# Double Multi-Head Attention for Speaker Verification

###### Abstract

Most state-of-the-art Deep Learning systems for speaker verification are based
on speaker embedding extractors. These architectures are commonly composed of
a feature extractor front-end together with a pooling layer to encode
variable-length utterances into fixed-length speaker vectors. In this paper we
present Double Multi-Head Attention pooling, which extends our previous
approach based on Self Multi-Head Attention. An additional self attention
layer is added to the pooling layer that summarizes the context vectors
produced by Multi-Head Attention into a unique speaker representation. This
method enhances the pooling mechanism by giving weights to the information
captured for each head and it results in creating more discriminative speaker
embeddings. We have evaluated our approach with the VoxCeleb2 dataset. Our
results show $6.09\%$ and $5.23\%$ relative improvement in terms of EER
compared to Self Attention pooling and Self Multi-Head Attention,
respectively. According to the obtained results, Double Multi-Head Attention
has shown to be an excellent approach to efficiently select the most relevant
features captured by the CNN-based front-ends from the speech signal.

Index Terms—  self multi-head attention, double attention, speaker
recognition, speaker verification

##  1 Introduction

Speaker verification aims to determine whether a pair of audios corresponds to
the same speaker. Given speech signals, speaker verification systems are able
to extract speaker identity patterns from the characteristics of the voice.
These patterns can be both statistically modelled or encoded into
discriminative speaker representations. Over the last few years, researchers
have put huge effort on encoding these speaker characteristics into more
discriminative speaker vectors. Current state-of-the-art speaker verification
systems are based on Deep Learning (DL) approaches. These architectures are
commonly trained as speaker classifiers in order to be used as speaker
embedding extractors. Speaker embeddings are fixed-length vectors extracted
from some of the last layers of these Deep Neural Networks (DNNs) [1]. The
most known representation is the x-vector [2], which has become state-of-the-
art for speaker recognition and has also been used for other tasks such as
language and emotion recognition [3, 4].

Most of the recent network architectures used for speaker embedding extraction
are composed of a front-end feature extractor, a pooling layer, and a set of
Fully Connected (FC) layers. Lately, there have been several architectures
proposed to encode audio utterances into speaker embeddings for different
choices of network inputs, such as [5, 6, 7, 8, 9]. Using Mel-Frequency
Cepstral Coefficient (MFCC) features, Time Delay Neural Network (TDNN) [5, 6]
is the most currently used architecture. TDNN is the x-vector front-end and
consists of a stack of 1-D dilated Convolutional Neural Networks (CNNs). The
idea behind the use of TDNNs is to encode a sequence of MFCC into a more
discriminative sequence of vectors by capturing long-term feature relations.
2-D CNNs have also shown competitive results for speaker verification. There
are Computer Vision architectures such as VGG [7, 10, 11] and ResNet [8, 12,
13] that have been adapted to capture speaker discriminative information from
the Mel-Spectrograms. In fact, Resnet34 has shown a better performance than
TDNN in the most recent speaker verification challenges [14, 15]. Finally,
there are also some other attempts to work directly on the raw signal instead
of using hand-crafted features [16, 17, 18].

Given the encoded sequence from the front-end, a pooling layer is adopted to
obtain an utterance-level representation. During the last few years, there are
several studies addressing different types of pooling strategies [19, 20, 21].
X-vector originally uses statistical pooling [6]. Self attention mechanisms
have been used to improve statistical pooling, such as [22]. In works like
[23], attention is used to extract better order features statistics. A wide
set of pooling layers based on self attention have been proposed improving
this vanilla self attention mechanism. In [22] several attentions are applied
over the same encoded sequence, producing multiple context vectors. In our
previous work [11], the encoded sequence is split into different heads and a
different attention model is applied over each head sub-sequence. Non self
attention mechanisms have also been proposed like [24], where a mutual
attention network is fed with the pair of utterances aimed to compare.

In this paper we present a Double Multi-Head Attention (MHA) pooling layer for
speaker verification. The use of this layer is inspired by [25], where Double
MHA is presented as a double self attention block which captures feature
statistics and makes adaptive feature assignment over images. In this work
this mechanism is used as a combination of two self attention pooling layers
to create utterance-level speaker embeddings. Given a sequence of encoded
representations from a CNN, Self MHA first concatenates the context vector
from $K$ head attentions applied over a $K$ sub-embedding sequences. An
additional self attention mechanism is then applied over the multi-head
context vector. This attention based pooling summarizes the set of head
context vectors into a global speaker representation. This representation is
pooled through a weighted average of the head context vectors, where the head
weights are produced with the self attention mechanism. On the one hand, this
approach allows the model to attend to different parts of the sequence,
capturing at the same time different subsets of encoded representations. On
the other hand, the pooling layer allows to select which head context vectors
are the most relevant to produce the global context vector. In comparison with
[25], the second pooling layer operates over the head context vectors produced
by a MHA instead of the global descriptors produced by a self multi attention
mechanism applied over an image.

##  2 Proposed Architecture

Our proposed system architecture is illustrated in Figure 1. It utilizes a
CNN-based front-end which takes in a set of variable length mel-spectrogram
features and outputs a sequence of speaker representations. These speaker
representations are further subject to a Double MHA pooling which is the main
contribution of this work. The Double MHA layer comprises a Self MHA pooling
and an additional Self Attention layer that summarizes the information of each
head context vector into a unique speaker embedding. The combination of Self
MHA pooling together with this Self Head Attention layer provides us with a
deeper self-attention pooling mechanism (Figure 2). The speaker embedding
obtained from the pooling layer is sent through a set of FC layers to predict
the speaker posteriors. This network architecture is trained with Additive
Margin Softmax (AMS) loss [26] as a speaker classifier so as to have a speaker
embedding extractor.

###  2.1 Front-End Feature Extractor

Our feature extractor network is a larger version of the adapted VGG proposed
in [11]. This CNN comprises four convolution blocks, each of which contains
two concatenated convolutional layers followed by a max pooling with a
$2\times 2$ stride. Hence given a spectrogram of $N$ frames, the VGG performs
a down-sampling reducing its output into a sequence of $N/16$ representations.
The output of the VGG $h\in\mathbb{R}^{M\times N/16\times D^{\prime}}$ is a
set of $M$ feature maps with $N/16\times D^{\prime}$ dimension. These feature
maps are concatenated into a unique vector sequence. This reshaped sequence of
hidden states can now be defined as $h\in\mathbb{R}^{N/16\times D}$, where
$D=MD^{\prime}$ corresponds to the hidden state dimension.

