* U.S. stocks lower in early New York trading * Gold prices rally * U.S. bond prices fall (Updates with early U.S. markets; changes dateline; previous LONDON) NEW YORK, Jan 4 (Reuters) - A global stock index fell on Monday, the first trading day of the new year, pulling back after recent gains to record highs, and the U.S. dollar was near flat. U.S. stocks were down sharply by late morning in New York, with the S&P 500 nearly 1.4% lower. Stocks hit records early in the session as investors focused on the rollout of COVID-19 vaccines. But investors quickly turned cautious over the path of the virus, which continues to spread amid the discovery of a new variant. The outcome of runoff elections on Tuesday in Georgia for two U.S. Senate seats added to investor nervousness. “It’s a small reversal, but I think people came in the morning to start the year with some optimism, but the cloudy political picture in the U.S. is weighing down the markets a little bit,” said Rick Meckler, partner at Cherry Lane Investments in New Vernon, New Jersey. MSCI’s All-Country World Index, which tracks stocks across 49 countries, was down 0.6% after earlier hitting a record high. The Dow Jones Industrial Average fell 535.35 points, or 1.75%, to 30,071.13, the S&P 500 lost 59.08 points, or 1.57%, to 3,696.99 and the Nasdaq Composite dropped 183.74 points, or 1.43%, to 12,704.55. The pan-European STOXX 600 index rose 0.687%, while Britain’s FTSE 100 index was up 1.7% on its first post-Brexit trading day. Britain began vaccinating its population with the COVID-19 shot developed by Oxford University and AstraZeneca on Monday. With the lag between a full vaccine rollout and a global economic recovery, investors will count on central banks to keep money cheap. Minutes of the Federal Reserve’s December meeting are due on Wednesday and should offer more detail on discussions about making the Fed’s forward policy guidance more explicit and the chance of a further increase in asset buying this year. Friday brings the U.S. December payroll report. In currencies, the U.S. dollar recovered after falling to its lowest level since April 2018. The dollar index rose 0.088%, with the euro up 1.03% to $1.2262. The Japanese yen strengthened 0.03% versus the greenback at 103.17 per dollar, while Sterling was last trading at $1.3561, down 0.80% on the day. Gold prices were up more than 2%, U.S. crude futures and Brent futures were lower. Benchmark 10-year Treasury notes last fell 4/32 in price to yield 0.9248%.