Does slow eccentric training build muscle?
It requires a greater external force (20-60%) to lengthen a muscle (eccentric action) than it does to shorten a muscle (concentric action). In other words, your muscles are stronger during eccentric actions. Not only does this make more sense physiologically, but it is also a defence mechanism of the muscle. Our muscle fibres work by creating "links" or "cross-bridges" as they lengthen or shorten.
Think of someone climbing a ladder, it requires less overall energy/force for the hands and legs to climb the rungs (concentric) than if you were to grab said person by the legs and try and pull them down from the ladder (eccentric). On the way down, the person would be fighting to hold on to each rung, and even if they let go of 1 rung, they could grab the next one on the way down, some rungs may even break on the way down, this is like filament damage in the muscles that can lead to hypertrophy (crude example but hopefully you can see the analogy).
Anyways, time under tension does appear to have carry-over to increases in muscular size (hypertrophy) because it causes an increased metabolic demand as well as muscular damage. It is also hypothesized that fast-twitch muscle fibres are recruited first during an eccentric contraction. For overall strength increases nothing beats lifting more weight forcefully, however. If one wishes to increase muscular size, then increased time under tension may have some benefits, and eccentric training can improve overall concentric and eccentric strength, but not to the extent lifting heavier loads can. It should be noted, however, that strength training (less reps, no focus on time under tension) can also cause muscular growth.
Hollander DB, Kraemer RR, Kilpatrick MW, Ramadan ZG, Reeves GV, Francois M, Hebert EP, Tryniecki JL. Maximal eccentric and concentric strength discrepancies between young men and women for dynamic resistance exercise. J Strength Cond Res 21: 34–40, 2007.