What is timelike geodesic? I have searched the internet for the definition of timelike geodesic curves. But I am not getting a consistent definition. In some places I saw the geodesic maximises the proper time and in some places  I saw the it is the curve whose tangent at each point is greater than zero.
I am confused about the definition of timelike geodesic curves. Could you clarify the proper definition and if possible explain what is the difference in finding the timelike geodesic equation from finding the geodesic equation given a metric.
 A: 
In some places I saw the geodesic maximises the proper time

This is the definition of a geodesic.

[...] in some places I saw the it is the curve whose tangent at each point is greater than zero.

This is what it means for a curve to be timelike, assuming that you mean $g_{\mu\nu} u^\mu u^\nu >0$, and that the metric has "mostly-minus" signature (+---).
A timelike geodesic is a curve which has both of these properties.
A: Let $(M,g)$ be a Lorentzian manifold with mostly plus signature $(-,+,\dots,+)$, and let $\nabla$ be the associated Levi-Civita connection.

*

*A (smooth) curve in $M$ refers to a smooth mapping $\gamma:I\to M$ (in these contexts, a curve doesn't mean the image set $\gamma(I)$; the parametrization is important as well).


*A smooth curve $\gamma:I\to M$ is said to be timelike if for all $s\in I$, the tangent vector $\dot{\gamma}(s)\in T_{\gamma(s)}M$ is a timelike vector; in our signature convention, this means $g_{\gamma(s)}(\dot{\gamma}(s),\dot{\gamma}(s))<0$. This condition is briefly written as $g(\dot{\gamma},\dot{\gamma})<0$. Similarly, you get the definition of a null curve and spacelike curve by replacing $<$ with $=$ and $>$ respectively.


*A (affinely-parametrized) geodesic in $(M,g)$ is a smooth curve $\gamma:I\to M$ such that $\nabla_{\dot{\gamma}}\dot{\gamma}=0$; this condition is also written $\frac{D\gamma}{ds}=0$. In a coordinate chart, letting $x^i$ be the $i^{th}$ component of $\gamma$, the geodesic equation reads $\ddot{x}^i+\Gamma^i_{jk}\dot{x}^j\dot{x}^k=0$ for all $i$.


*So putting things together, an (affinely-parametrized) timelike geodesic in $(M,g)$ is a timelike curve $\gamma$ which is also a geodesic, i.e a curve $\gamma$ which satisfies $g(\dot{\gamma},\dot{\gamma})<0$, and $\nabla_{\dot{\gamma}}\dot{\gamma}=0$. A null geodesic and spacelike geodesic are defined as geodesics which are null curves, and spacelike curves respectively.

For example, consider the standard Minkowski space $M=\Bbb{R}^4$ and $g=\eta=-dt^2+dx^2+dy^2+dz^2$. Consider the parametrized curves $\gamma_1,\gamma_2,\gamma_3:\Bbb{R}\to\Bbb{R}^4$ described as $\gamma_1(s)=(s,0,0,0)$, $\gamma_2(s)=(s,s,0,0)$, and $\gamma_3(s)=(2s,\sin(s),0,0)$. Then, you can easily verify (by working in the trivial $(t,x,y,z)$ coordinate system) that

*

*$\gamma_1$ is a (affinely-parametrized) timelike geodesic.

*$\gamma_2$ is a (affinely-parametrized) is a geodesic but it's not timelike (it's actually a null geodesic).

*$\gamma_3$ is a timelike curve, but not a geodesic.

A: Spacelike, null and timelike geodesics correspond to geodesics with signature $-+++$ (depending on the author) with a tangent vector $u$ of positive, zero or negative norm, $|u| = g_{\mu\nu} u^\mu u^\nu$. Since the geodesics correspond to non accelerated motions, any deviation from geodesics will itself imply accelerated motions in spacetime. In the four dimensional Minkowski space, the timelike geodesic is the only on which connects any given pair of events, and for the timelike geodesic, this is the curve with the longest proper time between the two events. In curved spacetime, it is possible for a pair of widely separated events to have more than one timelike geodesic between them and is an important concept in causally linked events in spacetime.
