. Gravit. 10, 181 (1979).
  * [20] R. F. C. Vessot et al., Phys. Rev. Lett. 45, 2081 (1980).
  * [21] S. A. Werner, J. L. Staudenmann and R. Colella, Phys. Rev. Lett. 42, 1103 (1979).
  * [22] P. Wolf and G. Petit, Phys. Rev. A 56, 4405 (1997).
  * [23] G. D. Puccini and F. Selleri, Nuovo Cimento B 117, 283 (2002).
  * [24] F. Selleri, Found. Phys. 26, 641 (1996).
  * [25] A. Brillet and J. L. Hall, Phys. Rev. Lett. 42, 549 (1979).
  * [26] C. Braxmaier et al., Phys. Rev. Lett. 88, 010401 (2002).
  * [27] H. Muller et al., Phys. Rev. Lett. 91, 020401 (2003).
  * [28] P. Wolf et al., Phys. Rev. Lett. 90, 060402 (2003).
  * [29] P. Wolf et al., Phys. Rev. D 70, 051902(R) (2004).
  * [30] P. Wolf et al., Gen. Relat. Gravit. 36, 2351 (2004).
  * [31] P. Antonini, M. Okhapkin, E. Goklu and S. Schiller, Phys. Rev. A 71, 050101(R) (2005).
  * [32] P. L. Stanwix et al., Phys. Rev. Lett. 95, 040404 (2005).
  * [33] S. Herrmann et al., Phys. Rev. Lett. 95, 150401 (2005).
  * [34] P. L. Stanwix et al., Phys. Rev. D 74, 081101(R) (2006).
  * [35] H. Muller et al., Phys. Rev. Lett. 99, 050401 (2007).
  * [36] S. Herrmann et al., Phys. Rev. D 80, 105011 (2009).
  * [37] C. Eisele, A. Y. Nevsky and S. Schiller, Phys. Rev. Lett. 103, 090401 (2009).
  * [38] M. F. Ahmed, B. M. Quine, S. Sargoytchev and A. D. Stauffer, Indian J. Phys. 86, 835 (2012).
  * [39] C. M. Will, Phys. Rev. D 45, 403 (1992).
  * [40] V. C. Gurzadyan et al., Mod. Phys. Lett. A 20, 19 (2005).
  * [41] V. G. Gurzadyan et al., Nuovo Cimento B 122, 515 (2007).
  * [42] J. P. Bocquet et al., Phys. Rev. Lett. 104, 241601 (2010).
  * [43] D. C. Champeney, G. R. Isaak and A. M. Khan, Phys. Lett. 7, 241 (1963).
  * [44] E. Preikschat, The Mössbauer effect and tests of relativity, Ph.D. thesis, University of Birmingham (1968).
  * [45] K. C. Turner and H. A. Hill, Phys. Rev. B 134, 252 (1964).
  * [46] M. Ruderfer, Phys. Rev. Lett. 5, 191 (1960).
  * [47] M. Ruderfer, Phys. Rev. Lett. 7, 361 (1961).
  * [48] M. Ruderfer, Am. J. Phys. 43, 279 (1975).
  * [49] E. Riis et al., Phys. Rev. Lett. 60, 81 (1988).
  * [50] Z. Bay and J. A. White, Phys. Rev. Lett. 62, 841 (1989).
  * [51] E. Riis et al., Phys. Rev. Lett. 62, 842 (1989).
  * [52] T. P. Krisher et al., Phys. Rev. D 42, 731 (1990).
  * [53] D. R. Gagnon, D. G. Torr, P. T. Kolen and T. Chang, Phys. Rev. A 38, 1767 (1988).
  * [54] W. S. N. Trimmer, R. F. Baierlei, J. E. Faller and H. A. Hill, Phys. Rev. D 8, 3321 (1973).
  * [55] W. S. N. Trimmer, Phys. Rev. D 9, 2489 (1974).
  * [56] J. Byl, M. Sanderse and W. Vanderkamp, Am. J. Phys. 53, 43 (1985).
  * [57] H. Aspden, Am. J. Phys. 53, 1131 (1985).
  * [58] J. P. Cedarholm and C. H. Townes, Nature 184, 1350 (1959).
