How to send an email to a well-known professor for a possible PhD student position?
Your email may run something like this (names and subject areas can be changed, of course):


  Dear Professor Carroll,
  
  Are you accepting PhD students at the     moment? I am planning to apply to CalTech this year for a PhD in physics and I am really interested in cosmology and in particular, your work on the arrow of time. I am currently a 3rd year undergraduate at University of Somewhere and have taken courses in differential geometry and astrophysics. 
  
  Best wishes,
  
  mathvc_