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If you bring a cellphone with you in the past it will not work because there is no field...
You need a different explanation than time travel if you really think those people were actually using it.
I don't see problem with a group of time travelers setting up a cellular network for communications within their target time period.
No one asks them why are they digging all the streets to pass their cables?
No one notices the thousands of antennas they are placing all around?
Well obviously they wouldn't be setting up a nationwide infrastructure for millions of customers. A few strategically paced sites could be set up clandestinely.
How to connect those strategic places between them?
https://en.wikipedia.org/wiki/Cell_on_wheels
"COWs are used to provide expanded cellular network coverage and/or capacity at special events such as major sporting events (Super Bowl, World Series, Rose Bowl), major conventions, or in disaster areas where cellular coverage either was never present (e.g., in a wilderness area where firefighters have set up a command center during a major forest fire) or was compromised by the disaster (e.g., in the Gulf Coast after Hurricane Katrina). Following the September 11 attacks on New York City in 2001, 36 cellular COWs were deployed by September 14, 2001 in Lower Manhattan to support the U.S. Federal Emergency Management Agency (FEMA) and provide critical phone service to rescue and recovery workers."
Just stuff that baby into the way-back-machine and Bob's your uncle!
