# A note on Positive Partial Transpose Blocks

Mohammad Alakhrass  Department of Mathematics, University of Sharjah , Sharjah
27272, UAE<EMAIL_ADDRESS>
###### Abstract.

In this article, we study the class of PPT blocks. A simple proof yields the
following result: For $X,A,B\in\mathbb{M}_{n}$, if
$\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT, then

$|X|\leq\left(A\\#B\right)\\#U^{*}(A\\#B)U,$

for some unitary $U\in\mathbb{M}_{n}$. Several consequences of this inequality
are established. These consequences improve, generalize and unify many
corresponding results in the literature.

###### Key words and phrases:

Block matrices; Positive partial transpose matrices; Lieb functions; Loewner
order; Unitarily invariant norm

###### 2010 Mathematics Subject Classification:

15A18, 15A42, 15A45, 15A60.

##  1\. Introduction

Let $\mathbb{M}_{n}$ be the algebra of all $n\times n$ complex matrices. For
$X\in\mathbb{M}_{n}$, the notation $X\geq 0$ (resp. $X>0$) will be used to
mean that $X$ is positive semidefinite (resp. positive definite). If
$X,Y\in\mathbb{M}_{n}$ are two Hermitian matrices in $\mathbb{M}_{n}$, we
write $X\leq Y$ to mean $Y-X\geq 0$. The eigenvalues and singular values of
$X\in\mathbb{M}_{n}$ are denoted by $\lambda_{j}(X)$ and $s_{j}(X)$,
respectively, $j=1,2,...,n.$ It is assumed that the eigenvalues and singular
values of $X$ are arranged so that
$|\lambda_{1}(X)|\geq...\geq|\lambda_{n}(X)|$ and $s_{1}(X)\geq
s_{2}(X)\geq...\geq s_{n}(X)$. The unitarily invariant norm of
$X\in\mathbb{M}_{n}$ is denoted by $\|X\|$. Recall that a norm $\|\cdot\|$ on
$\mathbb{M}_{n}$ is said to be unitarily invariant if it satisfies the
property $\|UXV\|$ for all $X\in\mathbb{M}_{n}$ and all unitaries
$U,V\in\mathbb{M}_{n}.$ For positive definite $X,Y\in\mathbb{M}_{n}$, the
geometric mean of $X$ and $Y$ is defined as follows

$X\\#Y=X^{1/2}(X^{-1/2}YX^{-1/2})^{1/2}X^{1/2}.$

Let $A,B,X\in\mathbb{M}_{n}$. Throughout this note, we consider the $2\times
2$ block matrix $H$ in the form

$H=\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right).$

It is well known that $H$ positive if and only if the Schur complement of $A$
in $H$ is positive semidefinite provided that $A$ is strictly positive. That
is, $H\geq 0$ if and only if

$H/A=B-X^{*}A^{-1}X\geq 0.$ (1)

The $2\times 2$ blocks play an important role in studying matrices and
positive matrices in particular. Bhatia book [7] provides a comprehensive
survey about block matrices. Furthermore, a positive $2\times 2$ block can be
a very useful tool in studying sectorial matrices, see for example [1], [2]
and [3].

The the partial transpose of the block $H$ is defined by

$H^{\tau}=\left(\begin{array}[]{cc}A&X^{*}\\\ X&B\\\ \end{array}\right).$

It is quite clear that the positivity of $H$ does not, in general, imply the
positivity of $H^{\tau}$. The block $H$ is said to be positive partial
transpose, or PPT for short, if both $H$ and $H^{\tau}$ are positive
semidefinite. The Schure criterion for positivity implies that $H$ is PPT if
and only if

$B-X^{*}A^{-1}X\geq 0\quad\text{and}\quad B-XA^{-1}X^{*}\geq 0,$

provided that $A$ is strictly positive.

The PPT criterion ( also called Peres–Horodecki criterion ) plays an important
roll in the quantum information theory. For example, PPT condition is a
necessary condition for a mixed quantum state to be separable. Moreover, in
low dimensional composite spaces ( two and three) this condition (PPT) is also
sufficient. See [14] and [18].

The class of PPT matrices possess many interesting properties. Therefore, it
has attracted a huge interest. See [4, 11, 12, 13, 15, 16, 17].

Given a PPT block $H=\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\
\end{array}\right).$ It is well known that for any unitarily invariant norm

$||H||\leq||A+B||.$ (2)

This result due to Hiroshima [13]. See also [17] for related results. A
stronger result can be obtained when $X$ is Hermitian. In fact it was shown in
[9] that if $X$ is Hermitian, then

$H=\left(\begin{array}[]{cc}A&X\\\ X&B\\\
\end{array}\right)=\frac{1}{2}\left[U\left(A+B\right)U^{*}+V\left(A+B\right)V^{*}\right],$
(3)

for some isometries $U,V\in\mathbb{M}_{2n\times n}$. Here
$\mathbb{M}_{2n\times n}$ is the space of all $2n\times n$ complex matrices.
See also [10] for more details and more results in this direction. The
inequality (2) and the identity (3) present connections between the whole
block $H$ and its main diagonal components $A$ and $B$. Recently, some
interesting inequalities, connecting the main and the off-diagonal of the PPT
Block $H$, have been established. These inequalities include trace
inequalities, norm inequalities, singular value inequalities and also
inequalities in the sense of Loewner.

In [16], Lin has proved

$tr\left(X^{*}X\right)\leq tr\left(AB\right)$ (4)

and

$\prod_{j=1}^{k}s_{j}(X)\leq\prod_{j=1}^{k}s_{j}\left(A\\#B\right)\leq\prod_{j=1}^{k}s_{j}\left(A^{1/2}B^{1/2}\right),j=1,2,...,n.$
(5)

In the same article, Lin has given a counter-example showing that the
inequalities

$s_{j}(X)\leq s_{j}\left(\frac{A+B}{2}\right),j=1,2,...,n,$ (6)

are not true in general.

In the sense of Loewner, it has been proved, in [15], that for some unitary
$U\in\mathbb{M}_{n}$

$|X|\leq\frac{A\\#B+U^{*}(A\\#B)U}{2}.$ (7)

The author of [15], used (7) to give some weak versions of (6).

In this paper, we show that if $H$ is PPT, then

$|X|\leq\left(A\\#B\right)\\#U^{*}(A\\#B)U,$ (8)

for some unitary $U\in\mathbb{M}_{n}$. Then we present several consequences of
(8) including inequalities such as (4), (5) and (7). Moreover, we present an
interesting and direct proof for Theorem 3.2 which states that a slightly
weaker version of the inequalities in (6) hold true. Finally, we present some
inequalities that connect the diagonal components to the real part of the off-
diagonal components of H.

##  2\. Preliminaries

In this section we present some basic properties of positive and PPT blocks.
These properties are summarized in Proportion 2.1, 2.2 and 2.3. To make this
note self-contained, we outline the proofs of these propositions. We also
recall some important facts about geometric mean of two positive matrices.

######  Proposition 2.1.

If $H=\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)\geq 0,$
then

  1. (1)

$\left(\begin{array}[]{cc}A&-X\\\ -X^{*}&B\\\ \end{array}\right)\geq 0$ and
$\left(\begin{array}[]{cc}B&X^{*}\\\ X&A\\\ \end{array}\right)\geq 0.$

  2. (2)

$\left(\begin{array}[]{cc}0&X\\\ X^{*}&0\\\
\end{array}\right)\leq\frac{1}{2}H.$

###### Proof.

To see the first part, observe that

$\left(\begin{array}[]{cc}A&-X\\\ -X^{*}&B\\\
\end{array}\right)=\left(\begin{array}[]{cc}-I&0\\\ 0&I\\\
\end{array}\right)\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\
\end{array}\right)\left(\begin{array}[]{cc}-I&0\\\ 0&I\\\
\end{array}\right)\geq 0$

and

$\left(\begin{array}[]{cc}B&X^{*}\\\ X&A\\\
\end{array}\right)=\left(\begin{array}[]{cc}0&I\\\ I&0\\\
\end{array}\right)\left(\begin{array}[]{cc}A&X^{*}\\\ X&B\\\
\end{array}\right)\left(\begin{array}[]{cc}0&I\\\ I&0\\\
\end{array}\right)\geq 0.$

For the second part, notice that

$\frac{1}{2}H-\left(\begin{array}[]{cc}0&X\\\ X^{*}&0\\\
\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{cc}A&-X\\\ -X^{*}&B\\\
\end{array}\right)\geq 0.$

∎

######  Proposition 2.2.

If $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT,
then the following blocks are positive semidefinite.
$\left(\begin{array}[]{cc}A&\mp X\\\ \mp X^{*}&B\\\
\end{array}\right),\left(\begin{array}[]{cc}A&\mp X^{*}\\\ \mp X&B\\\
\end{array}\right),\left(\begin{array}[]{cc}B&\mp X^{*}\\\ \mp X&A\\\
\end{array}\right),\left(\begin{array}[]{cc}B&\mp X\\\ \mp X^{*}&A\\\
\end{array}\right).$

###### Proof.

The semi positivity of the first two blocks follows from the definition of PPT
and the first part of Proposition 2.1. The semi positivity of the second two
blocks results from conjugating the first two blocks by the unitary
$\left(\begin{array}[]{cc}0&I\\\ I&0\\\ \end{array}\right).$ ∎

######  Proposition 2.3.

Let $H=\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ be PPT.
Then

$\left(\begin{array}[]{cc}A&e^{i\theta}X\\\ e^{-i\theta}X^{*}&B\\\
\end{array}\right)\quad\text{and}\quad\left(\begin{array}[]{cc}\frac{A+B}{2}&X\\\
X^{*}&\frac{A+B}{2}\\\ \end{array}\right)\quad\text{are PPT.}\quad$

###### Proof.

Let $W=\left(\begin{array}[]{cc}e^{i\theta}I&0\\\ 0&I\\\ \end{array}\right).$
Notice that

$\left(\begin{array}[]{cc}A&e^{i\theta}X\\\ e^{-i\theta}X^{*}&B\\\
\end{array}\right)=W\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\
\end{array}\right)W^{*}\geq 0,$

and

$\left(\begin{array}[]{cc}A&e^{-i\theta}X^{*}\\\ e^{i\theta}X&B\\\
\end{array}\right)=W^{*}\left(\begin{array}[]{cc}A&X^{*}\\\ X&B\\\
\end{array}\right)W\geq 0.$

This implies that the first block is PPT.

Since $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT,
Proposition 2.2 implies that $\left(\begin{array}[]{cc}B&-X\\\ -X^{*}&A\\\
\end{array}\right)\geq 0.$ Therefore,

$H=\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\
\end{array}\right)\leq\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\
\end{array}\right)+\left(\begin{array}[]{cc}B&-X\\\ -X^{*}&A\\\
\end{array}\right)=\left(\begin{array}[]{cc}A+B&0\\\ 0&A+B\\\
\end{array}\right),$

and so

$\frac{1}{2}H\leq\left(\begin{array}[]{cc}\frac{A+B}{2}&0\\\
0&\frac{A+B}{2}\\\ \end{array}\right).$ (9)

The second part of Proposition 2.1 implies that

$\left(\begin{array}[]{cc}0&X\\\ X^{*}&0\\\
\end{array}\right)\leq\frac{1}{2}H.$ (10)

Hence, combining (9) and (10) gives

$\left(\begin{array}[]{cc}0&X\\\ X^{*}&0\\\
\end{array}\right)\leq\left(\begin{array}[]{cc}\frac{A+B}{2}&0\\\
0&\frac{A+B}{2}\\\ \end{array}\right).$

Consequently, $\left(\begin{array}[]{cc}\frac{A+B}{2}&-X\\\
-X^{*}&\frac{A+B}{2}\\\ \end{array}\right)\geq 0,$ and then, by Proposition
2.1, we have $\left(\begin{array}[]{cc}\frac{A+B}{2}&X\\\
X^{*}&\frac{A+B}{2}\\\ \end{array}\right)\geq 0.$ A similar argument implies
that $\left(\begin{array}[]{cc}\frac{A+B}{2}&X^{*}\\\ X&\frac{A+B}{2}\\\
\end{array}\right)\geq 0.$ This proves that
$\left(\begin{array}[]{cc}\frac{A+B}{2}&X\\\ X^{*}&\frac{A+B}{2}\\\
\end{array}\right)$ is PPT. ∎

In the following paragraph, we present the definition of the geometric mean of
two positive matrices and then we state some of its properties.

