If you are using ObjectMapper class of fasterxml, 
by default ObjectMapper do not understand the LocalDateTime class, so, you need to add another dependency in your gradle/maven : 
compile 'com.fasterxml.jackson.datatype:jackson-datatype-jsr310:2.7.3'
Now you need to register the datatype support offered by this library into you objectmapper object, this can be done by following : 
ObjectMapper objectMapper = new ObjectMapper();
objectMapper.findAndRegisterModules();
Now, in your jsonString, you can easily put your java.LocalDateTime field as follows :
{
    "user_id": 1,
    "score": 9,
    "date_time": "2016-05-28T17:39:44.937"
}
By doing all this, your Json file to Java object conversion will work fine, you can read the file by following : 
objectMapper.readValue(jsonString, new TypeReference<List<User>>() {
            });