The couple have been quietly seeing each other for a couple months, a source tells PEOPLE exclusively

After splitting for good from his ex-girlfriend Amber Heard in February, Tesla CEO and SpaceX founder Elon Musk has been quietly dating Canadian singer Grimes, PEOPLE confirms.

The new couple made their public debut at the Met Gala on Monday night. Musk wore a black tuxedo jacket with a Papal-style shirt underneath, black trousers and a white robe, while the singer went for a gothic look in a black-and-silver corseted gown, teamed with a matching choker, sheer gloves and an Elvira-style updo.

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“They’ve been seeing each other but have been keeping their relationship under the radar,” says a source about Grimes, 30, and Musk, 46.

The duo raised eyebrows last month when they had a flirty interaction on Twitter, and Musk replied to a fan who told him to “Just ask her out already” with a wink emoji.

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Musk also gave his girlfriend’s “Venus Fly” music video a shout-out in late March, telling his followers it’s the “best music video art I’ve seen in a while.”

Musk — who was married thrice before and has six children — briefly reunited with Heard earlier this year after they broke up last August.

“Elon decided it was time to end it and Amber agreed. They both still care deeply for each other but the timing wasn’t right,” a source close to the former couple previously told PEOPLE.

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Heard was also at the 2018 Met Gala in a long-sleeve red gown teamed with a golden burst crown.

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The actress and the billionaire mogul were first linked together in 2016 as each was weathering a public breakup: Heard and actor Johnny Depp finalized their divorce in January, while Musk filed for divorce from Westworld actress Talulah Riley in early 2015.