Grate the ginger on the finest holes of a box grater (cheese grater). With a spoon scrape any of the minced ginger from the back of the grater and into a bowl. Put the minced ginger in a cheesecloth and squeeze the juice into a bowl. If there’s no cheesecloth, the ginger can be squeezed through the hands.
Alternatively, process the ginger until fine in a food processor and then squeeze through a cheesecloth or just with your hands into a bowl.
Combine the ginger juice, lemon juice, and sweetener and add an even amount to 4 glasses filled with ice. Top with seltzer or mineral water and stir well.