###  2.2 Self Multi-Head Attention Pooling

The sequence of hidden states output from the front-end feature extractor can
be expressed as $h=[h_{1}h_{2}...h_{N}]$ with $h_{t}\in\mathbb{R}^{D}$. If we
consider a number of $K$ heads for the MHA pooling, now we can define the
hidden state as $h_{t}=[h_{t1}h_{t2}...h_{tK}]$ where
$h_{tj}\in\mathbb{R}^{D/K}$. Hence each feature vector is split into a set of
sub-feature vectors of size $D/K$. In the same way, we have also a trainable
parameter $u=[u_{1}u_{2}...u_{K}]$ where $u_{j}\in\mathbb{R}^{D/K}$. A self
attention operation is then applied over each head of the encoded sequences.
The weights of each head alignment are defined as:

$\centering
w_{tj}=\frac{\exp{\left(\frac{h^{T}_{tj}u_{j}}{\sqrt{d_{h}}}\right)}}{\sum^{N}_{l=1}\exp{\left(\frac{h^{T}_{lj}u_{j}}{\sqrt{d_{h}}}\right)}}\@add@centering$
(1)

where $w_{tj}$ corresponds to the attention weight of the head $j$ on the step
$t$ of the sequence and $d_{h}$ corresponds to hidden state dimension $D/K$.
If each head corresponds to a subspace of the hidden state, the weight
sequence of that head can be considered as a probability density function
(pdf) from that subspace features over the sequence. We then compute a new
pooled representation for each head in the same way than vanilla self
attention:

$\centering c_{j}=\sum^{N}_{t=1}h^{T}_{tj}w_{tj}\@add@centering$ (2)

where $c_{j}\in\mathbb{R}^{D/K}$ corresponds to the utterance level
representation from head $j$. The final utterance level representation is then
obtained with the concatenation of the utterance level vectors from all the
heads $c=[c_{1}c_{2}...c_{k}]$. This method allows the network to extract
different kinds of information over different regions of the network.

###  2.3 Double Multi-Head Attention

Fig. 1: System Architecture.

The main disadvantage of Self MHA pooling is that it assumes uniform head
relevance. The output context vector is the concatenation of all head context
vectors and it is used as input of the following dense layers. Double MHA does
not assume that. Therefore, each utterance context vector is computed as a
different linear combination of head context vectors. A summarized vector $c$
is then defined as a weighted average over the set of head context vectors
$c_{i}$. Self attention is then used to pool the set of head context vectors
$c_{i}$ and obtain an overall context vector $c$.

$\centering
w^{\prime}_{i}=\frac{\exp{(c^{T}_{i}u^{\prime})}}{\sum^{K}_{l=1}\exp{(c^{T}_{l}u^{\prime})}}\@add@centering$
(3)

$\centering c=\sum^{K}_{i=1}c^{T}_{i}w^{\prime}_{i}\@add@centering$ (4)

Fig. 2: An example of Double MHA Pooling with $5$ heads.

where $w^{\prime}_{i}$ corresponds to the aligned weight of each head and
$u^{\prime}\in\mathbb{R}^{D/K}$ is a trainable parameter. The context vector
$c$ is then computed as the weighted average of the context vectors among
heads. With this method, each utterance context vector is created scaling the
information of the most/least relevant heads. Considering the whole pooling
layer, Double MHA allows to capture different kind of speaker patterns in
different regions of the input, and at the same time allows to weight the
relevance of each of these patterns for each utterance.

The number of heads used for this pooling defines both the context vector
dimension and how the VGG feature maps are grouped. Considering the number of
$M$ channels and $K$ heads, for each head we would create a $c_{i}$ context
vector of $D^{\prime}M/K$ dimension which contains a subset of $M/K$ feature
maps. Therefore, as the number of heads grows larger, it allows Double MHA to
consider more subsets of features while decreases the dimension of the final
utterance-level context vector. This implies a trade-off between the number of
features subsets we can create and how much compressed are these features in
the context vector subspace.

###  2.4 Fully-Connected Layers

The utterance-level speaker vector obtained from the pooling layer is fed into
a set of four FC layers (Figure 1). Each of the first two FC layers is
followed by a batch normalization [27] and Rectified Linear Unit (ReLU)
activations. A dense layer is adopted for the third FC layer and the last FC
corresponds to the speaker classification layer. Since AMS is used to train
the network, the third layer is set up without activation and batch
normalization as proposed in [26]. Once the network is trained, we can extract
a speaker embedding from one of the intermediate FC layers. According to [26],
we consider the second layer as the speaker embedding instead of the third
one. The output of this FC layer then corresponds to the speaker
representation that will be used for the speaker verification task.

##  3 Experimental Setup

The proposed system111Models are available at:
https://github.com/miquelindia90/DoubleAttentionSpeakerVerification in this
work has been assessed by VoxCeleb dataset [28, 7]. VoxCeleb is a large
multimedia database that contains more than one million 16kHz audio utterances
for more than 6K celebrities. VoxCeleb has two different versions with several
evaluation protocols. For our experiments, VoxCeleb2 development partition
with no augmentation have been used to train both baseline and presented
approaches. On the other hand, the performance of these systems have been
evaluated with Vox1 test, Vox1-E, and Vox1-H conditions. These protocols
contain sets of 37,611, 581,480, and 552,536 of Vox1 random pairs,
respectively. Vox1 test only uses the test set, Vox1-E uses the whole
development + test corpus and Vox1-H is restricted to audio pairs from same
nationality and gender speakers.

Table 1: CNN Architecture. In and Out Dim. refers to the input and output feature maps of the layer. Feat Size refers to the dimension of each one of this output feature maps. Layer | Size | In Dim. | Out Dim. | Stride | Feat Size
---|---|---|---|---|---
conv11 | 3x3 | 1 | 128 | 1x1 | Nx80
conv12 | 3x3 | 128 | 128 | 1x1 | Nx80
mpool1 | 2x2 | - | - | 2x2 | N/2x40
conv21 | 3x3 | 128 | 256 | 1x1 | N/2x40
conv22 | 3x3 | 256 | 256 | 1x1 | N/2x40
mpool2 | 2x2 | - | - | 2x2 | N/4x20
conv31 | 3x3 | 256 | 512 | 1x1 | N/4x20
conv32 | 3x3 | 512 | 512 | 1x1 | N/4x20
mpool3 | 2x2 | - | - | 2x2 | N/8x10
conv41 | 3x3 | 512 | 1024 | 1x1 | N/8x10
conv42 | 3x3 | 1024 | 1024 | 1x1 | N/8x10
mpool4 | 2x2 | - | - | 2x2 | N/16x5
flatten | - | 1024 | 1 | - | N/16x5120

Two different baselines have been considered to compare with the presented
approach. Double MHA pooling have been evaluated against two self attentive
based pooling methods: vanilla Self Attention and Self MHA. In order to
evaluate them, these mechanisms have replaced the pooling layer of the system
(Figure 1) without modifying any other block or parameter from the network.
The speaker embeddings used for the verification tests have been extracted
from the same FC layer for each of the pooling methods. Cosine distance has
been used to compute the scores between pairs of speaker embeddings.