For positive definite $X,Y\in\mathbb{M}_{n}$, the geometric mean of $X$ and
$Y$ is defined as follows

$X\\#Y=X^{1/2}(X^{-1/2}YX^{-1/2})^{1/2}X^{1/2}.$

It is well-known that

$X\\#Y=Y\\#X\leq\frac{X+Y}{2}.$ (11)

See [7, Chapter 4].

An extremal property of the geometric mean of positive definite
$X,Y\in\mathbb{M}_{n}$ is given as follows

$X\\#Y=\max\left\\{Z:Z=Z^{*},\left(\begin{array}[]{cc}X&Z\\\ Z&Y\\\
\end{array}\right)\geq 0\right\\}.$ (12)

See [7, Theorem 4.1.3]. We have for $k=1,2,...,n$

$\prod_{j=1}^{k}s_{j}(X\\#Y)\leq\prod_{j=1}^{k}s_{j}(Y^{1/4}X^{1/2}Y^{1/4})\leq\prod_{j=1}^{k}s_{j}(X^{1/2}Y^{1/2}).$
(13)

Moreover, for every unitarily invariant norm

$||X\\#Y||\leq||X^{1/2}Y^{1/2}||\leq||\frac{1}{2}(X+Y)||.$ (14)

See [6].

##  3\. Main result

We start this section by the following two lemmas. The proof of the first
lemma can be found in [4]. The second lemma results from the first lemma and
Proposition 2.2. See also [16, Lemma 4.2].

######  Lemma 3.1.

If $\left(\begin{array}[]{cc}A_{j}&X\\\ X^{*}&B_{j}\\\ \end{array}\right)\geq
0\quad(j=1,2),$ then $\left(\begin{array}[]{cc}A_{1}\\#A_{2}&X\\\
X^{*}&B_{1}\\#B_{2}\\\ \end{array}\right)\geq 0.$

######  Lemma 3.2.

If $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT,
then $\left(\begin{array}[]{cc}A\\#B&X\\\ X^{*}&A\\#B\\\ \end{array}\right)$
is PPT.

We recall that the absolute value of $X\in\mathbb{M}_{n}$ is defined as
$|X|=(X^{*}X)^{1/2}$.

The main result can be stated as follows.

######  Theorem 3.1.

Let $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ be PPT and
let $X=U|X|$ be the polar decomposition of $X$. Then

$|X|\leq\left(A\\#B\right)\\#U^{*}(A\\#B)U.$

###### Proof.

Let $X=U|X|$ be the polar decomposition of $X$. Let $W$ be the unitary defined
as $W=\left(\begin{array}[]{cc}U&0\\\ 0&I\\\ \end{array}\right).$ Since
$\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT, Lemma
3.2 implies that $\left(\begin{array}[]{cc}A\\#B&X\\\ X^{*}&A\\#B\\\
\end{array}\right)\geq 0.$ Therefore,

$W^{*}\left(\begin{array}[]{cc}A\\#B&X\\\ X^{*}&A\\#B\\\
\end{array}\right)W=\left(\begin{array}[]{cc}U^{*}(A\\#B)U&|X|\\\ |X|&A\\#B\\\
\end{array}\right)\geq 0.$

By the extremal property of the geometric mean (12) we get

$|X|\leq\left(A\\#B\right)\\#U^{*}(A\\#B)U.$

This proves the first inequality. The second inequality follows directly from
(11). ∎

The following result, which can found in [15], follows directly from Theorem
3.1.

######  Corollary 3.1.

Let $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ be PPT and
let $X=U|X|$ be the polar decomposition of $X$. Then

$|X|\leq\frac{(A\\#B)+U^{*}(A\\#B)U}{2}.$

Before we proceed to the next result, we need to recall some facts about Lieb
functions.

A continuous function $\mathrm{L}:\mathbb{M}_{n}\longrightarrow\mathbb{C}$ is
called a Lieb function if it satisfies the following two properties:

  1. (1)

$0\leq\mathrm{L}(X)\leq\mathrm{L}(Y)$ whenever $0\leq X\leq Y$.

  2. (2)

$|\mathrm{L}(X^{*}Y)|^{2}\leq\mathrm{L}(X^{*}X)\mathrm{L}(Y^{*}Y)$ for all
$X,Y$.

It is well known that if $\mathrm{L}$ is a Lieb function, then for all
$X,Y,Z\in\mathbb{M}_{n}$

$|\mathrm{L}(Z)|^{2}\leq\mathrm{L}(X)\mathrm{L}(Y)\quad\text{whenever}\quad\left(\begin{array}[]{cc}X&Z^{*}\\\
Z&Y\\\ \end{array}\right)\geq 0.$ (15)

See [8, p. 270]. Moreover, for two positive matrices $X,Y\in\mathbb{M}_{n}$,
the extremal property in (12) implies $\left(\begin{array}[]{cc}X&X\\#Y\\\
X\\#Y&Y\\\ \end{array}\right)\geq 0.$ Hence (15) gives

$L(X\\#Y)\leq\sqrt{L(X)L(Y)}.$ (16)

Examples of Lieb functions are trace, determinant, permanent, spectral radius,
unitarily invariant norms, and any elementary symmetric function of the
eigenvalues. For each $1\leq k\leq n$, the function
$f_{k}(X)=\prod_{j=1}^{k}s_{j}(X),X\in\mathbb{M}_{n}$ is a Lieb function.
Moreover, $g_{k}(X)=\prod_{j=1}^{k}\lambda_{j}(X),X\in\mathbb{M}_{n}$ is also
a Lieb function provided that the eigenvalues of $X\in\mathbb{M}_{n}$ are
arranged so that
$|\lambda_{1}(X)|\geq|\lambda_{2}(X)|\geq...\geq|\lambda_{n}(X)|$.

A lieb function $L$ is said to be unitarily invariant if $L(UXV)=L(X)$ for all
$X\in\mathbb{M}_{n}$ and all unitaries $U,V\in\mathbb{M}_{n}$.

Theorem 3.1 implies the following corollary.

######  Corollary 3.2.

Let $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ be PPT.
Let $L$ be a unitarily invariant Lieb function. Then

$L(|X|)\leq L\left(A\\#B\right).$

In what follows we present three corollaries which are consequences of Theorem
3.1. These corollaries can be found in literature, see for example [16].

######  Corollary 3.3.

If $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT,
then for every unitarily invariant norm $||\cdot||$

$\displaystyle||X||$ $\displaystyle\leq||A\\#B||$
$\displaystyle\leq||A^{1/2}B^{1/2}||$ $\displaystyle\leq\frac{1}{2}||A+B||.$

###### Proof.

The first inequality results from applying Corollary 3.2 to the Lieb function
$L(\cdot)=||\cdot||$. The other inequalities follow from (14). ∎

If we square the inequalities in Corollary 3.3 and choose the Hilbert-Schmidt
norm, $||\cdot||_{2}$, we get the following result. It is an improvement of
the trace inequality (4). Recall that the Hilbert-Schmidt norm is defined as
$||X||_{2}^{2}=tr(X^{*}X)$. See also Corollary 2.6 in [15] for alternative
proof.

######  Corollary 3.4.

If $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT,
then

$\displaystyle tr(X^{*}X)$ $\displaystyle\leq tr(A\\#B)^{2}$
$\displaystyle\leq tr(AB).$ $\displaystyle\leq\frac{1}{2}tr(A+B)^{2}.$

######  Corollary 3.5.

If $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT,
then for $k=1,2,...,n$

$\prod_{j=1}^{k}|s_{j}(X)|\leq\prod_{j=1}^{k}s_{j}(A\\#B)\leq\prod_{j=1}^{k}s_{j}\left(A^{1/2}B^{1/2}\right)$
(17)

###### Proof.

The first inequality results from applying Corollary 3.2 to the Lieb function
$L(\cdot)=\prod_{j=1}^{k}s_{j}(\cdot)$. The other inequalities follow from
(13). ∎

We remark that using [6, Theorem 3] and [6, (17)] we can have stronger
versions of the inequalities in Corollary 3.3, Corollary 3.4 and Corollary
3.5.

In the following theorem, we present a slightly weaker version of the singular
value inequalities in (6). These inequalities, in fact, have been presented
implicitly in [15, Corollary 2.5] as a consequence of the inequality (7). To
make this note self-contained, we present a different and direct proof.

######  Theorem 3.2.

Let $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ be PPT.
Then

$s_{j}(X)\leq s_{[\frac{j+1}{2}]}\left(A\\#B\right)\leq
s_{[\frac{j+1}{2}]}\left(\frac{A+B}{2}\right),\quad j=1,2,...,n,$

where $[x]$ is the greatest integer $\leq x$.

###### Proof.

Since $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ be PPT,
Proposition 2.3 implies that $\left(\begin{array}[]{cc}\frac{A+B}{2}&X\\\
X^{*}&\frac{A+B}{2}\\\ \end{array}\right)\geq 0,$ and then, by Proposition
2.2, we have $\left(\begin{array}[]{cc}\frac{A+B}{2}&-X\\\
-X^{*}&\frac{A+B}{2}\\\ \end{array}\right)\geq 0.$ This is equivalent to

$\left(\begin{array}[]{cc}0&X\\\ X^{*}&0\\\
\end{array}\right)\leq\left(\begin{array}[]{cc}\frac{A+B}{2}&0\\\
0&\frac{A+B}{2}\\\ \end{array}\right).$

Therefore, by Weyl’s monotonicity principle, we have

$\lambda_{j}\left(\begin{array}[]{cc}0&X\\\ X^{*}&0\\\
\end{array}\right)\leq\lambda_{j}\left(\begin{array}[]{cc}\frac{A+B}{2}&0\\\
0&\frac{A+B}{2}\\\ \end{array}\right),\quad\text{for $j=1,2,,...,2n$}.$

Note that, the eigenvalues of $\left(\begin{array}[]{cc}0&X\\\ X^{*}&0\\\
\end{array}\right)$ are

$s_{1}(X)\geq s_{2}(X)\geq...\geq s_{n}(X)\geq 0\geq-s_{n}(X)\geq-
s_{n-1}(X)\geq...\geq-s_{1}(X),$

and the eigenvalues of $\left(\begin{array}[]{cc}\frac{A+B}{2}&0\\\
0&\frac{A+B}{2}\\\ \end{array}\right)$ are

$s_{1}(X)\geq s_{1}(X)\geq s_{2}(X)\geq s_{2}(X)\geq...\geq s_{n}(X)\geq
s_{n}(X).$

Therefore, we have shown that if $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\
\end{array}\right)$ is PPT, then

$s_{j}(X)\leq s_{[\frac{j+1}{2}]}\left(\frac{A+B}{2}\right),\quad
j=1,2,...,n.$ (18)

Since $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ is PPT,
Lemma 3.2 implies that $G=\left(\begin{array}[]{cc}A\\#B&X\\\ X^{*}&A\\#B\\\
\end{array}\right)$ is PPT. Applying (18) to $G$ implies the first inequality
of the theorem. The second inequality of the theorem follows directly by (11).
∎

Finally, we study the connection between the diagonal components and the real
part of the off-diagonal components of the PPT block H. Before doing so, we
recall that every $X\in\mathbb{M}_{n}$ admits what is called the cartesian
decomposition

$X=Re(X)+iIm(X),$

where $Re(X)$ and $Im(X)$ are the Hermitian matrices defined as
$Re(X)=\frac{X+X^{*}}{2},Im(X)=\frac{X-X^{*}}{2i}$ and are known,
respectively, as the real and the imaginary parts of $X$.

