Rabbi Joseph Telushkin explains why Jews do not believe that Jesus is the Messiah:
From Judaism’s perspective, Jesus did not fulfill the messianic prophecies and therefore is not regarded as the Messiah. The best-known of the prophecies concerning the messianic days is that “Nation shall not lift up sword against nation, nor shall they learn war anymore” (Isaiah 2:4). Since world peace must accompany the Messiah, and world peace (or, for the past 2,000 years, anything remotely approaching it) has not come, clearly the Messiah has not come either. In addition, Jewish tradition teaches that the Messiah will enable the Jews to lead a peaceful and independent existence in Israel. This, too, was not achieved by Jesus.
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Though it has been apparent for almost 2,000 years that the messianic days of peace have not arrived, Christians still assume that Jesus was the Messiah. How do they explain this? By arguing that there will be a Second Coming, during which Jesus will return to Earth, and fulfill the messianic functions originally expected of him. For Jews, however, this argument is unconvincing, since the idea of a Second Coming is nowhere found in the Hebrew Bible (what Christians refer to as the Old Testament). This idea seems to have been unknown to Jesus as well, since the New Testament cites him as telling his followers that some of them will still be alive when all the messianic prophecies will be fulfilled (see Mark 9:1 and 13:30). I would guess that the idea of a second coming was formulated by later Christians to explain Jesus’ failure to fulfill the messianic prophecies. In short, from Judaism’s perspective, to call someone who does not bring about the messianic era the Messiah does not make sense.
Telushkin’s answer raises an obvious question: Why did so many Jews convert to Christianity in the first century? Presumably, they were well-versed in the Hebrew Bible and would be skeptical of the claims about a Messiah who disappeared after coming back to life after dying by crucifixion. Why did they believe in Christ? Is the second coming theory a plausible explanation?