The proposed network has been trained to classify variable-length speaker
utterances. As input features we have used $80$ dimension log Mel Spectrograms
with $25$ms length Hamming windows and $10$ms window shift. The audio features
have been normalized with Cepstral Mean Normalization (CMN). The CNN encoder
is then fed with $N\times 80$ spectrograms to obtain a sequence of $N/16\times
5120$ encoded hidden representations. For training we have used batches of
$N=350$ frames audio chunks but for test the whole utterances have been
encoded. The setup of the CNN feature extractor can be found on Table 1. For
the pooling layer we have tuned the number of heads for both Self MHA and
Double MHA. For the presented CNN setup we have considered $8$, $16$, and $32$
heads, which implies a head context vector $c_{i}$ of $640$, $320$, and $160$,
respectively. Models with $64$ heads have been discarded due to the
instability of their training. The last block of the system consists of four
consecutive FC layers. The first three dense layers have $400$ dimension. The
last FC layer has $5994$ dimension, which corresponds to the number of
training speaker labels. Batch normalization has been applied only on the
first two dense layers as mentioned in subsection 2.4. The network has been
trained with AMS loss with $s=30$ and $m=0.4$ hyper-parameters. Batch size
have been set to $128$ and Adam optimizer has been used to train all the
models with a 1e-4 learning rate and a 1e-3 weight decay. Models have been
trained for $100$ epochs using a learning rate annealing strategy. After each
15 epochs without validation improvement, learning rate was decayed by a $0.5$
factor.

##  4 Results

Table 2: Evaluation results on VoxCeleb 1 protocols. Head and global context vectors are referred to as $c_{i}$ and $c$. Approach | Pooling Setup | Vox1 Test | Vox1-E | Vox1-H
---|---|---|---|---
Heads |  $c_{i}$ dimension |  $c$ dimension | EER | DCF | EER | DCF | EER | DCF
Attention | $1$ | $5120$ | $5120$ | $3.42$ | $0.0031$ | $3.42$ | $0.0029$ | $4.89$ | $0.0038$
MHA | $8$ | $640$ | $5120$ | $3.36$ | $0.0029$ | $3.44$ | $0.0029$ | $5.04$ | $0.004$
MHA | $16$ | $320$ | $5120$ | $3.43$ | $0.0032$ | $3.4$ | $0.0029$ | $4.9$ | $0.004$
MHA | $32$ | $160$ | $5120$ | $3.64$ | $0.0032$ | $3.68$ | 0.0031 | $5.35$ | $0.0042$
Double MHA | $8$ | $640$ | $640$ | $3.27$ | $0.0028$ | $3.23$ | $0.0028$ | $4.69$ | $0.0037$
Double MHA | $16$ | $320$ | $320$ | 3.19 | 0.0027 | $3.22$ | $0.0027$ | $4.67$ | $0.0038$
Double MHA | $32$ | $160$ | $160$ | $3.23$ | $0.0028$ | 3.18 | 0.0026 | 4.61 | 0.0036

The proposed approach has been evaluated against different attention methods
in the VoxCeleb text-independent speaker verification task. Performance is
evaluated using Equal Error Rate (EER) and Detection Cost Function (DCF)
calculated using $C_{FA}=1$, $C_{M}=1$, and $P_{T}=0.01$. The results of this
task are presented in Table 2. Besides the mentioned metrics, both head
$c_{i}$ and $c$ context vectors dimensions are shown for each presented
pooling approach.

Self Attention pooling has shown very similar results compared to Self MHA
approaches. In comparison to Self Attention, Self MHA has shown better results
in Vox1-test and Vox1-E protocols with 8 heads and 16 heads, respectively. The
relative improvement of these approaches compared to Self Attention are
$1.75\%$ in terms of EER in Vox1 Test for 8 heads and $0.58\%$ in terms of EER
in Vox1-E for the 16 heads model. Otherwise, Self Attention showed the best
baseline result in Vox1-H with a $4.89\%$ EER and a 0.0038 DCF. Compared to
the best Self MHA approach, Self attention has only shown a relative
improvement of $0.58\%$ in terms of EER. This similarity in the results
indicates that Self MHA has not led to a noticeable performance improvement
compared to vanilla Self Attention pooling. Double MHA have shown better
results for all head values compared with both Self Attention and Self MHA
approaches. In average, the $32$ heads model has outperformed all the baseline
systems considering both EER and DCF metrics. For Vox1 Test, Vox1-E and
Vox1-H, $32$ heads Double MHA have shown a $11.26\%$, $13.58\%$ and $13.83\%$
of EER relative improvement in comparison to the $32$ heads Self MHA model,
respectively. This performance increase has also been shown for $8$ and $16$
heads in average for all the protocols with a $5,23\%$ and $5,62\%$
improvement, respectively. Hence Double MHA has provided the best results and
it has shown to be more effective than Self MHA for all the head values.

As the results have shown, best performance in Double MHA based models has
been achieved with $16$ and $32$ heads. We have included head and global
context vector dimensions in Table 2 in order to analyze the relation between
the number of heads and the models performance. As it was discussed in
subsection 2.3, $c_{i}$ in Self MHA and both $c_{i}$ and $c$ dimensions in
Double MHA are inversely proportional to the number of heads. Therefore, there
is a trade-off between the number of attentions used over the encoded sequence
and the amount of speaker information each attention is able to capture. Worst
performance with Double MHA was achieved with $8$ heads. This setup implies
that both $c_{i}$ and $c$ dimensions are $640$. Current state-of-the-art
speaker embeddings have a dimension range between $200$ and $1500$
approximately. This means that there is still some margin to reduce the $c$
dimension and increase the number of attentions used. Increasing the number of
heads has led to a better verification performance. The best Double MHA model
has $32$ heads, whose context vector dimension $c$ is $160$. Here the speaker
information was encoded into a lower dimension representation and the pooling
was allowed to attend to $32$ different sub-sets of CNN channels. Models with
a larger number of heads have also been considered. However these models could
not have been trained due to the narrow $c$ dimension, which led us to
unstable training. Therefore Double MHA can be considered as a regularized
extension of Self MHA that works as a bottleneck layer. On the other hand, the
head number selection is also related to the CNN setup. The results indicate
that CNN output feature maps are more efficiently grouped in subsets of
$M/K=32$ channels, which correspond to sub-sequences of $160$ dimension
embeddings. Considering these sets of $32$ context vectors pooled in that
layer, these representations are efficiently averaged with Double MHA into
unique $160$ dimension utterance-level speaker representations.

##  5 Conclusion

In this paper we have implemented a Double Multi-Head Attention mechanism to
obtain speaker embeddings at utterance level by pooling short-term
representations. The proposed pooling layer is composed of a Self Multi-Head
Attention pooling and a Self Attention mechanism that summarizes the context
vectors of each head into a unique speaker vector. This pooling layer has been
tested in a neural network based on a CNN. The CNN maps the spectrograms into
sequences of speaker vectors. These vectors are the inputs to the proposed
pooling layer, whose output activations are then connected to a set of dense
layers. The network is trained as a speaker classifier and a bottleneck layer
from these fully connected layers is used as speaker embedding. We have
evaluated this approach with other pooling methods for the text-independent
speaker verification task using the speaker embeddings and applying the cosine
distance. The presented approach has outperformed both vanilla Self Attention
and Self Multi-Head Attention poolings.

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# Can a pure vector gravitational wave mimic a pure tensor one?

Bruce Allen  MPI for Gravitational Physics, Callinstrasse 38, Hannover,
Germany

###### Abstract

In the general theory of relativity, gravitational waves have two possible
polarizations, which are transverse and traceless with helicity $\pm 2$. Some
alternative theories contain additional helicity $0$ and helicity $\pm 1$
polarization modes. Here, we consider a hypothetical “pure vector” theory in
which gravitational waves have only two possible polarizations, with helicity
$\pm 1$. We show that if these polarizations are allowed to rotate as the wave
propagates, then for certain source locations on the sky, the strain outputs
of three ideal interferometric gravitational wave detectors can exactly
reproduce the strain outputs predicted by general relativity.