######  Theorem 3.3.

Let $\left(\begin{array}[]{cc}A&X\\\ X^{*}&B\\\ \end{array}\right)$ be PPT.
Then

$\mp Re(X)\leq A\\#B\quad\text{and}\quad\mp Im(X)\leq A\\#B.$

###### Proof.

By Lemma 3.2, we have

$\left(\begin{array}[]{cc}A\\#B&X\\\ X^{*}&A\\#B\\\ \end{array}\right)\geq
0,\quad\text{and}\quad\left(\begin{array}[]{cc}A\\#B&X^{*}\\\ X&A\\#B\\\
\end{array}\right)\geq 0.$

Therefore,

$\left(\begin{array}[]{cc}A\\#B&Re(X)\\\ Re(X)&A\\#B\\\
\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{cc}A\\#B&X\\\
X^{*}&A\\#B\\\
\end{array}\right)+\frac{1}{2}\left(\begin{array}[]{cc}A\\#B&X^{*}\\\
X&A\\#B\\\ \end{array}\right)\geq 0.$

Let $J=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}I&-I\\\ I&I\\\
\end{array}\right).$ Then

$\left(\begin{array}[]{cc}A\\#B+Re(X)&0\\\ 0&A\\#B-Re(X)\\\
\end{array}\right)=J^{*}\left(\begin{array}[]{cc}A\\#B&Re(X)\\\ Re(X)&A\\#B\\\
\end{array}\right)J\geq 0.$ (19)

This implies the first inequality. To prove the second inequality just
applying the first inequality to the block
$G=\left(\begin{array}[]{cc}A&-iX\\\ iX^{*}&B\\\ \end{array}\right).$ Note
that $G$ is PPT by Proposition 2.3. ∎

######  Corollary 3.6.

Let $\left(\begin{array}[]{cc}A&X\\\ X&B\\\ \end{array}\right)\geq 0.$ If $X$
is Hermitian, then

$\mp X\leq A\\#B.$

## References

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# On the total energy conservation of the Alcubierre spacetime

F. L. Carneiro<EMAIL_ADDRESS>Instituto de Física, Universidade de
Brasília, 70910-900, Brasília, DF, Brazil  S. C. Ulhoa<EMAIL_ADDRESS>Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF, Brazil
International Center of Physics, Instituto de Física, Universidade de
Brasília, 70.910-900, Brasilia, DF, Brazil  Canadian Quantum Research Center,
204-3002 32 Ave Vernon, BC V1T 2L7 Canada  J. W. Maluf<EMAIL_ADDRESS>Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF, Brazil
J. F. da Rocha-Neto<EMAIL_ADDRESS>Instituto de Física, Universidade de
Brasília, 70910-900, Brasília, DF, Brazil

###### Abstract

In this article, we consider the Alcubierre spacetime, such a spacetime
describes a “bubble” that propagates with arbitrary global velocity. This
setting allows movement at a speed greater than that of light. There are some
known problems with this metric, e.g., the source’s negative energy and the
violation of the source’s energy conservation when the bubble accelerates. We
address these two issues within the realm of the Teleparallel Equivalent of
General Relativity (TEGR). The energy conservation problem can be solved when
considering the energy of the gravitational field itself. The total energy of
the spacetime, gravitational plus source, is conserved even in accelerated
motion. We explicitly show the dependence of energy and gravitational energy
flux on the frame of reference, one adapted to a static observer and the other
to a free-falling one in the same coordinate system. Addressing the problem of
energy negativity of the source, we find that a static observer measures
positive energy of the source, while an Eulerian observer measures a negative
one. Thus, we surmise that negative energy may be a reference problem.

##  I Introduction

Alcubierre spacetime was first introduced by Alcubierre in 1994
alcubierre1994warp  by what is now called “metric engineering”
white2003discussion . Some solutions to Einstein equations were obtained by
establishing symmetries and then solving the equations themselves, including
the first known solution, that of Schwarzschild. Alcubierre spacetime was not
obtained by integration of a energy-momentum tensor of a source but was
constructed by geometric considerations and requirements. Alcubierre’s main
objective in establishing such a solution was to obtain a geometry that would
allow a spacecraft to travel at arbitrarily high speeds. It should be noted
that special relativity prohibits massive objects from reaching the speed of
light thus from this point of view they cannot exceed this speed. Within the
realm of General Relativity (GR) there is also a similar prohibition, that is,
local travelers cannot be faster than the speed of light. Thus the Alcubierre
metric seeks to describe theoretically how travelers can have arbitrary global
speeds. Although there are certain features considered undesirable in this
spacetime, such as the violation of energy conditions and the presence of so-
called closed timelike curves (CTCs), it determines a class of valid solutions
to Einstein equations. As we will see, some of these features can be
reinterpreted to increase the scope of what would be physically acceptable
solutions. Particularly the violation of energy conservation can be avoided
when taking into account gravitational energy.

The motion of test particles in Alcubierre spacetime is attained by
constructing a non-trivial geometric configuration in the form of a bubble,
where inside the bubble the spacetime is conformally flat and outside is
Minkowski spacetime. The bubble may propagate at any global velocity
$\displaystyle v_{s}$ and the proper time within the bubble is the same as the
proper time of an observer at spatial infinity (for a definition see
ashtekar1992spatial ), e.g., a static observer on Earth. This important
feature allows an observer to travel to a distant star and return without
problems related to time dilation. Albeit time dilatation is not the only
physiological problem that the traveler inside the bubble must deal with. The
bubble’s interaction with external particles results in their acceleration and
radiation emission mcmonigal2012alcubierre . Alcubierre spacetime,
geodesically, consists of a congruence expansion converging in front (between
the bubble and the destination) and diverging in the rear (between the bubble
and the origin), but the divergent congruence, or “geodesic expansion”, is not
a necessary feature natario2002warp .

Ever since the first appearance of Alcubierre spacetime, many authors reviewed
this topic, including Alcubierre himself alcubierre2017warp . Clark and
collaborators studied the effect of the spacetime on null geodesics
clark1999null . McMonigal and collaborators treated the interaction of the
bubble with massive and null particles mcmonigal2012alcubierre . Addressing
some causality problems in the Alcubierre spacetime, Krasnikov proposed
modifications to the original spacetime, describing what is now called
“Krasnikov tube” krasnikov1998hyperfast ; everett1997superluminal ;
gravel2004simple . The source of the gravitational field was recently
addressed by Santos-Pereira and collaborators who considered dust as the
source of the gravitational field, obtaining a zero energy density in case of
non-violation of the strong energy condition. Interestingly, in the latter
case, an equation was obtained and that resembles the Burgers equation, which
describes shock waves moving in a fluid. Latter, a perfect fluid and a charged
dust were also considered as a source santos2021fluid ; santos2021charged .

There are many known physical limitations of the Alcubierre spacetime
lobo2004fundamental ; everett1996warp , with several debates regarding the
violation of energy conditions lobo2003weak . Recently Bobrick and Martire
presented an algorithm for constructing physical “warp drives” that avoids
some of the problems present in Alcubierre spacetime bobrick2021introducing .
It was noted in this former paper that the Alcubierre spacetime has a problem
in real situations. A real spaceship is not expected to always have a constant
global velocity $\displaystyle v_{s}$, it is necessary for the ship to
accelerate and decelerate, which implies that $\displaystyle v_{s}=v_{s}(t)$.
When there is a temporal dependence of velocity, the $\displaystyle 00$
component of the source’s energy-momentum tensor also varies with
$\displaystyle t$, so it was argued that an accelerating spacecraft violates
the energy conservation.

The main objective of the present paper is to address the energy conservation
of the original Alcubierre spacetime. Although the energy of the source of
Alcubierre spacetime varies with time in an accelerated motion, we find that
the gravitational energy also varies in the same way, as a consequence the
energy of the source and the gravitational energy exactly cancel each other at
long distances from the bubble. Thus, the total energy of the spacetime is
conserved even in accelerated motion. We also obtain positive energy for the
source measured by a static observer and negative energy measured by a free-
falling one. Therefore, we point out the possibility that the negative energy
of the source in Alcubierre spacetime might be a problem of the frame. We also
investigate the gravitational energy flux measured by static and free-falling
observers, in the same coordinate system, and find a non-zero flux measured by
both observers.

In order to evaluate the gravitational energy and gravitational energy flux of
the spacetime, we use the Teleparallel Equivalent of General Relativity (TEGR)
which, although it has the same dynamics as GR, allows the definition of an
adequate energy-momentum four-vector. In section II we review some aspects of
the description of gravity in terms of the TEGR. In section III, we briefly
review Alcubierre spacetime and some of its properties, while establishing the
coordinates that are used in this article. In section IV, we introduce a set
of tetrads, associated with the Alcubierre metric and adapted to a static
observer, and evaluate the $\displaystyle(0)$ component for the energy-
momentum four-vector measured by this observer. In section V, we apply a
Lorentz transformation to the tetrad of the static observer and evaluate the
$\displaystyle(0)$ component for the energy-momentum four-vector measured by
the free-falling observer and the gravitational energy flux. In section VI we
address the case of variable velocity. Finally, in section VII we present our
conclusions regarding the obtained results and conjecture the implication of
the results.

In this paper we have established the following notation. The Latin letters at
begging of the alphabet $\displaystyle a$, $\displaystyle b$,
$\displaystyle\ldots$ denote $\displaystyle SO(3,1)$ indices and local
coordinates are indicated by $\displaystyle(0)$, $\displaystyle(1)$,
$\displaystyle\ldots$. The Greek letters $\displaystyle\mu$,
$\displaystyle\nu$, $\displaystyle\ldots$ denote the spacetime indices and
spacetime coordinates are indicated by $\displaystyle 0$, $\displaystyle 1$,
$\displaystyle\ldots$. Both run from $\displaystyle 0$ to $\displaystyle 3$.
Time is indicated by $\displaystyle(0)$ and $\displaystyle 0$ and space by
$\displaystyle(i)$ and $\displaystyle i$, where $\displaystyle i=1,2,3$. The
flat Minkowski tensor $\displaystyle\eta_{ab}$ raises and lowers
$\displaystyle SO(3,1)$ indices; the metric tensor $\displaystyle g_{\mu\nu}$
raises and lowers the spacetime indices. The geometrized unit system is used,
i.e., $\displaystyle G=c=1$.

##  II Teleparallel Equivalent of General Relativity (TEGR)

The TEGR is a geometric description of gravity that yields field equations
equivalent to GR. In GR, physical quantities are established in terms of
pseudo-tensors landau2013course , these are coordinate dependent. It is well
known that there are difficulties in trying to establish a true energy-
momentum tensor for the gravitational field in GR. The main objective of this
paper is to analyze the conservation of the total energy, gravitational plus
source (matter-radiation fields), of Alcubierre spacetime. In order to obtain
consistent results, we need a properly established definition of gravitational
energy that does not depend on coordinates. The TEGR, which uses a tetrad
description of gravity, allows the definition of a proper energy-momentum
tensor for the gravitational field and, as a consequence, an energy-momentum
four-vector. Thus, in this section, we briefly review some aspects of the TEGR
that are used in this paper and present the expressions for energy due to
gravitational field plus matter and the gravitational energy flux. A complete
review of TEGR can be found in Ref. maluf2013teleparallel .

