The ginger syrup made in step one may be stored in an airtight container, refrigerated, for up to a week.
Combine ginger and sugar with 4 cups water, or enough to cover, in a medium saucepan. Place over medium-high heat until mixture comes to a boil. Reduce to a simmer; stir until sugar dissolves completely. Continue to simmer about 20 minutes. Remove from heat; let cool to room temperature, at least 2 hours. Strain out the ginger, and discard.
Combine the ginger syrup, lime juice, and seltzer in a pitcher. Pour into ice-filled glasses, and serve immediately.