###### pacs:

04.30.-w,04.30.Db,04.30.Nk

DOI: 10.1103/PhysRevD.97.124020

##  I Introduction

In Einstein’s general theory of relativity (GR), gravitational waves (GW) are
radiative perturbations in the geometry of space-time that propagate at the
speed of light Einstein (1916, 1918). These so-called “tensor” modes are
spin-2 transverse traceless metric perturbations, which have only helicity
$\pm 2$ degrees of freedom Wald (1984). In alternatives to GR, GWs can contain
additional helicity $\pm 1$ and helicity 0 degrees of freedom, which are often
called “vector” and “scalar” modes Misner  _et al._ (1973); Schutz (1985);
Will (2014).

Interferometric GW detectors such as the Laser Interferometer Gravitational-
wave Observatory (LIGO) Aasi  _et al._ (2015) and Virgo Acernese  _et al._
(2015) would respond differently to vector and scalar modes than they respond
to tensor modes. When more than two GW detectors 111We assume that these
detectors can only observe a single polarization mode, or equivalently only a
single linear combination of polarization modes. observe the same source, this
makes it possible, at least in principle, to measure the wave polarization
content and determine which helicity components are present Cadonati (2004);
Wen and Schutz (2005); Isi  _et al._ (2017); Eardley  _et al._ (1973a, b); Isi
_et al._ (2015); Nishizawa  _et al._ (2009).

On 14 August 2017, the two LIGO detectors in the USA and the Virgo detector in
Italy made the first simultaneous 3-detector direct observation of GWs Abbott
_et al._ (2017a); the signal was present in the band of the detectors for a
fraction of a second. The waves were emitted by the merging binary black hole
system GW170814, and excited the three detectors in a pattern that was
consistent with the pure tensor mode predictions of GR. The discovery paper
Abbott  _et al._ (2017a) also claimed that the observation strongly disfavored
a pure vector model, which would not match the excitation pattern observed in
the detectors.

The presentation in Abbott  _et al._ (2017a) suggests that this is a theory-
independent observational result. Here we show that this is not the case, by
explicitly constructing a pure vector gravitational wave which for certain
source locations on the sky can precisely mimic or match the predictions of
GR. We assume that the signal has short duration compared to the 24h rotation
period of the Earth, so that during the observation the detector orientations
relative to the line of sight to the source do not change significantly.

To construct this counterexample, we allow the “direction” of the wave
polarization to change as the GW propagates, in the same way that the Faraday
effect can rotate the polarization of an electromagnetic wave Jackson (1998);
Fowles (1968); Rybicki and Lightman (1979); Born and Wolf (1999). With an
appropriate amount of rotation, the GW remains pure vector but exactly mimics
the detector response expected for tensor modes in GR.

The paper is organized as follows. In Sec. II we discuss the “pure tensor”
weak-field plane wave solution of GR. In Sec. III we show how an array of
three interferometric detectors would respond to these waves. In Sec. IV we
then consider the same questions for a conventional pure vector theory, and
discuss how null streams Wen and Schutz (2005) can distinguish these from
conventional GR. In Sec. V we construct an orthonormal basis for polarization
tensors which rotates about the propagation axis. In Sec. VI we use the
rotating basis to construct a pure vector gravitational wave which changes its
polarization as it propagates (behavior that does not happen in GR). By
correctly tuning the amount of rotation, we show that for some sky positions,
this pure vector theory would produce exactly the same detector outputs as GR.
This is followed by a short conclusion, and an Appendix giving quantities
needed to reproduce the results, such as the antenna response tensors for the
LIGO and Virgo detectors.

##  II Plane waves in general relativity

In GR, a GW propagating in the positive $z$-direction may be written in the
weak field limit as

$h_{ab}=w_{+}(t-z/c)e^{+}_{ab}+w_{\times}(t-z/c)e^{\times}_{ab}.$ (1)

Here $c$ is the speed of light and $w_{+}$ and $w_{\times}$ are arbitrary real
functions of one variable, which are the waveforms of the two different
polarization modes. The function $t-z/c$ is called “retarded time” 222This
retarded time is not referred to the source, but rather to the time at which
the corresponding wavefront passes the plane defined by $z=0$.. The transverse
traceless polarization tensors are

$\displaystyle e^{+}_{ab}$ $\displaystyle=x_{a}x_{b}-y_{a}y_{b}$ (2)
$\displaystyle e^{\times}_{ab}$ $\displaystyle=x_{a}y_{b}+y_{a}x_{b}$

where $x^{a}$ and $y^{a}$ are unit vectors 333Since we are in the weak field
limit, indices may be freely raised and lowered. in the x and y directions.
The complex combinations $e^{+}_{ab}\pm ie^{\times}_{ab}$ have helicity $\pm
2$, because under rotation of the orthogonal pair $x^{a},y^{a}$ through angle
$\phi$ in the plane orthogonal to the propagation direction $z$, the complex
polarizations acquire phase factors $\exp(\pm 2i\phi)$ Feynman .

The strain response $h(t)$ of an ideal interferometric gravitational wave
detector, with perpendicular arms of equal length is

$h(t)=d^{ab}h_{ab}(t,z_{0}),$ (3)

where $z_{0}$ is the z coordinate of the detector’s location, and

$d^{ab}=L_{1}^{a}L_{1}^{b}-L_{2}^{a}L_{2}^{b}$ (4)

is the detector response tensor. The quantities $L_{1}^{a}$ and $L_{2}^{a}$
are unit vectors along the two detector arms, and the detector is assumed to
be ideal. This means that it is free of instrumental noise, with a flat
frequency response that is broad enough to reproduce the spectral content of
the waveforms $w^{+}$ and $w^{\times}$.

##  III The case of three detectors

Now suppose that we have three ideal detectors, at locations with z coordinate
$z_{i}$ and with detector response tensors $d^{ab}_{i}$, for $i=1,\cdots,3$.
We assume that the GW source is at a distance from the detectors which is much
larger than their size, than the separation between them, and than the
characteristic wavelength of the radiation. We assume that the source’s sky
location is known precisely, and (without loss of generality) that it lies on
the negative $z$ axis. We also assume that the signal is visible in the
detectors for a time that is brief compared to the Earth’s spin period of 24h,
so that the instrument orientations may be treated as fixed.

In this case, the strain at the three detectors is given by combining (1) and
(3):

$h_{i}(t)=F^{+}_{i}w_{+}(t-z_{i}/c)+F^{\times}_{i}w_{\times}(t-z_{i}/c).$ (5)

The strain is seen to be a linear combination of the two different waveforms,
with weights $F_{i}^{+}$ and $F_{i}^{\times}$ that are determined by the
geometrical overlap between the detector response tensors and the polarization
tensors. These weights,

$F_{i}^{+}=e^{+}_{ab}d_{i}^{ab}\quad\text{and}\quad
F_{i}^{\times}=e^{\times}_{ab}d_{i}^{ab},$ (6)

are called “antenna functions” and are easily computed. Here, they should be
thought of as dimensionless numbers, whose fixed values are determined _a
priori_ by the orientations of the detector arms Allen  relative to the GW
source. (For completeness, the Appendix gives expressions for $d_{i}^{ab}$ for
the LIGO and Virgo detectors.)