Let us consider the spacetime as an orientable manifold in which, at any event
$\displaystyle x^{\mu}$, we may establish a local frame in the tangent
spacetime parametrized by $\displaystyle x^{a}$. The projection of spacetime
quantities may be achieved through a tetrad field $\displaystyle
e^{a}\,_{\mu}$, e.g., for a vector $\displaystyle V^{\mu}$ we have
$\displaystyle V^{a}=e^{a}\,_{\mu}V^{\mu}$. Whereas in the absence of gravity
the spacetime and the tangent space coincide and the quantities $\displaystyle
dx^{a}$ are holonomic, in the presence of gravity, and non-trivial geometries,
the quantities $\displaystyle dx^{a}$ are non-holonomic. When considering the
world line of an observer $\displaystyle x^{\mu}(\tau)$, where
$\displaystyle\tau$ is the proper time, the components $\displaystyle
e_{(0)}\,^{\mu}$ are always tangent to the observer’s worldline. Thus we
identify $\displaystyle e_{(0)}\,^{\mu}$ with the observer’s four-velocity
$\displaystyle u^{\mu}$ in its rest frame.

In the TEGR, the gravitational fundamental variables are the tetrads
$\displaystyle e^{a}\,_{\mu}$. The metric tensor can be obtained from the
tetrads as $\displaystyle g_{\mu\nu}=e^{a}\,_{\mu}e_{a\nu}$. While the metric
tensor has ten independent components, the tetrads have sixteen. Thus, any
quantity that is a function of the metric is also a function of the tetrads,
but the inverse is not true. By choosing the six additional components, a
reference system is established. The theory is constructed on the geometry of
absolute parallelism. Therefore,

$\nabla_{\mu}e^{a}\,_{\nu}=\partial_{\mu}e^{a}\,_{\nu}-\Gamma^{\lambda}\,_{\mu\nu}e^{a}\,_{\lambda}=0\,.$
(1)

The connection $\displaystyle\Gamma^{\lambda}\,_{\mu\nu}$ guarantees the
parallelism and is not symmetric in the permutation of the last two indices.
Isolating $\displaystyle\Gamma^{\lambda}\,_{\mu\nu}$ in Eq. (1), we obtain

$\Gamma^{\lambda}\,_{\mu\nu}=e_{a}\,^{\lambda}\partial_{\mu}e^{a}\,_{\nu}\,.$
(2)

The above connection is the Weitzenböck connection and yields the non-null
torsion tensor

$T^{\lambda}\,_{\mu\nu}=\Gamma^{\lambda}\,_{\mu\nu}-\Gamma^{\lambda}\,_{\nu\mu}\,.$
(3)

Using Eqs. (2) and (3), we arrive at

$T^{a}\,_{\mu\nu}=\partial_{\mu}e^{a}\,_{\nu}-\partial_{\nu}e^{a}\,_{\mu}\,.$
(4)

In TEGR, the frame is characterized from its acceleration $\displaystyle
a^{(i)}$. Assuming the observer carries orthonormal tetrads $\displaystyle
e^{a}\,_{\mu}$ along his world line $\displaystyle x^{\mu}(\tau)$ we may
evaluate the acceleration of the frame as mashhoon2002length ;
mashhoon2003vacuum

$a^{\mu}=u^{\alpha}\nabla_{\alpha}u^{\mu}=u^{\alpha}\nabla_{\alpha}e_{(0)}\,^{\mu}=\frac{d^{2}x^{\mu}}{d\tau^{2}}+\mathring{\Gamma}^{\mu}\,_{\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}\,,$
(5)

where $\displaystyle\mathring{\Gamma}^{\mu}\,_{\alpha\beta}$ are the
Christoffel symbols. It can be shown that maluf2013teleparallel

$a^{(i)}=e^{(i)}\,_{\mu}a^{\mu}=\eta^{(i)b}\phi_{(0)b}=\frac{1}{2}T_{(0)(0)}\,^{(i)}\,,$
(6)

where
$\displaystyle\phi_{a}\,^{b}=e^{b}\,_{\mu}u^{\lambda}\nabla_{\lambda}e_{a}\,^{\mu}$.
When the frame is in free fall, $\displaystyle u^{\mu}$ represents a geodesic
and $\displaystyle a^{\mu}=0$. Thus, we may identify
$\displaystyle\phi_{ab}\leftrightarrow(\bf{a},\bf{\Omega})$, where
$\displaystyle\bf{a}$ is the translational acceleration and
$\displaystyle\bf{\Omega}$ the rotational frequency of the local spatial frame
relative to a non-rotating transported frame mashhoon2002length ;
mashhoon2003vacuum . The establishment of the static nature of the observer is
made at space infinity, for asymptotically flat spacetimes, by requiring that

$\lim_{r\rightarrow\infty}e^{a}\,_{\mu}=\delta^{a}_{\mu}\,,$ (7)

where $\displaystyle r$ represents the distance from the observer to the
source of the gravitational field.

The Lagrangian formulation of the theory is constructed from the Lagrangian
density maluf2013teleparallel

$\mathcal{L}=-ke\Sigma^{abc}T_{abc}-\mathcal{L}_{M}\,,$ (8)

where $\displaystyle k=1/16\pi$, $\displaystyle\mathcal{L}_{M}$ is the
Lagrangian density of the matter-radiation fields and

$\Sigma^{abc}\equiv\frac{1}{4}\left(T^{abc}+T^{bac}-T^{cab}\right)+\frac{1}{2}\left(\eta^{ac}T^{b}-\eta^{ab}T^{c}\right)\,.$
(9)

The field equations derived from (8) are maluf2013teleparallel

$e_{a\lambda}e_{b\mu}\partial_{\nu}\left(e\Sigma^{b\lambda\nu}\right)-e\left(\Sigma^{b\nu}\,_{a}T_{b\nu\mu}-\frac{1}{4}e_{a\mu}T_{bcd}\Sigma^{bcd}\right)=\frac{1}{4k}eT_{a\mu}\,,$
(10)

where $\displaystyle T_{\mu\nu}=e^{a}\,_{\mu}T_{a\nu}$ is the mater-radiation
energy-momentum tensor. The field equations (10) are co-variant under
coordinate transformations, local and global Lorentz transformations, and can
be written as $\displaystyle
R_{a\mu}(e)-\frac{1}{2}e_{a\mu}R(e)=\frac{1}{2k}T_{a\mu}$
maluf2013teleparallel , where $\displaystyle R_{a\mu}$ and $\displaystyle R$
are, respectively, the projected Ricci tensor and Ricci scalar of the Riemann
space constructed from the Weitzenböck torsion.

In the Hamiltonian formulation of TEGR, the gravitation energy-momentum is
well established and can be formally obtained (see Ref. maluf2013teleparallel
for a complete discussion). However, the same result can be easily obtained
directly from the Lagrangian formulation of the theory and suffices for the
purpose of this paper. The field equations (10) can be written as

$\partial_{\nu}\left(e\Sigma^{a\lambda\nu}\right)=\frac{1}{4k}ee^{a}\,_{\mu}\left(t^{\lambda\mu}+T^{\lambda\mu}\right)\,,$
(11)

where

$t^{\lambda\nu}=k\left(4\Sigma^{bc\lambda}T_{bc}\,^{\mu}-g^{\lambda\mu}\Sigma^{bcd}T_{bcd}\right)\,.$
(12)

Using the antisymmetry of $\displaystyle\Sigma^{a\lambda\nu}$, we obtain from
Eq. (11)

$\partial_{\lambda}\left[ee^{a}\,_{\mu}\left(t^{\lambda\mu}+T^{\lambda\mu}\right)\right]=0\,.$
(13)

Integrating (13), we obtain the continuity equation

$\frac{d}{dx^{0}}\int_{V}{d^{3}x\left[ee^{a}\,_{\mu}\left(t^{0\mu}+T^{0\mu}\right)\right]}=-\oint_{S}{dS_{i}\left[ee^{a}\,_{\mu}\left(t^{i\mu}+T^{i\mu}\right)\right]}\,.$
(14)

Thus

$P^{a}\equiv\int_{V}{d^{3}xee^{a}\,_{\mu}\left(t^{0\mu}+T^{0\mu}\right)}\,,$
(15)

represents a conserved quantity. Identifying $\displaystyle t^{\mu\nu}$ as the
gravitational energy-momentum tensor in Eq. (11), we obtain

$P^{a}=4k\int_{V}{d^{3}x\,\partial_{i}(e\Sigma^{a0i}})=4k\oint_{S}dS_{i}(e\Sigma^{a0i})\,,$
(16)

where $\displaystyle S$ is the closed surface encompassing V. The quantity in
the above equation represent the total energy-momentum four-vector contained
in the volume V. Thus from Eq. (15), we identify

$P_{g}^{a}=\int_{V}{d^{3}x\,e(e^{a}\,_{\mu}t^{0\mu}})$ (17)

as the gravitational energy-momentum four-vector contained in the volume
$\displaystyle V$, and

$P_{m}^{a}=\int_{V}{d^{3}x\,(ee^{a}\,_{\mu}T^{0\mu}})$ (18)

as the energy-momentum four-vector of the source contained in the volume
$\displaystyle V$. From the spatial components of Eq. (14), we define the
gravitation energy-momentum flux as

$\Phi^{a}_{g}=\oint_{S}dS_{i}\,(ee^{a}\,_{\mu}t^{i\mu})\,.$ (19)

The four-vector $\displaystyle P^{a}$ is invariant under coordinate and global
Lorentz transformations and transforms as a four-vector under $\displaystyle
SO(3,1)$ symmetry. The zero component $\displaystyle P^{(0)}$ represents the
energy as in Special Relativity and is frame dependent. As mentioned before,
Eq. (16) is obtained from the vacuum equations of the Hamiltonian formulation
of the TEGR maluf2013teleparallel ; and the same expression is obtained
through the Noether’s charges emtsova2021conserved .

In sections IV and V we evaluate the quantities (16-19) for two distinct
observers.

##  III The Alcubierre spacetime

Alcubierre spacetime is a non-trivial geometry formed by a shell, where the
spacetime inside the shell is conformally flat and the exterior is Minkowski
spacetime. The shell is denoted as a “bubble”. The spacetime is asymptotically
flat mcmonigal2012alcubierre , therefore regularly represented by the
coordinates $\displaystyle x^{\mu}=(t,x,y,z)$ at space infinity. The bubble
propagates with a global velocity $\displaystyle v_{s}$, which can be greater
than unity in geometrized units, in the positive $\displaystyle z$ direction.
The coordinates $\displaystyle x^{\mu}$ are adapted to a static observer that
measures the velocity $\displaystyle v_{s}$ and the position $\displaystyle
r_{s}$ for the bubble, where $\displaystyle v_{s}=dr_{s}/dt$. Thus, the
quantity $\displaystyle r_{s}$ localize the bubble’s center in relation to a
static observer. In the $\displaystyle x^{\mu}$ coordinate system the line
element may be written as alcubierre2017warp

$ds^{2}=\big{(}f_{s}^{2}v_{s}^{2}-1\big{)}\,dt^{2}+dx^{2}+dy^{2}+dz^{2}-2f_{s}v_{s}dtdz\,,$
(20)

where $\displaystyle r_{s}=\sqrt{(z-v_{s}t)^{2}+x^{2}+y^{2}}$ and
$\displaystyle f_{s}=f_{s}(r_{s})$. The function $\displaystyle f_{s}$ is the
non-trivial geometric term and it may be, arbitrarily, chosen as long as some
conditions are satisfied, i.e., $\displaystyle f_{s}=constant$ inside the
bubble and $\displaystyle f_{s}=0$ outside of the bubble. The spacetime are of
Type $\displaystyle\mathcal{I}$ in Petrov classification and there is no
curvature singularities mattingly2021curvature . For the sake of simplicity,
we shall introduce polar coordinates into the two-dimensional space transverse
to the propagation direction. Thus, the line element (20) reads