  * [59] D. F. Comstock, Science 31, 767 (1910).
  * [60] W. Rindler, The relativity of simultaneity, in Essential Relativity$:$ Special, General, and Cosmological (Springer-Verlag, 1977), pp. 28–29.
  * [61] R. E. Scherr, P. S. Shaffer and S. Vokos, Am. J. Phys. 69, S24 (2001).
  * [62] N. Ashby, Relativistic effects in the global positioning system, in Proc. AAPT Topical Workshop$:$ Teaching General Relativity to Undergraduates (2006), https://www.aapt.org/doorway/TGRU/articles/Ashbyarticle.pdf.
  * [63] M. G. Sagnac, Comptes Rendus 157, 708 (1913).
  * [64] H. Witte, Verh. Dtsch. Z. 16, 142 (1914).
  * [65] A. A. Michelson, Astrophys. J. 61, 137 (1925).
  * [66] A. A. Michelson and H. G. Gale, Astrophys. J. 61, 140 (1925).
  * [67] D. W. Allan, M. A. Weiss and N. Ashby, Science 228, 69 (1985).
  * [68] N. Ashby, The Sagnac effect in the global positioning system, in Relativity in Rotating Frames, eds. G. Rizzi and M. L. Ruggiero (Kluwer, 2004), pp. 11–28.
  * [69] R. A. Nelson, Metrologia 48, S171 (2011).
  * [70] D. Adam, Nature 416, 10 (2002).
  * [71] W. Bertiger et al., Relative time and frequency alignment between two low earth orbiters, GRACE, in Proc. 2003 IEEE Int. Frequency Control Symp. and PDA Exhibition Jointly with 17th European Frequency and Time Forum (2003), pp. 273–279.
  * [72] K. M. Larson, N. Ashby, C. Hackman and W. Bertiger, Metrologia 44, 484 (2007).
  * [73] J. Kim, Simulation study of a low-low satellite-to-satellite tracking mission, Ph.D. thesis, University of Texas (2000).
  * [74] N. Ashby, Relativity in the global positioning system, in 100 Years of Relativity, ed. A. Ashtekar (World Scientific, 2005), pp. 257–289.
  * [75] G. Rizzi and A. Tartaglia, Found. Phys. 28, 1663 (1998).
  * [76] G. B. Malykin, Phys.-Usp. 45, 907 (2002).
  * [77] G. Rizzi and A. Serafini, Synchronization and desynchronization on rotating platforms, in Relativity in Rotating Frames, eds. G. Rizzi and M. L. Ruggiero (Kluwer, 2004), pp. 79–102.
  * [78] F. Sorge, Local and global anisotropy in the speed of light, in Relativity in Rotating Frames, eds. G. Rizzi and M. L. Ruggiero (Kluwer, 2004), pp. 155–165.
  * [79] J. F. Pascual-Sanchez, A. S. Miguel and F. Vicente, Isotropy of the velocity of light and the Sagnac effect, in Relativity in Rotating Frames, eds. G. Rizzi and M. L. Ruggiero (Kluwer Academic Publishers, 2004), pp. 167–178.
  * [80] B. J. Bloom et al., Nature 506, 71 (2014).
  * [81] N. Hinkley et al., Science 341, 1215 (2013).
  * [82] C. W. Chou, D. B. Hume, T. Rosenband and D. J. Wineland, Science 329, 1630 (2010).
  * [83] T. Rosenband et al., Science 319, 1808 (2008).
  * [84] M. A. Lombardi, Fundamentals of time and frequency, in The Mechatronics Handbook, ed. R. H. Bishop (CRC Press, 2002), pp. 17-1–17-18.
  * [85] R. Szplet, Time-to-digital converters, in Design, Modeling and Testing of Data Converters, eds. P. Carbone et al. (Springer-Verlag Berlin, 2014), pp. 211–246.
  * [86] D. W. Macarthur, Phys. Rev. A 33, 1 (1986).
  * [87] G. B. Malykin, Phys.-Usp. 47, 739 (2004).
  * [88] G. B. Malykin, Tech. Phys. 50, 1383 (2005).
  * [89] G. B. Malykin and V. I. Pozdnyakova, Tech. Phys. 52, 133 (2007).