I) The proof that the orthochronous Lorentz group $O^{+}(1,d; \mathbb{R})$ form a group (which is closed/stabile under multiplication and inversion) is given in this Phys.SE post.
II) Next we would like to prove the following.
Proposition. A Lorentz transformation takes a timelike vector
$\tilde{x}=\begin{bmatrix} x^0\cr x \end{bmatrix}$
with $|x| < |x^0|$ to a timelike vector
$\tilde{x}^{\prime}=\begin{bmatrix}x^{\prime 0}\cr x^{\prime}\end{bmatrix}$
with $|x^{\prime}| < |x^{\prime 0}|$.
Proof. This follows from the fact that a Lorentz transformation preserves the Minkowski norm. $\Box$
Proposition. An orthochronous Lorentz transformation takes a future timelike vector $\tilde{x}=\begin{bmatrix} x^0\cr x \end{bmatrix}$
with $|x| < x^0$ to a future timelike vector
$\tilde{x}^{\prime}=\begin{bmatrix}x^{\prime 0}\cr x^{\prime}\end{bmatrix}$ with $|x^{\prime}| < x^{\prime 0}$.
Proof. To prove that an orthochronous Lorentz transformation
$$\tag{1} \tilde{x}^{\prime}~=~\Lambda\tilde{x}, \qquad
\Lambda ~=~ \begin{bmatrix}a & b^t \cr c &R \end{bmatrix}~\in~O(1,d; \mathbb{R}), $$
(which by definition has $a=\Lambda^0{}_0>0$), takes a future timelike vector $\tilde{x}=\begin{bmatrix} x^0\cr x \end{bmatrix}$ with
$$\tag{2} |x| ~<~ x^0$$
to a future timelike vector
$\tilde{x}^{\prime}=\begin{bmatrix}x^{\prime 0}\cr x^{\prime}\end{bmatrix}$ with $|x^{\prime}| < x^{\prime 0}$, it is enough to prove that
$$\tag{3} 0~\stackrel{(4)}{<}~ a x^0 + b\cdot x~\stackrel{(1)}{=}~ x^{\prime 0}.$$
But the inequality (3) follows from the following inequality
$$\tag{4} -2\frac{b}{a} \cdot \frac{x}{x^0}~\leq~ \left(\frac{b}{a}\right)^2 +\left(\frac{x}{x^0}\right)^2 ~\stackrel{(2)}{<}~ \frac{a^2-1}{a^2} + 1 ~<~2. $$
Here we used the fact that $ b\cdot b =a^2-1$ and the inequality (2). $\Box$
III) Thus there only remains OP's last question:
What is the relation between $SL(2,\mathbb{R})$ and $SO^{+}(1,2;\mathbb{R})$?
Naturally our treatment will have some overlap with Trimok's correct answer. We use the sign convention $(+,-,-,-)$ for the Minkowski metric $\eta_{\mu\nu}$.
IV) First let us identify the Minkowski space $M(1,3;\mathbb{R})$ with the space of Hermitian $2\times2$ matrices $u(2)$. In detail, there is a bijective isometry from the Minkowski space $(M(1,3;\mathbb{R}),||\cdot||^2)$ to the space of Hermitian $2\times2$ matrices $(u(2),\det(\cdot))$,
$$\mathbb{R}^4~=~M(1,3;\mathbb{R}) ~\cong ~ u(2)
~:=~\{\sigma\in {\rm Mat}_{2\times 2}(\mathbb{C}) \mid \sigma^{\dagger}=\sigma \}
~=~ {\rm span}_{\mathbb{R}} \{\sigma_{\mu} \mid \mu=0,1,2,3\}, $$
$$ M(1,3;\mathbb{R})\ni\tilde{x}~=~(x^0,x^1,x^2,x^3) \quad\mapsto \quad\sigma~=~x^{\mu}\sigma_{\mu}~\in~ u(2), $$
$$\tag{5} ||\tilde{x}||^2 ~=~x^{\mu} \eta_{\mu\nu}x^{\nu} ~=~\det(\sigma), \qquad \sigma_{0}~:=~{\bf 1}_{2 \times 2},$$
see also this Phys.SE post.
