You can use a solution of bicarbonate of soda or baking soda in water. What you do, take a tablespoon of the baking soda, dissolve in the water then pour into the sink. Make sure the sink has a stopper. Then leave it for a while maybe 30 minutes or so. Then scrub gently with a cloth and you should be good to go.
Try not to use strong abrasives which will wear out and dull the surface. I use chlorine beach in a dilute soak once a month or on a saturated cloth to sit across the surface for tough discoloration (for white sinks only) Generally washing the sink w/ dish soap and water once a day and gently buffing w/ baking soda will keep it looking good.
You have to use neutral cleaners. Such materials can be delicate to clean so you have to make sure that you are careful about choosing the best products to use for cleaning purposes. You may also consider travertine as a good material for home improvement.