Overall, Froome is in 18th place, 55 seconds off the pace and still 44 behind 2017 Giro d'Italia victor Dutchman Tom Dumoulin, who with Team Sunweb only lost seven seconds to the Briton on Monday.
Stage victor Slovakia's Peter Sagan, wearing the overall leader's yellow jersey, celebrates on the podium after the second stage of the Tour de France.
Tonight's fourth stage is a flat, 195km stage in Brittany between La Baule and Sarzeau that is expected to finish in a bunch sprint.
Froome is aiming to join Jacques Anquetil, Eddy Merckx, Bernard Hinault and Miguel Indurain as the only riders to win the Tour five times.
The 35.5km teams trial held in and around Cholet was a key test for the general classification favourites.
BMC clocked 38min 46sec as their Belgian rider Greg van Avermaet took the overall leader's yellow jersey, with Sky in second place, four seconds behind.
But Gaviria was involved in a group pileup inside the three-kilometre zone that neutralizes the impact of accidents and could do nothing to stop Sagan from claiming a six-second overall lead.
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Squad leaders traded turns on the front of the team "trains", with the ability to constantly maintain a fast pace the decisive factor over a challenging - if not highly technical - route featuring a few minor hills. "It was a good sprint, slightly uphill in the end, but I had good legs and a good feeling", said Gaviria, only the second Colombian to wear the yellow jersey after Victor Hugo Pena in 2003.
"Obviously it was a tough one today but the big strong guys did a great job and it was a ideal team time trial", Porte told Cycling.com.
Chavanel, who has said this will be his last Tour, soaked up applause from French fans on his solo run, and he even had time to raise his arms in celebration as he passed through the crowds that lined the road midway through the stage.
The event returned for the first time since 2015, and time trial specialists BMC took full advantage as they set a time of 38 minutes, 46 seconds on the 35.5-kilometre course that started and ended in the town of Cholet in the Pays de la Loire region of France.
"It was a goal to get yellow in the first week", said van Avermaet. "It just shows that we are here, too, and let's get it on".
It will be van Avermaet's second stint in yellow after three days in the maillot jaune during the 2016 Tour de France.
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