V) There is a group action $\rho: SL(2,\mathbb{C})\times u(2) \to u(2)$ given by
$$\tag{6} g\quad \mapsto\quad\rho(g)\sigma~:= ~g\sigma g^{\dagger},
\qquad g\in SL(2,\mathbb{C}),\qquad\sigma\in u(2). $$
A straightforward calculation shows that the two groups $SL(2,\mathbb{R}) \equiv Sp(2,\mathbb{R})$ and
$$SU(1,1)~=~\left\{\left. \begin{bmatrix} a & b \\ b^{*} & a^{*} \end{bmatrix}\right| a,b\in \mathbb{C}, |a|^2-|b|^2=1\right\}$$
$$ \tag{7} ~=~\left\{\left. \begin{bmatrix} f\sqrt{|b|^2+1} & b \\ b^{*} & f^{*}\sqrt{|b|^2+1} \end{bmatrix} \right| f,b\in \mathbb{C}, |f|=1\right\}~\cong~S^1\times \mathbb{C}$$
are the stabilizer subgroups (also called the isotropy subgroups) of the $x^2$-coordinate and the $x^3$-coordinate, respectively. Since there is no spatially preferred direction, the two subgroups are isomorphic. (The explicit isomorphism is given in Ref. 1.) The two subgroups are path connected but not simply connected. In detail, the fundamental group is
$$\tag{8} \pi_1(SL(2,\mathbb{R}),*)~=~\pi_1(SU(1,1),*)~=~\pi_1(S^1\times \mathbb{C},*)$$
$$~=~\pi_1(S^1,*)\oplus \pi_1(\mathbb{C},*)~=~\mathbb{Z}.$$
VI) We now restrict attention to the $1+2$ dimensional case. Let us identity the Minkowski space $M(1,2;\mathbb{R})~\subseteq~ M(1,3;\mathbb{R})$ as the hyperplane $x^2=0$. The corresponding hyperplane in $u(2)$ is the set
$$\tag{9} s(2)~:=~\{ \sigma \in {\rm Mat}_{2\times 2}(\mathbb{R}) \mid \sigma^t =\sigma \} $$
of real symmetric $2\times2 $ matrices.
VII) There is a group action $\rho: SL(2,\mathbb{R})\times s(2) \to s(2)$ given by
$$\tag{10} g\quad \mapsto\quad\rho(g)\sigma~:= ~g\sigma g^t,
\qquad g\in SL(2,\mathbb{R}),\qquad\sigma\in s(2), $$
which is length preserving, i.e. $g$ is a pseudo-orthogonal (or Lorentz) transformation. In other words, there is a Lie group homomorphism
$$\tag{11} \rho: SL(2,\mathbb{R}) \quad\to\quad O(s(2),\mathbb{R})~\cong~ O(1,2;\mathbb{R}).$$
Since $\rho$ is a continuous map from a path connected set $SL(2,\mathbb{R})$, the image $\rho(SL(2,\mathbb{R}))$ is also path connected. We conclude that
Lie group homomorphism
$$\tag{12} \rho: SL(2,\mathbb{R}) \quad\to\quad SO^{+}(s(2),\mathbb{R})~\cong~ SO^{+}(1,2;\mathbb{R})$$
maps into the restricted Lorentz group $SO^{+}(1,2;\mathbb{R})$. [Here we have used the easily established fact that the Lorentz group $O(1,2;\mathbb{R})$ has at least four connected components because $\Lambda^0{}_{0}\neq 0$ and $\det(\Lambda)\neq 0$. We do not assume the fact that there is precisely four connected components.] It is trivial to check that the kernel is
$$\tag{13} {\rm ker}(\rho)~=~\rho^{-1}({\bf 1}_{s(2)})~=~\{\pm {\bf 1}_{2 \times 2}\}~\cong~\mathbb{Z}_{2}.$$
Let
$$\tag{14} \tilde{\rho}: SL(2,\mathbb{R})/\mathbb{Z}_2 \quad\to\quad SO^{+}(1,2;\mathbb{R})$$
denote the corresponding injective Lie group homomorphism. Thus if we could prove that $\rho$ is surjective/onto, i.e. that the image ${\rm Im}(\rho)\equiv\rho(SL(2,\mathbb{R}))$ is precisely the restricted Lorentz group, cf. Section X below, we would have proved that
Theorem. $SL(2,\mathbb{R})$ is the double cover of the restricted Lorentz group $SO^{+}(1,2;\mathbb{R})$.
Note that $SL(2,\mathbb{R})$ is not a universal cover, since we just saw in Section V that
$$\tag{15}\pi_1(SL(2,\mathbb{R}),*)~=~\mathbb{Z}.$$
The universal covering group $\overline{SL(2,\mathbf{R})}$ is an example of a finite-dimensional Lie group that is not a matrix group.
VIII) One may show that the exponential map $\exp: sl(2,\mathbb{R}) \longrightarrow SL(2,\mathbb{R})$ is not surjective
$$\tag{16}{\rm Im}(\exp)
~=~\left\{M\in SL(2,\mathbb{R}) \mid {\rm Tr}(M)> -2\right\} ~\cup~ \left\{-{\bf 1}_{2 \times 2}\right\} ~\subsetneq~ SL(2,\mathbb{R}).$$
It is a small miracle that plus/minus the exponential map $\pm \exp: sl(2,\mathbb{R}) \longrightarrow SL(2,\mathbb{R})$ is indeed surjective, which is enough for our purposes, cf. the $\mathbb{Z}_{2}$-kernel (13).