$ds^{2}=\big{(}f_{s}^{2}v_{s}^{2}-1\big{)}\,dt^{2}+d\rho^{2}+\rho^{2}d\phi^{2}+dz^{2}-2f_{s}v_{s}dtdz\,,$
(21)

where $\displaystyle r_{s}=\sqrt{(z-v_{s}t)^{2}+\rho^{2}}$, $\displaystyle
x=\rho\cos{\phi}$ and $\displaystyle y=\rho\sin{\phi}$. In the
$\displaystyle(t,\rho,\phi,z)$ coordinates, the non-zero components
$\displaystyle G^{\mu\nu}$ of the Einstein tensor are

$\displaystyle\displaystyle G^{00}$
$\displaystyle\displaystyle=-\frac{1}{4}\rho^{2}v_{s}^{2}\left(\frac{f^{{}^{\prime}}_{s}}{r_{s}}\right)^{2}\,,$
(22) $\displaystyle\displaystyle G^{01}$
$\displaystyle\displaystyle=-\frac{1}{2}\frac{\rho
v_{s}(tv_{s}-z)}{r_{s}}\frac{d}{dr_{s}}\left(\frac{f^{{}^{\prime}}_{s}}{r_{s}}\right)\,,$
(23) $\displaystyle\displaystyle G^{03}$
$\displaystyle\displaystyle=-\frac{1}{4}v_{s}\frac{\rho^{2}v_{s}^{2}r_{s}f_{s}f^{{}^{\prime}2}_{s}+2\left(\rho^{2}+2t^{2}v_{s}^{2}-4tv_{s}z+2z^{2}\right)f^{{}^{\prime}}_{s}+2\rho^{2}r_{s}f^{{}^{\prime\prime}}_{s}}{r_{s}^{3}}\,,$
(24) $\displaystyle\displaystyle G^{11}$
$\displaystyle\displaystyle=\frac{1}{4}\frac{v_{s}^{2}}{r_{s}^{2}}\bigg{[}\frac{4\rho^{2}}{r_{s}}(1-f_{s}-)f^{{}^{\prime}}_{s}+2\rho^{2}f^{{}^{\prime}2}_{s}+\big{[}4(tv_{s}-z)^{2}-\rho^{2}\big{]}+4(tv_{s}-z)^{2}(f_{s}-1)f^{{}^{\prime\prime}}_{s}\bigg{]}\,,$
(25) $\displaystyle\displaystyle G^{13}$
$\displaystyle\displaystyle=-\frac{1}{2}\frac{\rho(tv_{s}-z)}{r^{2}_{s}}\bigg{[}(1-2f_{s})f^{{}^{\prime}}_{s}+2r_{s}f^{{}^{\prime}2}_{s}+r_{s}(2f_{s}-1)f^{{}^{\prime\prime}}_{s}\bigg{]}\,,$
(26) $\displaystyle\displaystyle G^{22}$
$\displaystyle\displaystyle=-\frac{1}{4}\frac{v_{s}}{r_{s}^{3}}\bigg{[}4\rho^{2}(f_{s}-1)f^{{}^{\prime}}_{s}+r_{s}\big{[}4(-tv_{s}+z)^{2}+\rho^{2}\big{]}f^{{}^{\prime}2}_{s}+4(-tv_{s}+z)^{2}r_{s}(f_{s}-1)f^{{}^{\prime\prime}}_{s}\bigg{]}\,,$
(27) $\displaystyle\displaystyle G^{33}$
$\displaystyle\displaystyle=-\frac{1}{4}\frac{v_{s}^{2}}{r_{s}^{3}}\bigg{[}3\rho^{2}r_{s}f^{{}^{\prime}2}_{s}+v^{2}_{s}\rho^{2}r_{s}f_{s}^{2}f_{s}^{{}^{\prime}2}+4f_{s}\big{[}(2(-tv_{s}+z)^{2}+\rho^{2})f^{{}^{\prime}}_{s}+\rho^{2}r_{s}f^{{}^{\prime\prime}}_{s}\big{]}\bigg{]}\,,$
(28)

where the prime indicates a derivation with respect to $\displaystyle r_{s}$.

A test particle inside the bubble experiences a geodesic motion, i.e., it is
dragged with the bubble to any value of $\displaystyle v_{s}$
alcubierre2017warp . The four-velocity of this particle is alcubierre2017warp

$u^{\mu}=(1,0,0,v_{s}f_{s})\,.$ (29)

The time dilatation effect on the particle can be obtained simply by
considering this particle along the propagation axis, i.e.,
$\displaystyle\rho=0$, in the line element (21) and $\displaystyle f_{s}=1$.
Therefore, in the center of the bubble $\displaystyle dz=v_{s}dt$, thus we
have $\displaystyle d\tau=dt$. Note that realistically minor dilatation
effects may occur, but can be controlled by choosing a bubble large or thin
enough.

The expansion of timelike congruence can be defined as
$\displaystyle\Theta=\partial_{\mu}u^{\mu}$ poisson2004relativist , where
$\displaystyle u^{\mu}$ is the tangent field to the congruence. Hence, using
(29), we obtain alcubierre2017warp

$\Theta=v_{s}(\partial_{z}r_{s})f^{\prime}_{s}=v_{s}\frac{z-z_{0}}{r_{s}}f^{{}^{\prime}}_{s}\,,$
(30)

where $\displaystyle z_{0}=v_{s}t$ is the center of the bubble. We can see
that in the front of the bubble ($\displaystyle z>z_{0}$) we have
$\displaystyle\Theta>0$ and in the rear ($\displaystyle z<z_{0}$) we have
$\displaystyle\Theta<0$. Hence, the Alcubierre spacetime may be interpreted as
a “expansion” of the spacetime followed by a “contraction”. The expansion was
latter shown by Natário not to be a necessary property natario2002warp  in the
construction of his “Natário drives” as now called.

Some properties required for the source of this gravitational field can be
perceived through Einstein equations. The source energy-momentum component
$\displaystyle T^{00}$ reads

$T^{00}=2kG^{00}=-\frac{k}{2}\rho^{2}v_{s}^{2}\left(\frac{f^{{}^{\prime}}_{s}}{r_{s}}\right)^{2}\leq
0\,.$ (31)

We can see that this component implies a negative energy density, in
spacetime, for the source. In TEGR, the energy is the $\displaystyle(0)$
component of a four-vector and depends on the frame. Thus, in order to measure
a physical property, we must establish a frame, i.e., an observer. In section
VII, we compare the results obtained in section IV and V and address the
“negative energy problem”. It can be seen that when $\displaystyle
v_{s}=v_{s}(t)$ the component $\displaystyle T^{00}$ varies with time. We
address this in section VI.

The function $\displaystyle f_{s}(r_{s})$ may assume different forms, but in
general it has the dependency on two parameters related to the radius of the
bubble and its thickness. The original function considered by Alcubierre is
alcubierre1994warp

$f_{s}(r_{s})=\frac{\tanh{[\sigma(r_{s}+R)]-\tanh{[\sigma(r_{s}-R)]}}}{2\tanh{\sigma
R}}\,,$ (32)

where the parameter $\displaystyle R$ is related to the radius of the bubble
and $\displaystyle\sigma$ to the inverse of its thickness. Thus, we have three
regions I, II and III. I and III are flat and separated by the non-flat region
II.

In this paper, we shall consider the function given by (32) and, for
comparison, another form given by

$f_{s}(r_{s})=e^{-\sigma\,r_{s}^{2}}\,,$ (33)

where $\displaystyle\sigma$ is a term that regulates the smoothness of the
transition between the wall and the flat region. For the function (33), the
bubble wall consumes almost all of its interior, leaving the spacetime flat
only in the vicinity of the center of the bubble.

##  IV Static observer

In order to evaluate the energy of the spacetime, we need to establish a
tetrad field that is associated with the spacetime metric and adapted to an
observer. A frame of reference can be characterized by a congruence of
stationary observers (for instance, at spatial infinity). Each observer
carries along its worldline a set of tetrads that are orthonormal fields. Then
at the limit of $\displaystyle r\rightarrow\infty$ the tetrad field takes the
form $\displaystyle e^{a}\,_{\mu}\rightarrow\delta^{a}_{\mu}$ for stationary
observers. The component $\displaystyle e_{(0)}\,^{\mu}$ describes a
congruence of time-like vectors and is associated with the velocity field of
the observer $\displaystyle u^{\mu}$. Thus a field of observers is described
by its velocity field and by the acceleration of the frame as a whole. Such
acceleration is given by the absolute derivative of the tetrads with respect
to an affine parameter such as the proper time. That is

$\frac{De_{a}\,^{\mu}}{d\tau}=\phi_{a}\,^{b}\,e_{b}\,^{\mu}\,,$ (34)

where $\displaystyle\phi_{ab}$ is the acceleration tensor. We adopt the
interpretation presented in the reference [18], in which
$\displaystyle\phi_{(0)(i)}$ is associated to the translational acceleration
of the frame, while $\displaystyle\phi_{(i)(j)}$ is the frequency of rotation
with respect to a non-rotating Fermi- Walker transported frame. As it was
stated after eq. (6). In this section we consider a static observer, i.e.,
$\displaystyle u^{i}=e_{(0)}\,^{i}=0$. A static observer will only measure the
physical properties of the spacetime and not quantities related to his motion.
A set of tetrads that satisfies the requirements is

$e_{a\mu}=\left(\begin{array}[]{cccc}-A&0&0&-B\\\
0&\cos{\phi}&-\rho\sin{\phi}&0\\\ 0&\sin{\phi}&\rho\cos{\phi}&0\\\
0&0&0&1/A\\\ \end{array}\right)\,,$ (35)

where $\displaystyle A=\sqrt{1-f_{s}^{2}v_{s}^{2}}$ and $\displaystyle
B=f_{s}v_{s}/A$. The inverse tetrads read

$e^{a\mu}=\left(\begin{array}[]{cccc}-1/A&0&0&0\\\
0&\cos{\phi}&-\sin{\phi}/\rho&0\\\ 0&\sin{\phi}&\cos{\phi}/\rho&0\\\
-B&0&0&A\\\ \end{array}\right)\,.$ (36)

The determinant is $\displaystyle e=\rho$. It is easy to check that
$\displaystyle e_{(0)}\,^{\mu}=(1/A,0,0,0)$. At space infinity, we have
$\displaystyle\displaystyle{\lim_{r\rightarrow\infty}}e_{a}\,^{\mu}=\delta_{a}^{\mu}$,
where $\displaystyle r\equiv\sqrt{\rho^{2}+z^{2}}$, i.e., we have Minkowski
spacetime. The the tetrads (35) are adapted to a static observer. The metric
tensor can be easily recovered remembering that $\displaystyle
g_{\mu\nu}=e_{a\mu}e^{a}\,_{\nu}$. The non-vanishing components of the torsion
tensor $\displaystyle T^{\lambda\mu\nu}$ can be calculated using (4). They
read

$\displaystyle\displaystyle T^{001}$ $\displaystyle\displaystyle=\frac{\rho
v_{s}^{2}}{r_{s}}f_{s}f^{{}^{\prime}}_{s}\,,$ (37) $\displaystyle\displaystyle
T^{003}$
$\displaystyle\displaystyle=-\frac{v_{s}^{2}(tv_{s}-z)}{r_{s}A^{2}}(1-f_{s}+v_{s}^{2}f_{s}^{3})f^{{}^{\prime}}_{s}\,,$
(38) $\displaystyle\displaystyle T^{013}$
$\displaystyle\displaystyle=-\frac{\rho
v_{s}}{r_{s}A^{2}}f^{{}^{\prime}}_{s}\,,$ (39) $\displaystyle\displaystyle
T^{301}$ $\displaystyle\displaystyle=-\frac{\rho
v_{s}^{3}}{r_{s}A^{2}}f^{2}_{s}f^{{}^{\prime}}_{s}\,,$ (40)
$\displaystyle\displaystyle T^{303}$
$\displaystyle\displaystyle=\frac{v_{s}^{3}(tv_{s}-z)}{r_{s}A^{2}}f_{s}^{2}f^{{}^{\prime}}_{s}\,,$
(41)

where, again, the prime indicates a derivation with respect to $\displaystyle
r_{s}$.

