­Citing the risk to public safety posed by a federal judge’s ruling against New York’s stop, question, and frisk policing program, the city’s former mayor, Rudolph Giuliani, and a former U.S. attorney general, Michael Mukasey, are calling on an appeals court to stop the case in its tracks.

The two filed a motion Monday in the U.S. Court of Appeals for the Second Circuit, asking to be admitted as friends of court in the case. Their motion comes amid growing alarm among courthouse watchers that the city has not been moving aggressively enough where there are growing indications that time is of the essence.

In a report earlier this month, New York Post disclosed a spike in crime in the immediate aftermath of the ruling, including a 13% increase in shooting incidents in the first four weeks since Judge Scheindlin ruled unconstitutional the NYPD’s practice of stopping and questioning — and, in some instances, frisking — persons arousing an officer’s suspicions.

Judge Scheindlin’s ruling, which argues that the city’s Stop and Frisk procedures run afoul of constitutional standards, has already created a climate of fear within the police department, Messrs. Giuliani and Mukasey asserted in their filing. Members of the NYPD find themselves “in constant peril of being found to have violated applicable law,” they write. Judge Scheindlin’s ruling has placed officers “on notice that they must change their methods in undefined ways or be called to account.”

Citing their own experience in managing law enforcement agencies, Messrs. Giuliani and Mukasey argue that the stop, question, and frisk ruling, with its call for an extended, long-range process to develop new standards, has already run the risk of disrupting police work: “[W]hen the mandate for a fundamental shift in law enforcement approach has been put in place, both day-to-day performance and long-term planning and training are immediately affected,” they write.

The impact of the ruling has also left “police with little direction” regarding how to handle street stops, Messrs. Giuliani and Mukasey argue. They also note that the stop, question, and frisk program “is not a stand-alone procedure” that can be considered in isolation. The procedure is “only one element of a proactive approach to policing that stresses situational awareness and affirmative steps to prevent crime.” As a result of Judge Scheindlin’s ruling, “that entire approach has been cast in doubt,” they note.

“It is impossible,” the two say, “to exaggerate the impact” that Judge Scheindlin’s ruling “will have on planning and training within the NYPD, and on individual day-to-day policing decisions.”

The filing by the former mayor and former federal judge also argues that Judge Scheindlin is wrong on the law and that the city’s stop, question, and frisk policies are well within constitutional bounds. “Police may consider a person’s race in conjunction with other descriptive characteristics when investigating and preventing crime,” they said in the filing, which also argues that Judge Scheindlin’s ruling is an unwarranted intervention by the federal government on a local government. Her proposed interventions, they argue, “dramatically overstep the proper bounds of the judiciary in fashioning injunctive relief against a municipality.”

The former mayor and attorney general argue that their long careers in public service “in several of the highest public offices concerned with law enforcement and governance of the City of New York and the United States” make them “uniquely suited to provide insight into the impact” of Judge Scheindlin’s ruling against the city. One or the other of them has served as U.S. attorney general, chief judge of the federal court for the Southern District of New York, mayor of the city, and chief federal prosecutor for the Southern District.

Their filing also notes that Mr. Giuliani is “widely credited with improving the quality of life in the City, in large part due to the significant drop in crime under his administration.” During his mayoralty, they say, “New York’s crime rate fell by 57 percent and the Federal Bureau of Investigation rated New York City as America’s safest large city.”

Their filing is in response to Judge Scheindlin’s September 17 refusal to grant the city a stay on the case. She claimed the postponement “would encourage the NYPD to return to its former practice of conducting thousands upon thousands of improper stops — including those based merely on a person entering or exiting a building in which he or she resides.”

At that time the judge favorably noted the legal argument of the Democratic candidate for Mayor, Bill DeBlasio, who observed in a court filing of his own that “It is well-past time for the City to cease the meritless scare tactic of contending that conforming the use of stop and frisk to constitutional standards will increase crime or make the public unsafe.” Mr. DeBlasio went on to say that “there is simply no proof” that Judge Scheindlin’s ruling “will harm the public.”