At around 1.36am on Sunday (August 22) it is reported that a black Vauxhall Insignia was in collision with a pedestrian on Regent Street in the city centre.
The police said the vehicle did not stop at the scene and continued travelling in the direction of Portobello, then Brook Hill roundabout.
The pedestrian, a 58-year-old man, was found a short time later by members of the public and was taken to hospital, where he remains in a critical but stable condition.
The vehicle has since been recovered by officers, but enquiries are ongoing to locate the driver.
Acting Police Sergeant Rod McEnery who is leading the investigation said in a statement: “Officers are carrying out extensive enquiries to locate the driver of the vehicle so we can ascertain exactly what happened in the early hours of this morning.
“It’s vital that we hear from anyone who might have seen the vehicle prior to the collision, or afterwards, so we can piece together exactly what has happened.
“If you have any information, or saw anything that might help our investigation, I urge you to please contact us immediately. You can call us on 101, or submit information online using our online portal or live chat.”
Anyone with information is asked to call 101 by quoting incident number 176 of 22 August.