The only non-vanishing component $\displaystyle e\Sigma^{(0)01}$ constructed
from (9) and (37-41) is

$e\Sigma^{(0)01}=-\frac{\rho^{2}v_{s}^{2}f_{s}f^{\prime}_{s}}{4Ar_{s}}\,.$
(42)

Therefore, from (16), choosing a very long cylinder as the surface of
integration, we obtain the energy

$\displaystyle\displaystyle P^{(0)}$
$\displaystyle\displaystyle=-k\int_{S}dS_{1}\,\Bigg{(}\frac{\rho_{0}^{2}v_{s}^{2}f_{s}f^{\prime}_{s}}{r_{s}A}\Bigg{)}$
$\displaystyle\displaystyle=-\frac{\rho_{0}^{2}}{8}\int_{-L}^{L}dz\,\Bigg{(}\frac{v_{s}^{2}f_{s}f^{\prime}_{s}}{r_{s}A}\Bigg{)}\,.$
(43)

Eq. (IV) represents the total energy contained inside a spatial region of
$\displaystyle\rho=\rho_{0}=constant$, i.e., a cylinder of radius
$\displaystyle\rho_{0}$ and length $\displaystyle 2L$ in Euclidean space. In
order to evaluate only the gravitational energy, we need the components
$\displaystyle t^{\mu\nu}$ given by (12). They read

$\displaystyle\displaystyle t^{00}$
$\displaystyle\displaystyle=-k\frac{\rho^{2}v_{s}^{2}}{2r_{s}^{2}A^{2}}\left(1+v_{s}^{2}f_{s}^{2}\right)f_{s}^{{}^{\prime}2}\,,$
(44) $\displaystyle\displaystyle t^{03}$
$\displaystyle\displaystyle=-k\frac{\rho^{2}v_{s}^{3}}{2r_{s}^{2}A^{2}}f_{s}\left(1+v_{s}^{2}f^{2}_{s}\right)f^{{}^{\prime}2}_{s}\,.$
(45)

Then, from (17), we have

$\displaystyle\displaystyle P_{g}^{(0)}$
$\displaystyle\displaystyle=-\frac{k}{2}\int_{V}d^{3}x\,\Bigg{(}\frac{\rho^{3}v_{s}^{2}\left(1+v_{s}^{2}f_{s}^{2}\right)f^{{}^{\prime}2}_{s}}{r_{s}^{2}A^{3}}\Bigg{)}$
$\displaystyle\displaystyle=-\frac{k}{2}\int_{0}^{2\pi}d\phi\int_{-L}^{L}dz\int_{0}^{\rho_{0}}d\rho\,\Bigg{(}\frac{\rho^{3}v_{s}^{2}\left(1+v_{s}^{2}f_{s}^{2}\right)f^{{}^{\prime}2}_{s}}{r_{s}^{2}A^{3}}\Bigg{)}\,.$
(46)

The energy of the source, measured by the static observer, can be evaluated
using (18) with the aid of Einstein equations, (22-28) and (35). Thus,

$\displaystyle\displaystyle P_{m}^{(0)}$
$\displaystyle\displaystyle=-\int_{V}d^{3}x\,e\left(e_{(0)0}T^{00}+e_{(0)3}T^{03}\right)$
$\displaystyle\displaystyle=-2k\int_{V}d^{3}x\,e\left(e_{(0)0}G^{00}+e_{(0)3}G^{03}\right)$
$\displaystyle\displaystyle=-\frac{k}{2}\int_{0}^{2\pi}d\phi\int_{-L}^{L}dz\int_{0}^{\rho_{0}}d\rho\,\Bigg{(}\rho
v_{s}^{2}\frac{\rho^{2}r_{s}f^{{}^{\prime}2}_{s}+2f_{s}[\rho^{2}r_{s}f^{{}^{\prime\prime}}_{s}+(2r_{s}^{2}-\rho^{2})f_{s}^{{}^{\prime}}]}{r_{s}^{3}A}\Bigg{)}\,.$
(47)

The consistency of the calculations can be verified analytically by adding
Eqs. (46) and (47). By doing it, we can easily obtain (IV), so the Eq. (11) is
satisfied. The relation
$\displaystyle\partial_{\rho}f_{s}=(\rho/r_{s})f^{{}^{\prime}}_{s}$ may be
useful in achieving such an equality.

The integrals (IV,46,47) can be evaluated numerically and the results seen in
Figure 2 for distinct radius $\displaystyle\rho_{0}$ using the original
Alcubierre function (32).

Figure 1: Total (IV), gravitational (46) and source (47) energies
$\displaystyle E$ measured by a static observer for different radii
$\displaystyle\rho_{0}$. The integrals were calculated over an infinity
cylinder of radius $\displaystyle\rho_{0}$. The continuous curve represents
the total energy, the dashed the gravitational and the dotted the source. The
quantity $\displaystyle\rho_{0}$ is a parameter, not a variable. The remaining
parameters are $\displaystyle R=1$, $\displaystyle\sigma=0.01$, $\displaystyle
t=0$, $\displaystyle v_{s}=0.5$ and $\displaystyle f_{s}$ given by (32).

Figure 2: Total (IV), gravitational (46) and source (47) energies
$\displaystyle E$ measured by a static observer for different radii
$\displaystyle\rho_{0}$. The integrals were calculated over an infinity
cylinder of radius $\displaystyle\rho_{0}$. The continuous curve represents
the total energy, the dashed the gravitational and the dotted the source. The
quantity $\displaystyle\rho_{0}$ is a parameter, not a variable. The remaining
parameters are $\displaystyle R=50$, $\displaystyle\sigma=0.01$,
$\displaystyle t=0$, $\displaystyle v_{s}=0.5$ and $\displaystyle f_{s}$ given
by (32).

We can see that the gravitational and source energies converge to constant
values at space infinity, but the total energy is zero at space infinity. This
is an interesting behavior never before observed in TEGR. The total energy of
the Alcubierre spacetime is zero at infinity, but not around the ship. We can
see a peak and then a progressive decrease. Also interesting, the energy of
the source is positive, while the gravitational is negative. Negative
gravitational energies are not a new feature in TEGR, already obtained by
integrating the Schwarzschild spacetime from the event horizon to infinity;
and for the spacetime of plane gravitational waves maluf2008energy . The
effect of the radius $\displaystyle R$ on (32) can be perceived by comparing
Figures 2 and 2. When increasing the radius, the energy also increases, but
the qualitative behavior is not altered. The energy is more sensitive to the
thickness parameter $\displaystyle\sigma$ of the bubble than to its radius
$\displaystyle R$, as we can see a significant reduction in energy for a
smoother $\displaystyle\sigma$ in Figure 4.

Figure 3: Total (IV), gravitational (46) and source (47) energies
$\displaystyle E$ measured by a static observer for different radii
$\displaystyle\rho_{0}$. The integrals were calculated over an infinity
cylinder of radius $\displaystyle\rho_{0}$. The continuous curve represents
the total energy, the dashed the gravitational and the dotted the source. The
quantity $\displaystyle\rho_{0}$ is a parameter, not a variable. The remaining
parameters are $\displaystyle R=1$, $\displaystyle\sigma=0.04$, $\displaystyle
t=0$, $\displaystyle v_{s}=0.5$ and $\displaystyle f_{s}$ given by (32).

Figure 4: Total (IV), gravitational (46) and source (47) energies
$\displaystyle E$ measured by a static observer for different radii
$\displaystyle\rho_{0}$. The integrals were calculated over an infinity
cylinder of radius $\displaystyle\rho_{0}$. The continuous curve represents
the total energy, the dashed the gravitational and the dotted the source. The
quantity $\displaystyle\rho_{0}$ is a parameter, not a variable. The remaining
parameters are $\displaystyle R=1$, $\displaystyle\sigma=0.01$, $\displaystyle
t=0$, $\displaystyle v_{s}=0.7$ and $\displaystyle f_{s}$ given by (32).

Thus, a larger bubble with a thin border appears to be more “energy
efficient”. The energy distribution seems to be more localized for a thin
bubble, as the total energy tends to zero in a smaller radius. The energy has
a strong dependence on the velocity $\displaystyle v_{s}$ as can be perceived
from Figure 4. With an increase of only approximately $\displaystyle 40\%$ in
the velocity, the energy approximately doubles. This increase in the energy
with the velocity can be better seen in Figure 5, where a fixed integration
radius $\displaystyle\rho_{0}$ is chosen, and the total energy is evaluated
for different values of $\displaystyle v_{s}$. In Figure 5,
$\displaystyle\rho_{0}$ was chosen to be the radius around the peak of the
energy. The position of the peak does not depend on $\displaystyle v_{s}$.

Figure 5: Total energy (IV) measured by a static observer for different
$\displaystyle v_{s}$. The integral was calculated over an infinity cylinder
of radius $\displaystyle\rho_{0}=75$. The quantity $\displaystyle v_{s}$ is a
parameter, not a variable. The parameters are $\displaystyle R=1$,
$\displaystyle t=0$, $\displaystyle\sigma=0.01$ and $\displaystyle f_{s}$
given by (32). Figure 6: Total (IV), gravitational (46) and source (47)
energies $\displaystyle E$ measured by a static observer for different radii
$\displaystyle\rho_{0}$. The integrals were calculated over an infinity
cylinder of radius $\displaystyle\rho_{0}$. The continuous curve represents
the total energy, the dashed the gravitational and the dotted the source. The
quantity $\displaystyle\rho_{0}$ is a parameter, not a variable. The remaining
parameters are $\displaystyle\sigma=0.1$, $\displaystyle t=0$, $\displaystyle
v_{s}=0.5$ and $\displaystyle f_{s}$ given by (33).

The energy also depends on the form of the function $\displaystyle f_{s}$, as
we can see in Figure 6 that the smoother function (33) greatly reduces energy
requirements.

###  IV.1 Gravitational radiation measured by a static observer

In this subsection, we evaluate the gravitational flux measured by a static
observer, e.g., an observer on Earth that measures the gravitational flux as
the bubble moves away. As we are considering a ship moving in the positive
$\displaystyle z$ direction, the static observer is always behind the ship for
$\displaystyle t>0$. Defining the quantity
$\displaystyle\phi_{g}^{aj}=e^{a}\,_{\mu}t^{j\mu}$ we may write Eq. (19) as
$\displaystyle\Phi_{g}^{a}=\oint_{S}dS_{j}(e\phi^{aj}_{g})$. The evaluation of
the components $\displaystyle\phi_{g}^{(0)i}$ is straightforward from the
torsion tensor components (37-41). We obtain

$\displaystyle\displaystyle\phi_{g}^{(0)1}$
$\displaystyle\displaystyle=k\frac{\rho^{2}v_{s}^{3}(tv_{s}-z)}{r_{s}^{2}A^{3}}\big{[}1-f_{s}(1+v_{s}^{2}f_{s}^{2})\big{]}f_{s}^{{}^{\prime}2}\,,$
(48) $\displaystyle\displaystyle\phi_{g}^{(0)3}$
$\displaystyle\displaystyle=-k\frac{\rho^{3}v_{s}^{3}}{r_{s}^{2}A^{3}}f_{s}\big{(}1+v_{s}^{2}f_{s}^{2}\big{)}f^{{}^{\prime}2}_{s}\,,$
(49)

as the only non-zero components. The component $\displaystyle\Phi_{g}^{(0)}$
represents the gravitational energy flowing through a finite closed surface.
Observing the components (48,49) we may perceive that we have gravitational
energy flux normal to the surfaces $\displaystyle dS_{1}$ and $\displaystyle
dS_{3}$, i.e., through a “cylinder” with sides surface, base and top. The
total flux reads

$\displaystyle\displaystyle\Phi_{g}^{(0)}$ $\displaystyle\displaystyle=\int
dS_{1}\phi_{g}^{(0)1}+\int dS_{3}\phi_{g}^{(0)3}$
$\displaystyle\displaystyle=-\frac{1}{8}\Bigg{(}\int_{0}^{\rho_{0}}d\rho\,\frac{\rho^{3}v_{s}^{3}}{r_{s}^{2}A^{3}}f_{s}\big{(}1+v_{s}^{2}f_{s}^{2}\big{)}f^{{}^{\prime}2}_{s}\Bigg{|}_{z=L}+\int_{0}^{\rho_{0}}d\rho\,\frac{\rho^{3}v_{s}^{3}}{r_{s}^{2}A^{3}}f_{s}\big{(}1+v_{s}^{2}f_{s}^{2}\big{)}f^{{}^{\prime}2}_{s}\Bigg{|}_{z=-L}$
$\displaystyle\displaystyle-\int_{-L}^{L}dz\,\frac{\rho_{0}^{2}v_{s}^{3}(tv_{s}-z)}{r_{s}^{2}A^{3}}\big{[}1-f_{s}(1+v_{s}^{2}f^{2})\big{]}f_{s}^{{}^{\prime}2}\Bigg{)}\,.$
(50)

Expression (50) can be numerically integrated and the results are presented in
Figures 8 for different values of the velocity $\displaystyle v_{s}$.

