The M.T.A. and Related Companies will not, as originally expected, sign a contract this week for a $15 billion residential and commercial development above the M.T.A.’s West Side rail yards. The M.T.A. and Related, which is working with Goldman Sachs on the project, have agreed to a 90-day extension of the contract deadline. The M.T.A. imposed a 165-day deadline on signing a contract when it appointed Related the project developer in May, but the M.T.A. said its attorneys have not been able to finish the contract because of work related to the financial crisis. The M.T.A. had originally planned to close on the deal for the eastern side of the rail yards four months after signing a contract.