Was Arnold's initial hair style a wig?

When he first arrives in 1984 he has a longer hairstyle with combed bangs. When he runs through the fire before the big car chase scene, his hair gets burned down to a spiky punk looking haircut and his eyebrows are singed off. Every Terminator fan probably already knows this.

But here's my question. I just read that the first scene Schwarzenegger filmed was when he was driving the police car and searching for Sarah in that parking garage, hence he already had the burnt hair and no eyebrows. So, since this was filmed first, does that mean he was wearing a wig in all the scenes that took place before that (before his hair got burnt)?

I guess the obvious answer would be yes, unless they waited a while for his hair to grow back, which is highly unlikely. But I still find it strange because I've seen lots of pics of Arnold with the longer haircut seen at the beginning of the film throughout the 80's. It looked like his real hair and not a wig. Can anyone shed some light on this?

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