Figure 7: Gravitational flux (50) measured by a static observer for different
radii $\displaystyle\rho_{0}$. The integrals were calculated over a cylinder
of radius $\displaystyle\rho_{0}$ and length $\displaystyle 2L$. The quantity
$\displaystyle\rho_{0}$ is a parameter, not a variable. The remaining
parameters are $\displaystyle R=1$, $\displaystyle\sigma=0.01$, $\displaystyle
t=100$, $\displaystyle L=100$, and $\displaystyle f_{s}$ given by (32).

Figure 8: Gravitational flux (50) measured by a static observer as a function
of $\displaystyle t$. The integrals were calculated over a cylinder of radius
$\displaystyle\rho_{0}$ and length $\displaystyle 2L$. The parameters are
$\displaystyle R=1$, $\displaystyle\sigma=0.01$, $\displaystyle L=100$,
$\displaystyle\rho_{0}=100$, and $\displaystyle f_{s}$ given by (32).

A static observer measures a gravitational energy flux that increases (its
absolute value) with the radius of the cylinder until it reaches a constant
value. In Figure 8 the gravitational energy flux is plotted over a fixed
cylindrical surface as a function of time. The gravitational energy flux
decays to zero as time increases, which is expected as the bubble has
localized gravitational energy around itself and moves away from the observer.
The expressions for the flux are not regular for $\displaystyle v_{s}>1$ and
we believe that this feature is due to causality problems, since a static
observer cannot measure radiation from a supraluminal source. In subsection
V.1, we demonstrate that an observer moving with the bubble does not have this
problem. The flux has a peak in time. The higher the velocity, the sooner the
peak occurs. Since we are considering a surface fixed to the static observer,
an observer moving with the bubble would see the rear surface moving away and
the front moving closer. The peak occurs the moment that the bubble crosses
the front surface. Thus, this increase in flux is caused by the bubble itself
“flowing” through the surface.

##  V Free-falling observer

In section IV we obtained the gravitational energy and the total energy for
the Alcubierre spacetime measured by a static observer. In this section, we
discuss the gravitational energy and source energy of Alcubierre spacetime
measured by a free-falling observer, i.e., an observer that is in geodesic
motion. Such an observer moves with the bubble, so the results obtained in
this section can be interpreted as the results that an observer “inside” the
bubble measures. In order to accomplish this task, one can change the
coordinates by removing its origin from Earth and resetting it with the center
of the bubble as its origin. In this new coordinate system, the surface
equivalent to that used in (IV) would be considered to travel backward
relative to the origin. However, the tetrad approach allows the analysis to
take place in the same coordinate system, as we can transform the frames
independently of the coordinates. Applying a Lorentz transformation

$\Lambda^{a}\,_{b}=\left(\begin{array}[]{cccc}\gamma&0&0&\gamma v\\\
0&1&0&0\\\ 0&0&1&0\\\ \gamma v&0&0&\gamma\\\ \end{array}\right)\,,$ (51)

in the positive direction of the $\displaystyle z$ axis in the tetrads (36) we
obtain a new set of tetrads $\displaystyle
e^{{}^{\prime}a}\,_{\mu}=\Lambda^{a}\,_{b}e^{b}\,_{\mu}$ that is adapted to an
observer with velocity $\displaystyle v$ relative to the static observer and
the origin of the coordinate system remains fixed on Earth, where
$\displaystyle\gamma=(1-v^{2})^{1/2}$. In order to determine the value of
$\displaystyle v$ that leads to a free-falling observer, we evaluate the
components of the accelerating tensor (6) for the tetrads $\displaystyle
e^{{}^{\prime}a}\,_{\mu}$. They read

$\displaystyle\displaystyle\phi^{{}^{\prime}(0)(1)}$
$\displaystyle\displaystyle=-\frac{\gamma^{2}v_{s}\rho}{r_{s}A^{2}}\big{[}v-v_{s}(1+v^{2})f_{s}+vv_{s}^{2}f_{s}^{2}\big{]}f^{{}^{\prime}}_{s}\cos{\phi}\,,$
(52) $\displaystyle\displaystyle\phi^{{}^{\prime}(0)(2)}$
$\displaystyle\displaystyle=-\frac{\gamma^{2}v_{s}\rho}{r_{s}A^{2}}\big{[}v-v_{s}(1+v^{2})f_{s}+vv_{s}^{2}f_{s}^{2}\big{]}f^{{}^{\prime}}_{s}\sin{\phi}\,,$
(53) $\displaystyle\displaystyle\phi^{{}^{\prime}(0)(3)}$
$\displaystyle\displaystyle=-\frac{\gamma^{3}(tv_{s}-z)}{r_{s}A^{3}}\Bigg{\\{}v^{2}(1-v^{2})\big{[}1+v_{s}^{2}f_{s}^{3}-f_{s}(1+v_{s}v)\big{]}f^{{}^{\prime}}_{s}-A^{2}\big{[}v_{s}-(1+v_{s}^{2}f-v_{s}^{2}f^{2})v\big{]}v^{\prime}\Bigg{\\}}\,,$
(54)

where $\displaystyle v^{\prime}\equiv dv/dr_{s}$. The above components are the
components of the inertial acceleration necessary for an observer to remain in
its dynamical state. For a static observer ($\displaystyle v=0$), these are
the components of the acceleration necessary for the observer to remain static
and must vanish at space infinity, i.e., there is no inertial acceleration at
infinity. Thus, we may obtain the value $\displaystyle v$ necessary for the
observer to be in free fall requiring that all the components (52-54) vanish.
It is straightforward to see that $\displaystyle\phi_{(0)1}=0$ and
$\displaystyle\phi_{(0)2}=0$ lead to $\displaystyle v=v_{s}f_{s}$, and this
also satisfy $\displaystyle\phi_{(0)3}=0$. Thus, the tetrads

$e^{{}^{\prime}}_{a\mu}=\lambda_{a}\,^{b}e_{b\mu}=\left(\begin{array}[]{cccc}-1&0&0&0\\\
0&\cos{\phi}&-\rho\sin{\phi}&0\\\ 0&\sin{\phi}&\rho\cos{\phi}&0\\\
-v_{s}f_{s}&0&0&1\\\ \end{array}\right)\ $ (55)

are adapted to a free-falling observer. We may observe that $\displaystyle
u^{\mu}=e_{(0)}\,^{\mu}=(1,0,0,v_{s}f_{s})$, which is the four-velocity (29)
of the so-called Eulerian observers in the $\displaystyle 3+1$ formalism. It
should be noted that these observers are at rest with respect to the spatial
infinity, as $\displaystyle f_{s}\rightarrow 0$ when $\displaystyle
r\rightarrow\infty$. The same coordinate system is used for the observers
adapted to the tetrads (55) and for those adapted to the tetrads (35). The
torsion tensor components (4) can be evaluated for the tetrads (55) and the
components (9) obtained. Performing this, we obtain

$\Sigma^{{}^{\prime}(0)0\mu}=0\Rightarrow P^{{}^{\prime}(0)}=0\,,$ (56)

i.e., a free-falling observer measures zero total energy. The same result was
obtained in Ref. maluf2007reference  in the context of the Schwarzschild black
hole, i.e., an observer free-falling radially into the black hole does not
measure the energy of the black hole. In the latter Ref. this was interpreted
as a consequence of the equivalence principle. Although the observer measures
zero total energy, the energy of the source and gravitational energy are not
zero but are equal in modulus and opposites in signal. We return to the
discussion of the source and gravitational energy in section VII.

###  V.1 Gravitational radiation measured by a free-falling observer

The gravitational energy flux measured by the free-falling observer can be
obtained similarly to the case for the static observer. The non-zero
$\displaystyle t^{{}^{\prime}\mu\nu}$ components read

$\displaystyle\displaystyle
t^{{}^{\prime}00}=k\frac{\rho^{2}v_{s}^{2}}{2r_{s}^{2}}f^{{}^{\prime}2}_{s}\,,$
(57) $\displaystyle\displaystyle
t^{{}^{\prime}03}=k\frac{\rho^{2}v_{s}^{3}}{2r_{s}^{2}}f_{s}f^{{}^{\prime}2}_{s}\,.$
(58)

Thus, the non-vanishing component of $\displaystyle\phi_{g}^{{}^{\prime}(0)i}$
is

$\phi_{g}^{{}^{\prime}(0)3}=k\frac{\rho^{3}v_{s}^{3}}{2r_{s}^{2}}f_{s}f^{{}^{\prime}2}_{s}\,.$
(59)

Hence, the gravitational energy flux measured by the free-falling observer, on
the cylindrical surface, reads

$\Phi_{g}^{{}^{\prime}(0)}=\frac{1}{16}\Bigg{[}\int_{0}^{\rho_{0}}d\rho\,\Bigg{(}\frac{\rho^{3}v_{s}^{3}}{r_{s}^{2}}f_{s}f^{{}^{\prime}2}_{s}\Bigg{)}\Bigg{|}_{z=L}-\int_{0}^{\rho_{0}}d\rho\,\Bigg{(}\frac{\rho^{3}v_{s}^{3}}{r_{s}^{2}}f_{s}f^{{}^{\prime}2}_{s}\Bigg{)}\Bigg{|}_{z=-L}\Bigg{]}\,.$
(60)

The results for the numerical integration of (60) can be seen in Figure 9.

Figure 9: Gravitational flux (60) measured by a free-falling observer as a
function of $\displaystyle t$. The integrals were calculated over a cylinder
of radius $\displaystyle\rho_{0}$ and length $\displaystyle 2L$. The
parameters are $\displaystyle R=1$, $\displaystyle\sigma=0.01$, $\displaystyle
L=10$, $\displaystyle\rho_{0}=100$, and $\displaystyle f_{s}$ given by (32).

The gravitational energy flux decays over time as the bubble moves away from
the surface. In Figure 9, the flux increases and then decreases. This happens
when the bubble approaches the surface of the cylinder ahead and the flux
increases; and as it moves away, the flux decreases. The larger the velocity
$\displaystyle v_{s}$, the sooner the surface is reached by the bubble.
Remember that the coordinate system is centered at Earth, so the surface is
fixed with respect to Earth and moves with respect to the bubble. We can see
that the integrals (60) are no longer irregular when $\displaystyle v_{s}>1$,
as was the case with (50). An observer who moves with the bubble has no
causality problems when measuring physical properties of the bubble.

It is interesting for a co-moving observer to measure the radiation from a
source. This is not a unique feature of gravitation. It was found in Ref.
maluf2010electrodynamics  that an observer moving along an accelerated charge
measures electromagnetic radiation.