For later convenience, we introduce the non-negative lengths $F_{i}$ of these
three two-vectors

$F_{i}=\sqrt{(F^{+}_{i})^{2}+(F^{\times}_{i})^{2}}$ (7)

and the angles defined by the ratios of the two components,

$\phi_{i}=\arctan\frac{F^{\times}_{i}}{F^{+}_{i}}.$ (8)

Here, the arc-tangent is defined as the principal value of
$\arg(F^{+}+iF^{\times})$. In terms of these lengths and angles, we can write
the two-vectors $(F^{+}_{i},F^{\times}_{i})=F_{i}(\cos\phi_{i},\sin\phi_{i})$.
As before, $F_{i}$ and $\phi_{i}$ should be thought of as constants, which are
defined by the detector arm orientations relative to the direction to the
source.

To compare the strain outputs of the different detectors, it is helpful to
first shift the observed strains to a common fiducial arrival time, defining
new time series by $\bar{h}_{i}(t)=h_{i}(t+z_{i}/c)$. In GR these time-shifted
strains may be written as linear combinations of the two waveforms, with
weights determined by the antenna patterns of the three instruments,

$\left[\begin{array}[]{c}\bar{h}_{1}(t)\\\ \bar{h}_{2}(t)\\\
\bar{h}_{3}(t)\end{array}\right]=\begin{array}[]{c}F_{1}\\\ F_{2}\\\
F_{3}\end{array}\left[\begin{array}[]{cc}\cos\phi_{1}&\sin\phi_{1}\\\
\cos\phi_{2}&\sin\phi_{2}\\\
\cos\phi_{3}&\sin\phi_{3}\end{array}\right]\left[\begin{array}[]{c}w_{+}(t)\\\
w_{\times}(t)\\\ \end{array}\right],$ (9)

where the respective values of $F_{i}$ are understood to multiply the
corresponding row on the right-hand side (r.h.s.).

For generic detector orientations, the three two-vectors
$(F^{+}_{i},F^{\times}_{i})$ are nonvanishing. Hence they must be linearly
dependent, since a two-dimensional vector space can contain at most two
linearly independent vectors. Thus, the three time-shifted strains must also
be linearly dependent, with time-independent coefficients, since they are
linear combinations of two independent waveform functions $w_{+}(t)$ and
$w_{\times}(t)$.

This is the basis of “null tests”, which construct a linear combination of
these three time-shifted strains
$\bar{h}_{1}(t),\bar{h}_{2}(t),\bar{h}_{3}(t)$ which vanishes if GR is a
correct description of GW Wen and Schutz (2005). This vanishing linear
combination is called a “null stream”. By the same reasoning, if there are $N$
generically-oriented detectors, then $N-2$ “different” null stream
combinations may be constructed from them Wen and Schutz (2005).

##  IV The three-detector case in “pure vector gravity”

Now consider a fictitious pure vector theory of gravity, where the transverse
traceless polarization tensors of GR are replaced by a pair which for
convenience we denote with $L$ and $R$:

$\displaystyle e^{L}_{ab}$ $\displaystyle=x_{a}z_{b}+z_{a}x_{b}$ (10)
$\displaystyle e^{R}_{ab}$ $\displaystyle=y_{a}z_{b}+z_{a}y_{b}$

where $z^{a}$ is a unit vector in the positive $z$ direction. In this pure
vector case, the GW is described by

$h_{ab}=w_{L}(t-z/c)e^{L}_{ab}+w_{R}(t-z/c)e^{R}_{ab},$ (11)

where, as before, the waveforms $w_{L}$ and $w_{R}$ are arbitrary real
functions of one variable.

We assume that in this fictitious theory of gravity, the detectors respond as
before (3) so that the strain in the $i$th detector is

$h_{i}(t)=F^{L}_{i}w_{L}(t-z_{i}/c)+F^{R}_{i}w_{R}(t-z_{i}/c),$ (12)

where the weights $F$ that appear are again the overlaps between the detector
response tensors and the polarization tensors:

$F_{i}^{L}=e^{L}_{ab}d_{i}^{ab}\quad\text{and}\quad
F_{i}^{R}=e^{R}_{ab}d_{i}^{ab}.$ (13)

As before, after shifting to a common fiducial time, the three time series
$\bar{h}_{i}(t)=h_{i}(t+z_{i}/c)$ must be linearly dependent, because they are
linear combinations of the two functions $w_{L}(t)$ and $w_{R}(t)$.

As in the previous section, we introduce the non-negative lengths $f_{i}$ of
these three two-vectors

$f_{i}=\sqrt{(F^{L}_{i})^{2}+(F^{R}_{i})^{2}}$ (14)

and the angles defined by the ratios of the two components,

$\psi_{i}=\arctan\frac{F^{R}_{i}}{F^{L}_{i}},$ (15)

where the arc-tangent is defined as the principal value of
$\arg(F^{L}+iF^{R})$. In terms of these quantities, we can express the two-
vectors as $(F^{L}_{i},F^{R}_{i})=f_{i}(\cos\psi_{i},\sin\psi_{i})$, where
again $f_{i}$ and $\psi_{i}$ are constants defined by the detector arm
orientations relative to the direction to the source.

The time-shifted strains resulting from this pure vector theory are then

$\left[\begin{array}[]{c}\bar{h}_{1}(t)\\\ \bar{h}_{2}(t)\\\
\bar{h}_{3}(t)\end{array}\right]=\begin{array}[]{c}f_{1}\\\ f_{2}\\\
f_{3}\end{array}\left[\begin{array}[]{cc}\cos\psi_{1}&\sin\psi_{1}\\\
\cos\psi_{2}&\sin\psi_{2}\\\
\cos\psi_{3}&\sin\psi_{3}\end{array}\right]\left[\begin{array}[]{c}w_{L}(t)\\\
w_{R}(t)\\\ \end{array}\right],$ (16)

where the respective values of $f_{i}$ are understood to multiply the $i$th
row on the r.h.s. This should be compared with the corresponding Eq. (9) that
arises in GR.

For generic detector orientations, the first two rows of this system are
invertible, so the data at the first two detectors completely determines
$w_{L}(t)$ and $w_{R}(t)$. With that choice, $\bar{h}_{1}(t)$ and
$\bar{h}_{2}(t)$ will match the data perfectly. But if GR is correct, then
$\bar{h}_{3}(t)$ in (16) will be incorrect: it will not match the data from
the detector.

