The Obsidian Accelerator is a useful but confusing item in Destiny 2. If you have already gotten one, you might be tempted to “use” it when hovering over it in your Inventory. Definitely don’t do this, as the Obsidian Accelerator can be traded to Ada-1 for an Obsidian Radiance. The Radiance can be used to buff Forge weapons when they are used during a Forge event. “Using” the Accelerator simply wastes it, so hold on to it until the next time you go and visit Ada-1.
The only place to get the Obsidian Accelerator is from the Bergusia Forge, on Earth. Finishing the Forge will sometimes cause an Obsidian Accelerator to drop. It’s a guaranteed drop and is actually quite rare, so you may need to farm a lot to get your hands on one.
The only guaranteed Obsidian Accelerator in the game is from the Izanagi’s Burden quest, so you can complete that to get one if you haven’t done it yet. When you get one, you can go and visit Ada-1 in the Tower Annex, and trade the Accelerator for the Radiance. If you are having trouble finding it in her menu, make sure to check the second tab.
Now, inspect the Forge weapon that you wish to apply it to, and you should be able to place it in a box next to the Shaders. This will then buff your weapon when you are in any Forge, but the buff will not extend to any other activity.