##  VI The variable velocity case

So far, we have only considered the case $\displaystyle v_{s}=constant$. In
this case, the energy of the source and the gravitational energy are constant
in time. In a realistic situation, the ship, hence the bubble, must accelerate
from zero velocity to an arbitrary desired velocity. In the original article
alcubierre1994warp , Alcubierre considered the velocity to be a time-dependent
quantity. Considering $\displaystyle v_{s}=v_{s}(t)$ in the metric tensor, we
obtain the energy density of the source

$T^{00}=-\frac{1}{32\pi}\rho^{2}v_{s}(t)^{2}\left(\frac{f^{{}^{\prime}}_{s}}{r_{s}}\right)^{2}\,,$
(61)

which is a function of $\displaystyle t$. As pointed out in
bobrick2021introducing  this implies a violation of energy conservation for
the source, since its energy must increase with time in an accelerated motion.

In this section, we pursuit the main objective of this article, i.e., to
demonstrate that when the full gravitational system is considered, source plus
gravitational field, there is no violation of energy conservation. Considering
$\displaystyle v_{s}=v_{s}(t)$ in the tetrads (35), we may compute the torsion
tensor components and the total, gravitational and source energies, as in
section IV. By doing it, we analytically obtain the same expressions (IV),
(46) and (47), respectively, but now with $\displaystyle v_{s}=v_{s}(t)$,
i.e., a non-constant global velocity. Therefore, as in the case $\displaystyle
v_{s}=constant$, we have $\displaystyle P^{(0)}=0$ when the surface integral
is evaluated at infinity. Thus, as $\displaystyle P^{(0)}$ is not a function
of time, there is no (total) energy violation when the velocity is a function
of time. The total energy is still conserved in an accelerated motion.

Physically, the source energy (positive) varies with the same “frequency” as
the gravitational energy (negative), exactly canceling out the variations of
each other. In order to better elucidate what is happening, let us consider
two particular velocities $\displaystyle v_{s}(t)$. First, the velocity of a
particle with constant proper acceleration in special relativity

$v_{s}=\frac{\alpha t}{\sqrt{1+\alpha^{2}t^{2}}}\,,$ (62)

where $\displaystyle\alpha$ is its proper acceleration. In Figure 11 the
results for the total, gravitational, and source energies can be seen in a
region composed of an infinite cylinder with a very large radius
(approximately at infinity). The variation of the gravitational energy is
exactly the same as that of the source, but with the opposite sign. In the
case $\displaystyle v_{s}=constant$, we have straight lines. The curve of
total energy $\displaystyle P^{(0)}$ is zero, thus coinciding with the
$\displaystyle t$ axis. Second, we consider a bubble with a Gaussian
acceleration, i.e., which, starting from rest, accelerates to a maximum value
and then decelerates to rest again. Such physical situation can be described
by bobrick2021introducing

$v_{s}=v_{m}e^{-\left(\frac{t-t_{m}}{T}\right)^{2}}\,,$ (63)

where $\displaystyle v_{m}$ is the maximum value for the velocity,
$\displaystyle t_{m}$ the instant at which the maximum velocity is attained
and $\displaystyle 2T$ is related to the time length of the trajectory. The
energies for (63) can be seen in Figure 11. Again, the time variation of the
gravitational energy plus source energy is zero.

Figure 10: Total (IV), gravitational (46) and source (47) energies
$\displaystyle E$ measured by a static observer as a function of the time
$\displaystyle t$. The integrals were calculated over a infinite cylinder of
radius $\displaystyle\rho_{0}$. The parameters are $\displaystyle R=10$,
$\displaystyle\sigma=0.01$, $\displaystyle\rho_{0}=300$,
$\displaystyle\alpha=0.01$, with $\displaystyle f_{s}$ given by (32) and
$\displaystyle v_{s}$ by (62).

Figure 11: Total (IV), gravitational (46) and source (47) energies
$\displaystyle E$ measured by a static observer as a function of the time
$\displaystyle t$. The integrals were calculated over a infinite cylinder of
radius $\displaystyle\rho_{0}$. The parameters are $\displaystyle R=10$,
$\displaystyle\sigma=0.01$, $\displaystyle\rho_{0}=400$, $\displaystyle
v_{m}=0.9$, $\displaystyle a=100$, $\displaystyle t_{m}=200$, with
$\displaystyle f_{s}$ given by (32) and $\displaystyle v_{s}$ by (63).

##  VII Final considerations

Much attention has been devoted to the Alcubierre class of spacetimes,
sometimes called “warp drives”. Several related physical issues have been
highlighted regarding these solutions and a vast amount of papers have
attempted to address these problems. Many of these articles that have appeared
recently dealt with the problem of violation of the energy conditions.
Although the energy conditions are not a physical law, there is some intuition
on the attractive nature of gravity, which may be violated, in a spacetime
without rotation, according to the Raychaudhuri equation in the absence of the
energy conditions poisson2004relativist . Nevertheless, a violation of the
strong energy condition has already been speculated for the cosmological fluid
riess2004type . In addition, negative energy density has already been
predicted for the Casimir effect sopova2002energy ; ulhoa2019gravitational .
Several authors have proposed Alcubierre spacetime models that do not violate
some energy conditions lentz2021breaking ; fell2021positive , and Bobrick
introduced an algorithm to construct such spacetimes bobrick2021introducing .
Although it has been argued that these generalizations do not solve the
problem, as only one type of observer is considered santiago2021generic . Some
of these problems are unique to GR, as some of them do not seem to appear in
modified theories of gravity, such as conformal gravity varieschi2013conformal
. Indeed, there are many currently unexplained features of the Alcubierre
spacetime, but this spacetime is a valid class of solutions to Einstein
equations, and its comprehension may lead to the solution of many, if not all,
of these problems.

In the present paper, we addressed the problem of energy conservation of the
spacetime, but we also uncovered new properties of the spacetime, in
particular the gravitational energy flux measured by a static observer (50)
and by a free-falling observer (60). As far as we know, the total energy of
Alcubierre spacetime has never been obtained. The presence of negative energy
density is recurrent in the presence of supraluminal effects
olum1998superluminal , but we found positive energy for the source measured by
a static observer. The problem of negative energy may be a problem of the
frame, i.e., of the motion of an observer interfering with the measurement of
the energy. Evaluating the gravitational energy density $\displaystyle
e^{\prime}e^{{}^{\prime}(0)}\,_{\mu}t^{\prime 0\mu}$ for the tetrads (55), we
obtain

$P_{g}^{{}^{\prime}(0)}=\frac{k}{2}\int
d^{3}x\,\Bigg{(}\rho^{3}\frac{v_{s}^{2}}{r_{s}^{2}}f^{{}^{\prime}2}_{s}\Bigg{)}\geq
0\,,$ (64)

where $\displaystyle P_{g}^{\prime(0)}$ is the gravitational energy measured
by the free-falling observer. For the source, we have

$P_{m}^{{}^{\prime}(0)}=-2k\int
d^{3}x\,e^{{}^{\prime}}e^{{}^{\prime}}_{(0)0}G^{00}=-\frac{k}{2}\int
d^{3}x\,\Bigg{(}\rho^{3}\frac{v_{s}^{2}}{r_{s}^{2}}f^{{}^{\prime}2}_{s}\Bigg{)}\leq
0\,,$ (65)

where $\displaystyle P_{m}^{\prime(0)}$ is the energy of the source measured
by the free-falling observer. Thus, a free-falling observer does measure
negative energy for the source, but a static does not. We can see that (64)
and (65) satisfy (56), i.e.,

$P_{g}^{{}^{\prime}(0)}=-P_{m}^{\prime(0)}\,.$ (66)

This comparison is only possible when we treat the static observer and free-
falling one in the same coordinate system, as the tetrads (35) and (55)
represent the different observers in the same coordinate system. When an
observer is measuring physical properties, its dynamics affect the
measurement, as is the case of a free-falling observer in the Schwarzschild
spacetime, which does not measure the energy of the black hole due to its
dynamic state maluf2007reference . Therefore, only static observers can
measure reliable physical properties of spacetimes. The dependence of the
energy measurement on the choice of the observer is a characteristic of the
definition of the energy-momentum vector itself. This situation is not
exclusive to the expression obtained in the TEGR, it also exists in special
relativity. Thus, energy measurements tell us both the information about the
physical configuration and the reference system adopted.

When considering the effect of the gravitational field, in the realm of TEGR,
we found that the energy of the source is dominant around the bubble, but as
we radially move, the gravitational energy becomes relevant and, because it
has an opposite signal, cancels out exactly the source energy, thus resulting
in vanishing total energy at space infinity, as can be perceived in Figure 2.
This is true even when the velocity of the bubble is a function of
$\displaystyle t$, as demonstrated in section VI. This is a very interesting
behavior, as the Alcubierre spacetime is a spacetime with zero total energy.
The non-uniformity of the source (dominant closer to the bubble) and
gravitational (dominant furthest from the bubble) energies produces the
desired effect of the motion for the bubble. If we interpret the total energy
at infinity as the irreducible mass of the system, as is true in Schwarzschild
spacetime, the irreducible mass of Alcubierre spacetime is zero. Therefore,
the bubble as a whole (source and gravitational field) acts as a system of
null mass accelerated to supraluminal velocities. The presence of
gravitational energy flux corroborates the idea that the Alcubierre spacetime
is an entity composed of a local source and a local gravitational field. The
consideration of only the source renders the analysis incomplete. The presence
of gravitational energy flux may be of practical use. It is possible for
observers, static and co-moving, to measure this gravitational energy flux at
great distances from the bubble, thus making it possible, at least
theoretically, to detect such spacetimes at a distance. This spacetime has a
local “warp trail” characteristic that may even allow the direction of
movement to be determined. If we make $\displaystyle t\rightarrow-t$ in Figure
8, a bubble approaching the observer rather than moving away, we obtain a
reflected behavior in the $\displaystyle\rho_{0}$ axis, i.e., in the regions
where the flux is positive in Figure 8 it becomes negative and vice-versa.

If, on the one hand, the relativity of velocity is an old and well-founded
idea, even experimentally, on the other hand, there is controversy regarding
the application of this concept to acceleration. Regarding electromagnetic
radiation, we can mention the following result obtained in Ref. annalen . An
accelerated charge radiates when measured by a stationary observer, but an
accelerated observer does not observe electromagnetic radiation coming from a
particle at rest. This asymmetrical situation calls into question the
relativity of acceleration. In this article, we also look at an interesting
case involving acceleration. The flux of gravitational energy measured by a
stationary observer remains negative for most of the temporal evolution, while
for an accelerated observer the flux is positive. The total energy
conservation, that is, when gravitational energy is considered, occurs when
the spacecraft is accelerated when measured by a stationary observer. This
feature potentially reveals yet another hint of the absolute character of
acceleration.

In this article, we also were able to corroborate the extremely high energy
values associated with the Alcubierre spacetime. Let us consider a realistic
case for the parameters. Considering $\displaystyle R=100$ and
$\displaystyle\sigma=0.1$, we obtain $\displaystyle
P^{(0)}_{m}=-P^{(0)}_{g}\approx 8.32$ in geometrized units. This corresponds
to approximately $\displaystyle 10^{45}J$ in SI units. This is a relevant
percentage of the rest mass of the sun ($\displaystyle\approx 10^{47}\,J$).
High values for the energies involved in Alcubierre spacetime were also found
when applying quantum inequality restriction to the Alcubierre spacetime
pfenning1997unphysical . However, the values for the energies obtained in the
present article, even been beyond practical considerations, are not physically
unattainable. Besides, the metric can be modified in order to reduce the
energy requirements van1999awarp . An analysis of the energies of the modified
Alcubierre spacetimes is an interesting way to compare these spacetimes. This
will be pursued elsewhere.

## Acknowledgements

The author F. L. Carneiro is grateful to L. S. Barbosa for bringing this
matter to his attention.

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