##  V A rotating basis for polarization tensors

At this juncture, it is helpful to introduce a “rotating” vector polarization
basis 444This basis rotates in phase at frequency “one cycle per rotation”
($\phi$) as opposed to (say) frequency “two cycles per rotation” ($2\phi$) or
“no cycles per rotation” ($0\phi$). Since the polarization basis transforms as
a rank-(0,2) tensor, this implies that the vector polarizations are helicity
$\pm 1$ components of a spin-2 field. The name “vector” perturbations is
therefore somewhat misleading, since this implies a spin-1 object, which is a
rank-(0,1) or (1,0) tensor. However it has historically been used in this
context, so we perpetuate the misnomer. analogous to (10). For this, we first
define a basis that rotates with space and time as

$\displaystyle x^{\prime a}$
$\displaystyle=\cos\phi(t,z)x^{a}+\sin\phi(t,z)y^{a}$ (17) $\displaystyle
y^{\prime a}$ $\displaystyle=-\sin\phi(t,z)x^{a}+\cos\phi(t,z)y^{a}$
$\displaystyle z^{\prime a}$ $\displaystyle=z^{a}$

where the phase $\phi$ is an arbitrary real function of time $t$ and spatial
position $z$. The new basis vectors $(x^{\prime a},y^{\prime a})$ are obtained
by rotating the old basis vectors $(x^{a},y^{a})$ counterclockwise through
angle $\phi$.

Using this basis, we define new polarization tensors by

$\displaystyle e^{l}_{ab}$
$\displaystyle=x^{\prime}_{a}z^{\prime}_{b}+z^{\prime}_{a}x^{\prime}_{b}=\cos\phi\,e^{L}_{ab}+\sin\phi\,e^{R}_{ab}$
(18) $\displaystyle e^{r}_{ab}$
$\displaystyle=y^{\prime}_{a}z^{\prime}_{b}+z^{\prime}_{a}y^{\prime}_{b}=-\sin\phi\,e^{L}_{ab}+\cos\phi\,e^{R}_{ab}$

and express the gravitational wave amplitude as a sum over these two
polarizations:

$\displaystyle h_{ab}(t,z)$
$\displaystyle=w_{l}(t-z/c)e^{l}_{ab}+w_{r}(t-z/c)e^{r}_{ab}.$ (19)

As before, the waveform functions $w_{l}$ and $w_{r}$ are arbitrary real
functions of one variable. Note however that the left-hand side is a function
of retarded time $t-z/c$ if and only if the phase $\phi$ is also a function of
retarded time.

We stress that (17) is _not_ defining a new basis for the tensor space over
our space-time manifold. If it were, then scalar quantities such as the strain
$h$ at a particular detector would not be changed. This is because the
components of the field $h_{ab}$ would transform covariantly, but the
components of the detector tensors $d^{ab}$ would transform contravariantly,
and the scalar product $h_{ab}d^{ab}$ would be invariant. Here, we are
explicitly _changing_ the nature and physical content of the GW in ways that
we will discuss shortly.

By using (11), (13), (18), and (19), we can express the time-shifted detector
strains as the 2-dimensional vector inner products

${\bar{h}}_{i}(t)=\left[F^{L}_{i}\quad
F^{R}_{i}\right]\left[\begin{array}[]{cc}\cos\phi(t+z_{i}/c,z_{i})&-\sin\phi(t+z_{i}/c,z_{i})\\\
\sin\phi(t+z_{i}/c,z_{i})&\cos\phi(t+z_{i}/c,z_{i})\end{array}\right]\left[\begin{array}[]{c}w_{l}(t)\\\
w_{r}(t)\end{array}\right].$ (20)

If the phase $\phi$ only depends upon (a single variable, which is) the
retarded time $t-z/c$, then this is completely equivalent to the case
considered in the previous section with waveform functions $w_{L}(t)$ and
$w_{R}(t)$ defined by

$\left[\begin{array}[]{c}w_{L}(t)\\\
w_{R}(t)\end{array}\right]=\left[\begin{array}[]{cc}\cos\phi(t)&-\sin\phi(t)\\\
\sin\phi(t)&\cos\phi(t)\end{array}\right]\left[\begin{array}[]{c}w_{l}(t)\\\
w_{r}(t)\end{array}\right].$ (21)

As before, with three strain functions observed at the detectors, but only two
functional degrees of freedom available, the system is overconstrained.

##  VI Constructing a counterexample

In electrodynamics, when a wave is propagating in vacuum, then the
polarization state is “carried” by the wave-fronts. For an observer at a fixed
point in space, the direction of the electric field polarization vector may
change with time. But if we consider the polarization vector along the null
path followed by the wave, then the polarization is constant along that path.
In this sense, the wavefront carries the polarization state with it Mandel and
Wolf (1995); Born and Wolf (1999). However this is only true in a vacuum.

If instead, the wave is propagating through a material, then the polarization
vector can shift direction along the path followed by the wavefront. One
example is Faraday rotation, when the wave is propagating through a medium
containing free charges in the presence of a magnetic field Jackson (1998);
Rybicki and Lightman (1979); Fowles (1968).

GR is similar to vacuum electrodynamics, where by “polarization” we mean the
relative “mixture” of $e^{+}_{ab}$ and $e^{\times}_{ab}$ that is present in
the wave, for a fixed Cartesian basis. For a given fixed observer, this
mixture will vary with time as the GW passes by. But if we look along the null
path traced by a point on the wavefront, then it follows from (1) that the
mixture does not change. However there is no reason to assume that the
conjectured pure vector theory of gravity obeys this way!

To construct a pure vector gravitational wave which gives rise to the same
strains as would be found in pure tensor GR, we assume that in the pure vector
theory (for unspecified reasons) the polarization state can _change_ as the
wave propagates. For example, this might be due to the presence of the Earth,
which hypothetically could introduce a dependence on the variable $z$, so that

$\phi(t,z)=\Delta\psi(z).$ (22)

As can be seen from Eqs. (16) and (21), the effect is equivalent to rotating
the antenna pattern function through an additional angle
$\Delta\psi_{i}=\Delta\psi(z_{i})$ at the $i$th detector. Defining
$\tilde{\psi}_{i}=\psi_{i}+\Delta\psi_{i}$, the response of the detectors will
now be described by

$\left[\begin{array}[]{c}\bar{h}_{1}(t)\\\ \bar{h}_{2}(t)\\\
\bar{h}_{3}(t)\end{array}\right]=\begin{array}[]{c}f_{1}\\\ f_{2}\\\
f_{3}\end{array}\left[\begin{array}[]{cc}\cos\tilde{\psi}_{1}&\sin\tilde{\psi}_{1}\\\
\cos\tilde{\psi}_{2}&\sin\tilde{\psi}_{2}\\\
\cos\tilde{\psi}_{3}&\sin\tilde{\psi}_{3}\end{array}\right]\left[\begin{array}[]{c}w_{L}(t)\\\
w_{R}(t)\\\ \end{array}\right],$ (23)

where the factors $f_{i}$ multiply the corresponding lines on the r.h.s.

Now that we have introduced additional degrees of freedom in the
$\tilde{\psi}_{i}$, it will be possible _in some cases_ for this hypothetical
pure vector theory to exactly match the strains predicted or expected for GR.
However this is not always possible. Roughly speaking, by changing the
polarization angle shifts $\Delta\psi_{i}$ , we can vary the proportions of
the $w_{R}$ and $w_{L}$ waveforms that enter the detected strain at each site.
However if the vector response $f_{i}$ at a particular detector is too small
compared to the GR response at that detector, then it won’t be possible to
match the GR prediction, because varying only the phase $\Delta\psi_{i}$ can
not make the amplitude arbitrarily large. As an extreme example, suppose that
the source is located at a position in the sky for which the third detector
has no response to either of the pure vector modes, which means that $f_{3}$
vanishes. In this case there is no way for the pure vector theory to give the
same observed strain as GR at the third detector 555We assume here that
$F_{3}$ does not vanish, meaning that the GR strain in the third detector is
nonvanishing.

