Can an observer really catch up with light

TL;DR

The paper proves that observers cannot infinitesimally approach light along null geodesics because their proper acceleration becomes infinite, implying they must remain finitely separated to catch up with light.

Contribution

It demonstrates that approaching light arbitrarily closely requires infinite acceleration, clarifying limitations on observers trying to catch up with light.

Findings

Proper acceleration diverges near null geodesics

Observers must stay finitely separated to catch light

No observer can infinitesimally approach light

Abstract

Given a null geodesic $\gamma_0(\lambda)$ with a point $r$ in $(p,q)$ conjugate to $p$ along $\gamma_0(\lambda)$, there will be a variation of $\gamma_0(\lambda)$ which will give a time-like curve from $p$ to $q$. This is a well-known theory proved in the famous book\cite{2}. In the paper we prove that the time-like curves coming from the above-mentioned variation have a proper acceleration which approaches infinity as the time-like curve approaches the null geodesic. This means no observer can be infinitesimally near the light and begin at the same point with the light and finally catch the light. Only separated from the light path finitely, does the observer can begin at the same point with the light and finally catch the light.