IX) Next let us consider the following Lemma for arbitrary spatial dimensions $d$.
Lemma. Any restricted Lorentz transformation is a product of a pure rotation and a pure boost.
Proof. Let us decompose a Lorentz matrix $\Lambda$ into 4 blocks
$$\tag{17} \Lambda ~=~ \begin{bmatrix}a & b^t \cr c &R \end{bmatrix} ,$$
where $a=\Lambda^0{}_0\neq 0$ is a real number; $b$ and $c$ are real $d\times 1$ column vectors; and $R$ is a real $d\times d$ matrix. First argue from $\Lambda^t \eta \Lambda=\eta$, or equivalently from $\Lambda \eta^{-1} \Lambda^t=\eta^{-1}$, that
$$\tag{18} a^2~=~b^tb+1,\qquad c~=~\frac{Rb}{a}, \qquad b~=~\frac{R^tc}{a}. $$
Next argue that
$$\tag{19} B(b)~:=~ \begin{bmatrix} a & b^t \\ b & {\bf 1}_{d\times d}+\frac{bb^t}{a+1} \end{bmatrix}, \qquad a~:=~\sqrt{b^tb+1}~\geq~1, $$
is a Lorentz matrix with an inverse matrix
$$\tag{20} B(-b)~=~\begin{bmatrix} a & -b^t \\ -b & {\bf 1}_{d\times d}+\frac{bb^t}{a+1} \end{bmatrix}, \qquad B(b)B(-b)~=~{\bf 1}. $$
Such matrices correspond to pure (finite) boosts. Use this to prove the Lemma. Hint: The matrix $\Lambda B(-b)$ is on block diagonal form. $\Box$
Also note that we may conjugate a pure boost matrix with a pure rotation matrix to obtain a pure boost matrix in a preferred direction. The Lorentz algebra is
$$\tag{21} so(1,d;\mathbb{R})~=~\left. \left\{ \begin{bmatrix} 0 & b^t \\ b & r \end{bmatrix} \right| r^t ~=~ -r \right\}. $$
The exponential map is surjective on the set of pure boost:
$$\tag{22} \exp \begin{bmatrix} 0 & b^t \\ b & {\bf 0}_{d\times d} \end{bmatrix}
~=~ B\left(\frac{\sinh|b|}{|b|}b\right), \qquad |b|~:=~\sqrt{b^tb}~\geq~0.$$
Moreover, one may prove that the exponential map $\exp: so(d,\mathbb{R})\to SO(d,\mathbb{R})$ for pure rotations is surjective. For $d=2$ this is trivial.
[Below we only consider the case $d=2$.]
X) Finally, we are able to prove the following Lemma.
Lemma. The group homomorphism $\rho: SL(2,\mathbb{R})\to SO^{+}(1,2;\mathbb{R})$ is surjective.
Proof. Note that boosts along the $x^3$-axis correspond to
$$\tag{23} g(\beta)~:=~\begin{bmatrix}\exp\left(\frac{\beta}{2}\right) & 0 \cr 0 &\exp\left(-\frac{\beta}{2}\right) \end{bmatrix}\in SL(2,\mathbb{R}),$$
while rotations correspond to
$$\tag{24} g(\theta)~:=~\begin{bmatrix}\cos\frac{\theta}{2} & \sin\frac{\theta}{2} \cr -\sin\frac{\theta}{2} &\cos\frac{\theta}{2} \end{bmatrix}\in SL(2,\mathbb{R}).$$
Given an arbitrary restricted Lorentz matrix $\Lambda\in SO^+(1,2;\mathbb{R})$, we saw in Section IX that it can be decomposed as (rotation)(boosts along the $x^3$-axis)(rotation'). Hence it can be hit by the $\tilde{\rho}$ group homomorphism
$$\tag{25} \Lambda ~=~ \tilde{\rho}\left(\tilde{g}(\theta)\tilde{g}(\beta)\tilde{g}(\theta^{\prime})\right).$$
$\Box$
XI) We have the following commutative diagram
$$\tag{26} \begin{array}{rlcrl} &&\tilde{\rho}_{\ast}&& \cr
&sl(2,\mathbb{R}) & \longrightarrow &so(1,2;\mathbb{R}) \cr
\pm\exp &\downarrow &\circlearrowright&\downarrow&\exp\cr
&SL(2,\mathbb{R})/\mathbb{Z}_{2} & \longrightarrow &SO^+(1,2;\mathbb{R})\cr
&&\tilde{\rho}&&\end{array} $$
All horizontal arrows are bijections. In particular, the above shows the following theorem.
Theorem. The exponential map $\exp: so(1,2;\mathbb{R}) \to SO^+(1,2;\mathbb{R})$ is surjective.
References:
- V. Bargmann, Irreducible Unitary Representations of the Lorentz Group, Ann. Math. 48 (1947) 568-640. The pdf file is available here. We mostly use results from p. 589-591.