Can we pick $\Delta\psi_{i}$ so that the pure vector theory masquerades as
ordinary GR? It turns out that this is possible if and only if the antenna
patterns obey a triangle inequality. This will only be satisfied for some
incoming wave directions.

To understand what directions, invert the first two rows of Eq. (9) to obtain
$w_{+}$ and $w_{\times}$ as linear functions of $\bar{h}_{1}$ and
$\bar{h}_{2}$. Then use the first two rows of the pure vector Eq. (23) to
replace $\bar{h}_{1}$ and $\bar{h}_{2}$, obtaining expressions for $w_{+}$ and
$w_{\times}$ as linear functions of $w_{L}$ and $w_{R}$. With these choices,
we have ensured that the pure vector model will produce strains at detectors 1
and 2 which agree with those of GR.

Can we pick the $\Delta\psi_{i}$ (or equivalently, $\psi_{i}$) so that the
strain at the third detector would agree for GR and the pure vector model?
This implies that the third rows of (9) and (23) agree for all choices of
$w_{R}$ and $w_{L}$, which is possible if and only if

$f_{3}(\cos\tilde{\psi}_{3}w_{L}+\sin\tilde{\psi}_{3}w_{R})=F_{3}(\cos\phi_{3}w_{+}+\sin\phi_{3}w_{\times}).$
(24)

Replacing $w_{+}$ and $w_{\times}$ on the r.h.s. with the linear combinations
of $w_{L}$ and $w_{R}$ obtained above, multiplying by a factor of
$F_{1}F_{2}\cos(\phi_{2}-\phi_{1})$, and collecting terms, we obtain

$\displaystyle w_{L}(t)$
$\displaystyle\bigl{[}f_{1}F_{2}F_{3}\sin(\phi_{3}-\phi_{2})\cos\tilde{\psi}_{1}+F_{1}f_{2}F_{3}\sin(\phi_{1}-\phi_{3})\cos\tilde{\psi}_{2}+F_{1}F_{2}f_{3}\sin(\phi_{2}-\phi_{1})\cos\tilde{\psi}_{3}\bigr{]}+$
(25) $\displaystyle w_{R}(t)$
$\displaystyle\bigl{[}f_{1}F_{2}F_{3}\sin(\phi_{3}-\phi_{2})\sin\tilde{\psi}_{1}+F_{1}f_{2}F_{3}\sin(\phi_{1}-\phi_{3})\sin\tilde{\psi}_{2}+F_{1}F_{2}f_{3}\sin(\phi_{2}-\phi_{1})\sin\tilde{\psi}_{3}\bigr{]}=0.$

If it is possible to pick $\Delta\psi_{i}$ (or equivalently
$\tilde{\psi}_{i}$) so that both of the quantities in square brackets vanish,
then the pure vector theory can masquerade as GR.

The problem may be visualized as follows. Each of the three $\tilde{\psi}_{i}$
define a unit-length vector
$\hat{n}_{i}=(\cos\tilde{\psi}_{i},\sin\tilde{\psi}_{i})$ in the $(L,R)$
plane. The quantities in square brackets will vanish if and only if the
directions of these three unit vectors can be chosen so that the sum

$\displaystyle
f_{1}F_{2}F_{3}\sin(\phi_{3}-\phi_{2})\hat{n}_{1}+F_{1}f_{2}F_{3}\sin(\phi_{1}-\phi_{3})\hat{n}_{2}$
(26) $\displaystyle+$ $\displaystyle
F_{1}F_{2}f_{3}\sin(\phi_{2}-\phi_{1})\hat{n}_{3}$

vanishes. This in turn is possible if and only if the three lengths that
multiply these unit vectors obey the triangle inequality: each of the three
must not be larger than the sum of the other two.

If we use the definitions of the $\phi_{i}$, then this can be re-expressed in
terms of the original antenna pattern weights. This postulated pure vector
theory can masquerade as GR provided that the three cyclically-related
quantities

$\displaystyle f_{1}(F_{2}^{+}F_{3}^{\times}-F_{2}^{\times}F_{3}^{+}),$ (27)
$\displaystyle f_{2}(F_{3}^{+}F_{1}^{\times}-F_{3}^{\times}F_{1}^{+}),$ and
$\displaystyle f_{3}(F_{1}^{+}F_{2}^{\times}-F_{1}^{\times}F_{2}^{+})$

obey the triangle inequality: the sum of any two must be greater than or equal
to the third.

Note that if a particular incoming wave direction $\hat{z}$ satisfies this
inequality, then so does the antipodal direction $-\hat{z}$. This is because,
under the antipodal transformation $\vec{x}\rightarrow-\vec{x}$, the antenna
weights $F^{+}_{i}$ and $F^{\times}_{i}$ are invariant, as are the $f_{i}$. So
a sky map of the regions where the triangle inequality is satisfied is
symmetric under antipodal transformations.

At the moment, there are three sensitive interferometric gravitational wave
detectors in operation: LIGO Hanford, LIGO Livingston, and Virgo. In Fig. 1 we
show the regions of the sky where this triangle inequality is satisfied. Pure
vector theory sources in those directions could mimic the spin-2 transverse
traceless perturbations of GR. In particular, the area of the sky in which
GW170814 was likely located, overlaps these directions.

|
---|---
|
|
Figure 1:  These plots show directions on the sky, relative to the currently-
operating Earth-based gravitational wave detectors. The LIGO Hanford, LIGO
Livingston, and Virgo detector arms are shown in red (not to scale). Sky
directions in which the triangle inequality is _not_ satisfied are shown in
light blue; these are invariant under antipodal reflection. The remaining sky
directions (white) are those in which the pure vector model can mimic GR. The
dark blue path outlines the most likely region (at 90% Bayesian confidence)
from which the GW170814 event originated. In part of that region the pure
vector model presented here can give detector signals in the three instruments
which are the same as those from GR.

##  VII Conclusion

What has been presented here is a cautionary note which demonstrates that
ruling out a pure vector polarization model, in the absence of a theory which
defines its behavior, is difficult. We have constructed a pure vector model
which, for certain source locations, can exactly mimic the signal predicted by
GR in the LIGO and Virgo detectors. Of course the recent LIGO Abbott  _et al._
(2016a, b, 2017b) and LIGO-Virgo Abbott  _et al._ (2017a, c) observations and
many other experimental tests indicate that GR is an excellent description of
nature, so the author disavows any belief in the contrived pure vector model
presented here. It is only intended as a counterexample.

The pure vector model presented here is able to mimic GR for a three-detector
network because the polarization mixture “rotates” as the gravitational wave
propagates, so it demonstrates that such polarization rotation is _sufficient_
to mimic GR. While we have not proved that such rotation is _necessary_ to
mimic GR, that may well be true.

Counting free parameters suggests that it should be possible to construct
similar contrived pure vector models which would match the prediction of GR
for short-duration signals for networks containing more than three detectors.
However the set of allowed sky directions where this is possible would be
further constrained (smaller total sky area).

At each detector location (labeled by $i$) this pure vector model introduces
an additional angle $\Delta\psi_{i}$. So one might well ask: does it have any
predictive ability at all? Can it be falsified? The answer to both questions
is yes. Recall that the helicity of a field is defined by the properties of
the complex phase under rotations Feynman . Imagine that we have a set of co-
located gravitational wave detectors, whose arms are all in the same plane
tangent to the Earth, but which are rotated with respect to one another. Then
the helicity can be directly measured by comparing the phase of the strain
signals between the different instruments. For example suppose that we had 360
co-located interferometric detectors, each of which had its arms offset by
$1^{\circ}$ from the previous one, so that they spanned the entire
$360^{\circ}$ circle in $1^{\circ}$ increments. In GR, the phase of the signal
will go through two complete cycles (at any instant in time) as you move
around the “circle of detectors”. In the pure vector theory there would be
only one cycle of phase change. This thought-experiment demonstrates that the
pure vector model presented here does make falsifiable predictions.

Similar reasoning indicates that this pure vector model would have difficulty
in mimicking GR for long-duration signals. This is because the Earth’s
rotation changes the orientation of the source relative to the detectors. For
example a detector located at the North Pole with its arms in the plane
tangent to the Earth would undergo a complete $360^{\circ}$ rotation each day.
If a gravitational wave signal is observed over that entire time span, then
models like the one presented here could then be constrained or falsified
Abbott  _et al._ (2018).

The pure vector model presented here has one undetermined phase
$\Delta\psi_{i}$ for each detector. We have shown that for specific values of
these phases, the model can exactly reproduce the observed instrumental
strains expected from general relativity. It follows that for a real
instrument, in which detector noise makes it impossible to identify the signal
exactly, some range of these values would be compatible with data. So there is
a temptation to use Bayesian methods to compare the odds that this model (with
uniform uninformed priors on $\Delta\psi_{i}\in[0,2\pi]$) describes Nature,
versus the predictions of general relativity. But this would miss the point:
the model presented here is merely intended to demonstrate how difficult it is
to rule something out, in the absence of a specific predictive model. One
could compute odds ratios for the model presented here, but the number of
alternative models is only limited by the imagination. Without definite
predictions, their odds ratios are not computable.

###### Acknowledgements.

The author is grateful for helpful discussions with Patrick Sutton and Alan
Weinstein, useful comments from B.S. Sathyaprakash and Andrea Vicere, and
assistance with the plots from Reed Essick, Oliver Bock, and Heinz-Bernd
Eggenstein.

## Appendix: Detector response tensors for LIGO and Virgo

Here we list the quantities needed to easily reproduce Fig. 1. We work in a
right-handed Cartesian coordinate system $[u,v,w]$ which is fixed to the Earth
(so rotates with it) and whose origin is at the center of the Earth. The
positive $u$-axis passes through the line of Greenwich at zero longitude where
it crosses the equator, and the positive $w$-axis passes through the North
Pole. The Earth is assumed to be spherical.

The first quantities of interest are unit vectors $n$ pointing from the center
of the Earth to the detectors, which may be obtained from the geographical
information cataloged in Allen . These are

$\displaystyle n_{\rm LLO}$
$\displaystyle=-0.01157{\,\hat{u}}-0.86102{\,\hat{v}}+0.50844{\,\hat{w}}$
$\displaystyle n_{\rm LHO}$
$\displaystyle=-0.33833{\,\hat{u}}-0.60020{\,\hat{v}}+0.72477{\,\hat{w}}$
$\displaystyle n_{\rm Virgo}$
$\displaystyle=\,\,\,\,\,0.71169{\,\hat{u}}+0.13190{\,\hat{v}}+0.69000{\,\hat{w}}.$

The next useful quantities are unit vectors $L_{1}$ and $L_{2}$ along the two
detector arms, derived from information in Allen :

$\displaystyle\rm LLO:$ $\displaystyle
L_{1}=-0.95308{\,\hat{u}}-0.14432{\,\hat{v}}-0.26609{\,\hat{w}}$
$\displaystyle
L_{2}=\,\,\,\,\,0.30249{\,\hat{u}}-0.48767{\,\hat{v}}-0.81895{\,\hat{w}}$
$\displaystyle\rm LHO:$ $\displaystyle
L_{1}=-0.23684{\,\hat{u}}+0.79971{\,\hat{v}}+0.55169{\,\hat{w}}$
$\displaystyle
L_{2}=-0.91073{\,\hat{u}}+0.01500{\,\hat{v}}-0.41272{\,\hat{w}}$
$\displaystyle\rm Virgo:$ $\displaystyle
L_{1}=-0.70121{\,\hat{u}}+0.19275{\,\hat{v}}+0.68641{\,\hat{w}}$
$\displaystyle
L_{2}=-0.04246{\,\hat{u}}-0.97234{\,\hat{v}}+0.22967{\,\hat{w}}.$

Working in the same $u,v,w$ basis, the components of the detector response
tensors (4) are given by:

$\displaystyle d^{ab}_{\rm LLO}=$
$\displaystyle\left[\begin{array}[]{rrr}\;\;\,0.81686&0.28507&0.50134\\\
\;\;\,0.28507&-0.21699&-0.36097\\\
\;\;\,0.50134&-0.36097&-0.59988\end{array}\right]$ $\displaystyle d^{ab}_{\rm
LHO}=$ $\displaystyle\left[\begin{array}[]{rrr}-0.77334&-0.17575&-0.50654\\\
-0.17575&0.63931&0.44739\\\ -0.50654&0.44739&0.13403\end{array}\right]$
$\displaystyle d^{ab}_{\rm Virgo}=$
$\displaystyle\left[\begin{array}[]{rrr}\;\;\,0.48989&-0.17644&-0.47156\\\
-0.17644&-0.90830&0.35562\\\ -0.47156&0.35562&0.41841\end{array}\right].$

Please note that since the Earth is not spherical, these quantities differ
from the actual values by fractional amounts of order $10^{-3}$. For our
purposes, this may be neglected.

For a given sky location, the pure tensor polarization tensors
$e^{+,\times}_{ab}$ in Eq. (2) and the pure vector polarization tensors
$e^{L,R}_{ab}$ in Eq. (10) may be easily constructed from the right-handed
basis

$\displaystyle x=$ $\displaystyle-\sin L{\,\hat{u}}+\cos L{\,\hat{v}}$
$\displaystyle y=$ $\displaystyle\sin\ell\cos L{\,\hat{u}}+\sin\ell\sin
L{\,\hat{v}}-\cos\ell{\,\hat{w}}$ $\displaystyle z=$
$\displaystyle-\cos\ell\cos L{\,\hat{u}}-\cos\ell\sin
L{\,\hat{v}}-\sin\ell{\,\hat{w}},$

where (positive) $\ell\in[-90^{\circ},90^{\circ}]$ is lattitude North of the
equator, and (positive) $L\in[-180^{\circ},180^{\circ}]$ is longitude East of
Greenwich. (The published version of this paper has $\sin\ell$ and $\cos\ell$
interchanged in the equations defining $x$, $y$, and $z$. This is corrected
above.)

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